chapter 5 resource masters - ktl math...

Chapter 5Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 5 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-828008-7 Algebra 2Chapter 5 Resource Masters

1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02

Glenc oe/ M cGr aw-H ill

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 5-1Study Guide and Intervention . . . . . . . . 239–240Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 241Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 242Reading to Learn Mathematics . . . . . . . . . . 243Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 244









Chapter 5 AssessmentChapter 5 Test, Form 1 . . . . . . . . . . . . 293–294Chapter 5 Test, Form 2A . . . . . . . . . . . 295–296Chapter 5 Test, Form 2B . . . . . . . . . . . 297–298Chapter 5 Test, Form 2C . . . . . . . . . . . 299–300Chapter 5 Test, Form 2D . . . . . . . . . . . 301–302Chapter 5 Test, Form 3 . . . . . . . . . . . . 303–304Chapter 5 Open-Ended Assessment . . . . . . 305Chapter 5 Vocabulary Test/Review . . . . . . . 306Chapter 5 Quizzes 1 & 2 . . . . . . . . . . . . . . . 307Chapter 5 Quizzes 3 & 4 . . . . . . . . . . . . . . . 308Chapter 5 Mid-Chapter Test . . . . . . . . . . . . . 309Chapter 5 Cumulative Review . . . . . . . . . . . 310Chapter 5 Standardized Test Practice . . 311–312

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A38

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 5 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 5 Resource Masters includes the core materials neededfor Chapter 5. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 5-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 5Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 282–283. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

55

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 5.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

binomial

coefficient

KOH·uh·FIH·shuhnt

complex conjugates

KAHN·jih·guht

complex number

degree

extraneous solution

ehk·STRAY·nee·uhs

FOIL method

imaginary unit

like radical expressions

like terms

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

monomial

nth root

polynomial

power

principal root

pure imaginary number

radical equation

radical inequality

rationalizing the denominator

synthetic division

sihn·THEH·tihk

trinomial

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

55

Study Guide and InterventionMonomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

© Glencoe/McGraw-Hill 239 Glencoe Algebra 2

Less

on

5-1

Monomials A monomial is a number, a variable, or the product of a number and one ormore variables. Constants are monomials that contain no variables.

Negative Exponent a�n � and � an for any real number a � 0 and any integer n.

When you simplify an expression, you rewrite it without parentheses or negativeexponents. The following properties are useful when simplifying expressions.

Product of Powers am � an � am � n for any real number a and integers m and n.

Quotient of Powers � am � n for any real number a � 0 and integers m and n.

For a, b real numbers and m, n integers:(am)n � amn

Properties of Powers(ab)m � ambm

� �n� , b � 0

� ��n� � �n

or , a � 0, b � 0

Simplify. Assume that no variable equals 0.

bn�an

b�a

a�b

an�bn

a�b

am�an

1�a�n

1�an

ExampleExample

a. (3m4n�2)(�5mn)2

(3m4n�2)(�5mn)2 � 3m4n�2 � 25m2n2

� 75m4m2n�2n2

� 75m4 � 2n�2 � 2

� 75m6

b.

�

� �m12 � 4m4

� �4m16

�m12��4m

14�

(�m4)3�(2m2)�2

(�m4)3��(2m2)�2

ExercisesExercises


1. c12 � c�4 � c6 c14 2. b6 3. (a4)5 a20

4. 5. � ��16. � �2

7. (�5a2b3)2(abc)2 5a6b8c2 8. m7 � m8 m15 9.

10. 2 11. 4j(2j�2k2)(3j3k�7) 12. m23�2

2mn2(3m2n)2��

12m3n424j2�

k523c4t2�22c4t2

2m2�

n8m3n2�4mn3

1�5

x2�y4

x2y�xy3

b�a5

a2b�a�3b2

y2�x6

x�2y�x4y�1

b8�b2


Scientific Notation

Scientific notation A number expressed in the form a � 10n, where 1 � a 10 and n is an integer

Express 46,000,000 in scientific notation.

46,000,000 � 4.6 � 10,000,000 1 � 4.6 10

� 4.6 � 107 Write 10,000,000 as a power of ten.

Evaluate . Express the result in scientific notation.

� �

� 0.7 � 106

� 7 � 105

Express each number in scientific notation.

1. 24,300 2. 0.00099 3. 4,860,0002.43 � 104 9.9 � 10�4 4.86 � 106

4. 525,000,000 5. 0.0000038 6. 221,0005.25 � 108 3.8 � 10�6 2.21 � 105

7. 0.000000064 8. 16,750 9. 0.0003696.4 � 10�8 1.675 � 104 3.69 � 10�4

Evaluate. Express the result in scientific notation.

10. (3.6 � 104)(5 � 103) 11. (1.4 � 10�8)(8 � 1012) 12. (4.2 � 10�3)(3 � 10�2)1.8 � 108 1.12 � 105 1.26 � 10�4

13. 14. 15.

2.5 � 109 9 � 10�8 7.4 � 103

16. (3.2 � 10�3)2 17. (4.5 � 107)2 18. (6.8 � 10�5)2

1.024 � 10�5 2.025 � 1015 4.624 � 10�9

19. ASTRONOMY Pluto is 3,674.5 million miles from the sun. Write this number inscientific notation. Source: New York Times Almanac 3.6745 � 109 miles

20. CHEMISTRY The boiling point of the metal tungsten is 10,220°F. Write thistemperature in scientific notation. Source: New York Times Almanac 1.022 � 104

21. BIOLOGY The human body contains 0.0004% iodine by weight. How many pounds ofiodine are there in a 120-pound teenager? Express your answer in scientific notation.Source: Universal Almanac 4.8 � 10�4 lb

4.81 � 108��6.5 � 104

1.62 � 10�2��

1.8 � 1059.5 � 107��3.8 � 10�2

104�10�2

3.5�5

3.5 � 104��5 � 10�2

3.5 � 104��5 � 10�2

Study Guide and Intervention (continued)

Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeMonomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1


Less

on

5-1


1. b4 � b3 b7 2. c5 � c2 � c2 c9

3. a�4 � a�3 4. x5 � x�4 � x x2

5. (g4)2 g8 6. (3u)3 27u3

7. (�x)4 x4 8. �5(2z)3 �40z3

9. �(�3d)4 �81d4 10. (�2t2)3 �8t6

11. (�r7)3 �r 21 12. s3

13. 14. (�3f 3g)3 �27f 9g3

15. (2x)2(4y)2 64x2y2 16. �2gh( g3h5) �2g4h6

17. 10x2y3(10xy8) 100x3y11 18.

19. � 20.


21. 53,000 5.3 � 104 22. 0.000248 2.48 � 10�4

23. 410,100,000 4.101 � 108 24. 0.00000805 8.05 � 10�6


25. (4 � 103)(1.6 � 10�6) 6.4 � 10�3 26. 6.4 � 10109.6 � 107��1.5 � 10�3

2q2�p2

�10pq4r��5p3q2r

c7�6a3b

�6a4bc8��36a7b2c

8z2�w2

24wz7�3w3z5

1�k

k9�k10

s15�s12

1�a7



1. n5 � n2 n7 2. y7 � y3 � y2 y12

3. t9 � t�8 t 4. x�4 � x�4 � x4

5. (2f 4)6 64f 24 6. (�2b�2c3)3 �

7. (4d2t5v�4)(�5dt�3v�1) � 8. 8u(2z)3 64uz3

9. � 10. �

11. 12. � �2

13. �(4w�3z�5)(8w)2 � 14. (m4n6)4(m3n2p5)6 m34n36p30

15. � d2f 4�4�� d5f�3 �12d23f 19 16. � ��2

17. 18. �


19. 896,000 20. 0.000056 21. 433.7 � 108

8.96 � 105 5.6 � 10�5 4.337 � 1010


22. (4.8 � 102)(6.9 � 104) 23. (3.7 � 109)(8.7 � 102) 24.

3.312 � 107 3.219 � 1012 3 � 10�5

25. COMPUTING The term bit, short for binary digit, was first used in 1946 by John Tukey.A single bit holds a zero or a one. Some computers use 32-bit numbers, or strings of 32 consecutive bits, to identify each address in their memories. Each 32-bit numbercorresponds to a number in our base-ten system. The largest 32-bit number is nearly4,295,000,000. Write this number in scientific notation. 4.295 � 109

26. LIGHT When light passes through water, its velocity is reduced by 25%. If the speed oflight in a vacuum is 1.86 � 105 miles per second, at what velocity does it travel throughwater? Write your answer in scientific notation. 1.395 � 105 mi/s

27. TREES Deciduous and coniferous trees are hard to distinguish in a black-and-whitephoto. But because deciduous trees reflect infrared energy better than coniferous trees,the two types of trees are more distinguishable in an infrared photo. If an infraredwavelength measures about 8 � 10�7 meters and a blue wavelength measures about 4.5 � 10�7 meters, about how many times longer is the infrared wavelength than theblue wavelength? about 1.8 times

2.7 � 106��9 � 1010

4v2�m2

�20(m2v)(�v)3��

5(�v)2(�m4)15x11�

y3(3x�2y3)(5xy�8)��

(x�3)4y�2

y6�4x2

2x3y2��x2y5

4�3

3�2

256�wz5

4�9r4s6z12

2�3r2s3z6

27x6�

16�27x3(�x7)��

16x4

s4�3x4

�6s5x3�18sx7

4m7y2�

312m8y6��9my4

20d 3t2�

v5

8c9�b6

1�x4

Practice (Average)

Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

Reading to Learn MathematicsMonomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1


Less

on

5-1

Pre-Activity Why is scientific notation useful in economics?

Read the introduction to Lesson 5-1 at the top of page 222 in your textbook.

Your textbook gives the U.S. public debt as an example from economics thatinvolves large numbers that are difficult to work with when written instandard notation. Give an example from science that involves very largenumbers and one that involves very small numbers. Sample answer:distances between Earth and the stars, sizes of molecules and atoms

Reading the Lesson

1. Tell whether each expression is a monomial or not a monomial. If it is a monomial, tellwhether it is a constant or not a constant.

a. 3x2 monomial; not a constant b. y2 � 5y � 6 not a monomial

c. �73 monomial; constant d. not a monomial

2. Complete the following definitions of a negative exponent and a zero exponent.

For any real number a � 0 and any integer n, a�n � .

For any real number a � 0, a0 � .

3. Name the property or properties of exponents that you would use to simplify eachexpression. (Do not actually simplify.)

a. quotient of powers

b. y6 � y9 product of powers

c. (3r2s)4 power of a product and power of a power

Helping You Remember

4. When writing a number in scientific notation, some students have trouble rememberingwhen to use positive exponents and when to use negative ones. What is an easy way toremember this? Sample answer: Use a positive exponent if the number is10 or greater. Use a negative number if the number is less than 1.

m8�m3

1

�a1n�

1�z


Properties of ExponentsThe rules about powers and exponents are usually given with letters such as m, n,and k to represent exponents. For example, one rule states that am � an � am � n.

In practice, such exponents are handled as algebraic expressions and the rules of algebra apply.

Simplify 2a2(an � 1 � a4n).

2a2(an � 1 � a4n) � 2a2 � an � 1 � 2a2 � a4n Use the Distributive Law.

� 2a2 � n � 1 � 2a2 � 4n Recall am � an � am � n.

� 2an � 3 � 2a2 � 4n Simplify the exponent 2 � n � 1 as n � 3.

It is important always to collect like terms only.

Simplify (an � bm)2.

(an � bm)2 � (an � bm)(an � bm)F O I L

� an � an � an � bm � an � bm � bm � bm The second and third terms are like terms.

� a2n � 2anbm � b2m

Simplify each expression by performing the indicated operations.

1. 232m 2. (a3)n 3. (4nb2)k

4. (x3aj )m 5. (�ayn)3 6. (�bkx)2

7. (c2)hk 8. (�2dn)5 9. (a2b)(anb2)

10. (xnym)(xmyn) 11. 12.

13. (ab2 � a2b)(3an � 4bn)

14. ab2(2a2bn � 1 � 4abn � 6bn � 1)

12x3�4xn

an�2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

Example 1Example 1

Example 2Example 2

Study Guide and InterventionPolynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2


Less

on

5-2

Add and Subtract Polynomials

Polynomial a monomial or a sum of monomials

Like Terms terms that have the same variable(s) raised to the same power(s)

To add or subtract polynomials, perform the indicated operations and combine like terms.

Simplify �6rs � 18r2 � 5s2 � 14r2 � 8rs � 6s2.

�6rs � 18r2 � 5s2 �14r2 � 8rs� 6s2

� (18r2 � 14r2) � (�6rs � 8rs) � (�5s2 � 6s2) Group like terms.

� 4r2 � 2rs � 11s2 Combine like terms.

Simplify 4xy2 � 12xy � 7x2y � (20xy � 5xy2 � 8x2y).

4xy2 � 12xy � 7x2y � (20xy � 5xy2 � 8x2y)� 4xy2 � 12xy � 7x2y � 20xy � 5xy2 � 8x2y Distribute the minus sign.

� (�7x2y � 8x2y ) � (4xy2 � 5xy2) � (12xy � 20xy) Group like terms.

� x2y � xy2 � 8xy Combine like terms.

Simplify.

1. (6x2 � 3x � 2) � (4x2 � x � 3) 2. (7y2 � 12xy � 5x2) � (6xy � 4y2 � 3x2)2x2 � 4x � 5 3y2� 18xy � 8x2

3. (�4m2 � 6m) � (6m � 4m2) 4. 27x2 � 5y2 � 12y2 � 14x2

�8m2 � 12m 13x2 � 7y2

5. (18p2 � 11pq � 6q2) � (15p2 � 3pq � 4q2) 6. 17j2 � 12k2 � 3j2 � 15j2 � 14k2

3p2 � 14pq � 10q2 5j2 � 2k2

7. (8m2 � 7n2) � (n2 � 12m2) 8. 14bc � 6b � 4c � 8b � 8c � 8bc20m2 � 8n2 14b � 22bc � 12c

9. 6r2s � 11rs2 � 3r2s � 7rs2 � 15r2s � 9rs2 10. �9xy � 11x2 � 14y2 � (6y2 � 5xy � 3x2)24r2s � 5rs2 14x2 � 4xy � 20y2

11. (12xy � 8x � 3y) � (15x � 7y � 8xy) 12. 10.8b2 � 5.7b � 7.2 � (2.9b2 � 4.6b � 3.1)7x � 4xy � 4y 7.9b2 � 1.1b � 10.3

13. (3bc � 9b2 � 6c2) � (4c2 � b2 � 5bc) 14. 11x2 � 4y2 � 6xy � 3y2 � 5xy � 10x2

�10b2 � 8bc � 2c2 x2 � xy � 7y2

15. x2 � xy � y2 � xy � y2 � x2 16. 24p3 � 15p2 � 3p � 15p3 � 13p2 � 7p

� x2 � xy � y2 9p3 � 2p2 � 4p3�4

7�8

1�8

3�8

1�4

1�2

1�2

3�8

1�4

Example 1Example 1

Example 2Example 2

ExercisesExercises


Multiply Polynomials You use the distributive property when you multiplypolynomials. When multiplying binomials, the FOIL pattern is helpful.

To multiply two binomials, add the products ofF the first terms,

FOIL Pattern O the outer terms,I the inner terms, andL the last terms.

Find 4y(6 � 2y � 5y2).

4y(6 � 2y � 5y2) � 4y(6) � 4y(�2y) � 4y(5y2) Distributive Property

� 24y � 8y2 � 20y3 Multiply the monomials.

Find (6x � 5)(2x � 1).

(6x � 5)(2x � 1) � 6x � 2x � 6x � 1 � (�5) � 2x � (�5) � 1First terms Outer terms Inner terms Last terms

� 12x2 � 6x � 10x � 5 Multiply monomials.

� 12x2 � 4x � 5 Add like terms.

Find each product.

1. 2x(3x2 � 5) 2. 7a(6 � 2a � a2) 3. �5y2( y2 � 2y � 3)6x3 � 10x 42a � 14a2 � 7a3 �5y4 � 10y3 � 15y2

4. (x � 2)(x � 7) 5. (5 � 4x)(3 � 2x) 6. (2x � 1)(3x � 5)x2 � 5x � 14 15 � 22x � 8x2 6x2 � 7x � 5

7. (4x � 3)(x � 8) 8. (7x � 2)(2x � 7) 9. (3x � 2)(x � 10)4x2 � 35x � 24 14x2 � 53x � 14 3x2 � 28x � 20

10. 3(2a � 5c) � 2(4a � 6c) 11. 2(a � 6)(2a � 7) 12. 2x(x � 5) � x2(3 � x)�2a � 27c 4a2 � 10a � 84 x3 � x2 � 10x

13. (3t2 � 8)(t2 � 5) 14. (2r � 7)2 15. (c � 7)(c � 3)3t4 � 7t2 � 40 4r2 � 28r � 49 c2 � 4c � 21

16. (5a � 7)(5a � 7) 17. (3x2 � 1)(2x2 � 5x)25a2 � 49 6x4 � 15x3 � 2x2 � 5x

18. (x2 � 2)(x2 � 5) 19. (x � 1)(2x2 � 3x � 1)x4 � 7x2 � 10 2x3 � x2 � 2x � 1

20. (2n2 � 3)(n2 � 5n � 1) 21. (x � 1)(x2 � 3x � 4)2n4 � 10n3 � 5n2 � 15n � 3 x3 � 4x2 � 7x � 4


Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticePolynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2


Less

on

5-2

Determine whether each expression is a polynomial. If it is a polynomial, state thedegree of the polynomial.

1. x2 � 2x � 2 yes; 2 2. no 3. 8xz � y yes; 2

Simplify.

4. (g � 5) � (2g � 7) 5. (5d � 5) � (d � 1)3g � 12 4d � 4

6. (x2 � 3x � 3) � (2x2 � 7x � 2) 7. (�2f 2 � 3f � 5) � (�2f 2 � 3f � 8)3x2 � 4x � 5 �4f 2 � 6f � 3

8. (4r2 � 6r � 2) � (�r2 � 3r � 5) 9. (2x2 � 3xy) � (3x2 � 6xy � 4y2)5r2 � 9r � 3 �x2 � 3xy � 4y2

10. (5t � 7) � (2t2 � 3t � 12) 11. (u � 4) � (6 � 3u2 � 4u)2t2 � 8t � 5 �3u2 � 5u � 10

12. �5(2c2 � d2) 13. x2(2x � 9)�10c2 � 5d2 2x3 � 9x2

14. 2q(3pq � 4q4) 15. 8w(hk2 � 10h3m4 � 6k5w3)6pq2 � 8q5 8hk2w � 80h3m4w � 48k5w4

16. m2n3(�4m2n2 � 2mnp � 7) 17. �3s2y(�2s4y2 � 3sy3 � 4)�4m4n5 � 2m3n4p � 7m2n3 6s6y3 � 9s3y4 � 12s2y

18. (c � 2)(c � 8) 19. (z � 7)(z � 4)c2 � 10c � 16 z2 � 3z � 28

20. (a � 5)2 21. (2x � 3)(3x � 5)a2 � 10a � 25 6x2 � 19x � 15

22. (r � 2s)(r � 2s) 23. (3y � 4)(2y � 3)r2 � 4s2 6y2 � y � 12

24. (3 � 2b)(3 � 2b) 25. (3w � 1)2

9 � 4b2 9w2 � 6w � 1

1�2

b2c�d4


Determine whether each expression is a polynomial. If it is a polynomial, state thedegree of the polynomial.

1. 5x3 � 2xy4 � 6xy yes; 5 2. � ac � a5d3 yes; 8 3. no

4. 25x3z � x�78� yes; 4 5. 6c�2 � c � 1 no 6. � no

Simplify.

7. (3n2 � 1) � (8n2 � 8) 8. (6w � 11w2) � (4 � 7w2)11n2 � 7 �18w2 � 6w � 4

9. (�6n � 13n2) � (�3n � 9n2) 10. (8x2 � 3x) � (4x2 � 5x � 3)�9n � 4n2 4x2 � 8x � 3

11. (5m2 � 2mp � 6p2) � (�3m2 � 5mp � p2) 12. (2x2 � xy � y2) � (�3x2 � 4xy � 3y2)8m2 � 7mp � 7p2 �x2 � 3xy � 4y2

13. (5t � 7) � (2t2 � 3t � 12) 14. (u � 4) � (6 � 3u2 � 4u)2t2 � 8t � 5 �3u2 � 5u � 10

15. �9( y2 � 7w) 16. �9r4y2(�3ry7 � 2r3y4 � 8r10)�9y2 � 63w 27r 5y9 � 18r7y6 � 72r14y2

17. �6a2w(a3w � aw4) 18. 5a2w3(a2w6 � 3a4w2 � 9aw6)�6a5w2 � 6a3w5 5a4w9 � 15a6w5 � 45a3w9

19. 2x2(x2 � xy � 2y2) 20. � ab3d2(�5ab2d5 � 5ab)

2x4 � 2x3y � 4x2y2 3a2b5d7 � 3a2b4d 2

21. (v2 � 6)(v2 � 4) 22. (7a � 9y)(2a � y)v4 � 2v2 � 24 14a2 � 11ay � 9y2

23. ( y � 8)2 24. (x2 � 5y)2

y2 � 16y � 64 x4 � 10x2y � 25y2

25. (5x � 4w)(5x � 4w) 26. (2n4 � 3)(2n4 � 3)25x2 � 16w2 4n8 � 9

27. (w � 2s)(w2 � 2ws � 4s2) 28. (x � y)(x2 � 3xy � 2y2)w3 � 8s3 x3 � 2x2y � xy2 � 2y3

29. BANKING Terry invests $1500 in two mutual funds. The first year, one fund grows 3.8%and the other grows 6%. Write a polynomial to represent the amount Terry’s $1500grows to in that year if x represents the amount he invested in the fund with the lessergrowth rate. �0.022x � 1590

30. GEOMETRY The area of the base of a rectangular box measures 2x2 � 4x � 3 squareunits. The height of the box measures x units. Find a polynomial expression for thevolume of the box. 2x3 � 4x2 � 3x units3

3�5

6�s

5�r

12m8n9��(m � n)2

4�3

Practice (Average)

Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2

Reading to Learn MathematicsPolynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2


Less

on

5-2

Pre-Activity How can polynomials be applied to financial situations?


Suppose that Shenequa decides to enroll in a five-year engineering programrather than a four-year program. Using the model given in your textbook,how could she estimate the tuition for the fifth year of her program? (Donot actually calculate, but describe the calculation that would be necessary.)Multiply $15,604 by 1.04.

Reading the Lesson

1. State whether the terms in each of the following pairs are like terms or unlike terms.

a. 3x2, 3y2 unlike terms b. �m4, 5m4 like termsc. 8r3, 8s3 unlike terms d. �6, 6 like terms

2. State whether each of the following expressions is a monomial, binomial, trinomial, ornot a polynomial. If the expression is a polynomial, give its degree.

a. 4r4 � 2r � 1 trinomial; degree 4 b. �3x� not a polynomialc. 5x � 4y binomial; degree 1 d. 2ab � 4ab2 � 6ab3 trinomial; degree 4

3. a. What is the FOIL method used for in algebra? to multiply binomialsb. The FOIL method is an application of what property of real numbers?

Distributive Propertyc. In the FOIL method, what do the letters F, O, I, and L mean?

first, outer, inner, lastd. Suppose you want to use the FOIL method to multiply (2x � 3)(4x � 1). Show the

terms you would multiply, but do not actually multiply them.

F (2x)(4x)

O (2x)(1)

I (3)(4x)

L (3)(1)


4. You can remember the difference between monomials, binomials, and trinomials bythinking of common English words that begin with the same prefixes. Give two wordsunrelated to mathematics that start with mono-, two that begin with bi-, and two thatbegin with tri-. Sample answer: monotonous, monogram; bicycle, bifocal;tricycle, tripod


Polynomials with Fractional CoefficientsPolynomials may have fractional coefficients as long as there are no variables in the denominators. Computing with fractional coefficients is performed in the same way as computing with whole-number coefficients.

Simpliply. Write all coefficients as fractions.

1. ��35�m � �

27�p � �

13�n� � ��

73�p � �

52�m � �

34�n�

2.��32�x � �

43�y � �

54�z� � ��

14�x � y � �

25�z� � ��

78�x � �

67�y � �

12�z� �

38

�

3.��12�a2 � �

13�ab � �

14�b2� � ��

56�a2 � �

23�ab � �

34�b2� �

43

4.��12�a2 � �

13�ab � �

14�b2� � ��

13�a2 � �

12�ab � �

56�b2� �

5.��12�a2 � �

13�ab � �

14�b2� � ��

12�a � �

23�b� �

14

�

6.��23�a2 � �

15�a � �

27��

23�a3 � �

15�a2 � �

27�a� �

7.��23�x2 � �

34�x � 2� � ��

45�x � �

16�x2 � �

12��

8.��16� � �

13�x � �

16�x4 � �

12�x2� � ��

16�x3 � �

13� � �

13�x� �

31

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2

Study Guide and InterventionDividing Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3


Less

on

5-3

Use Long Division To divide a polynomial by a monomial, use the properties of powersfrom Lesson 5-1.

To divide a polynomial by a polynomial, use a long division pattern. Remember that onlylike terms can be added or subtracted.

Simplify .

� � �

� p3�2t2�1r1�1 � p2�2qt1�1r2�1 � p3�2t1�1r1�1

� 4pt �7qr � 3p

Use long division to find (x3 � 8x2 � 4x � 9) � (x � 4).

x2 � 4x � 12x � 4�x�3�� 8�x�2�� 4�x� �� 9�

(�)x3 � 4x2

�4x2 � 4x(�)�4x2 � 16x

�12x � 9(�)�12x � 48

�57The quotient is x2 � 4x � 12, and the remainder is �57.

Therefore � x2 � 4x � 12 � .

Simplify.

1. 2. 3.

6a2 � 10a � 10 5ab � � 7a4

4. (2x2 � 5x � 3) (x � 3) 5. (m2 � 3m � 7) (m � 2)

2x � 1 m � 5 �

6. (p3 � 6) (p � 1) 7. (t3 � 6t2 � 1) (t � 2)

p2 � p � 1 � t2 � 8t � 16 �

8. (x5 � 1) (x � 1) 9. (2x3 � 5x2 � 4x � 4) (x � 2)

x4 � x3 � x2 � x � 1 2x2 � x � 2

31�t � 2

5�p � 1

3�m � 2

4b2�

a6n3�m

60a2b3 � 48b4 � 84a5b2��

12ab224mn6 � 40m2n3��

4m2n318a3 � 30a2��3a

57�x � 4

x3 � 8x2 � 4x � 9��x � 4

9�3

21�3

12�3

9p3tr�3p2tr

21p2qtr2��

3p2tr12p3t2r�

3p2tr12p3t2r � 21p2qtr2 � 9p3tr��

3p2tr

12p3t2r � 21p2qtr2 � 9p3tr��

3p2trExample 1Example 1

Example 2Example 2

ExercisesExercises


Use Synthetic Division

Synthetic divisiona procedure to divide a polynomial by a binomial using coefficients of the dividend and the value of r in the divisor x � r

Use synthetic division to find (2x3 � 5x2 � 5x � 2) (x � 1).

Step 1 Write the terms of the dividend so that the degrees of the terms are in 2x3 � 5x2 � 5x � 2 descending order. Then write just the coefficients. 2 �5 5 �2

Step 2 Write the constant r of the divisor x � r to the left, In this case, r � 1. 1 2 �5 5 �2Bring down the first coefficient, 2, as shown.

2

Step 3 Multiply the first coefficient by r, 1 � 2 � 2. Write their product under the 1 2 �5 5 �2 second coefficient. Then add the product and the second coefficient: 2 �5 � 2 � � 3. 2 �3

Step 4 Multiply the sum, �3, by r : �3 � 1 � �3. Write the product under the next 1 2 �5 5 �2 coefficient and add: 5 � (�3) � 2. 2 �3

2 �3 2

Step 5 Multiply the sum, 2, by r : 2 � 1 � 2. Write the product under the next 1 2 �5 5 �2 coefficient and add: �2 � 2 � 0. The remainder is 0. 2 �3 2

2 �3 2 0

Thus, (2x3 � 5x2 � 5x � 2) (x � 1) � 2x2 � 3x � 2.

Simplify.

1. (3x3 � 7x2 � 9x � 14) (x � 2) 2. (5x3 � 7x2 � x � 3) (x � 1)

3x2 � x � 7 5x2 � 2x � 3

3. (2x3 � 3x2 � 10x � 3) (x � 3) 4. (x3 � 8x2 � 19x � 9) (x � 4)

2x2 � 3x � 1 x2 � 4x � 3 �

5. (2x3 � 10x2 � 9x � 38) (x � 5) 6. (3x3 � 8x2 � 16x � 1) (x � 1)

2x2 � 9 � 3x2 � 5x � 11 �

7. (x3 � 9x2 � 17x � 1) (x � 2) 8. (4x3 � 25x2 � 4x � 20) (x � 6)

x2 � 7x � 3 � 4x2 � x � 2 �

9. (6x3 � 28x2 � 7x � 9) (x � 5) 10. (x4 � 4x3 � x2 � 7x � 2) (x � 2)

6x2 � 2x � 3 � x3 � 2x2 � 3x � 1

11. (12x4 � 20x3 � 24x2 � 20x � 35) (3x � 5) 4x3 � 8x � 20 ��65�3x � 5

6�x � 5

8�x � 6

5�x � 2

10�x � 1

7�x � 5

3�x � 4


Dividing Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3

ExercisesExercises

Skills PracticeDividing Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3


Less

on

5-3

Simplify.

1. 5c � 3 2. 3x � 5

3. 5y2 � 2y � 1 4. 3x � 1 �

5. (15q6 � 5q2)(5q4)�1 6. (4f 5 � 6f4 � 12f 3 � 8f 2)(4f 2)�1

3q2 � f 3 � � 3f � 2

7. (6j2k � 9jk2) 3jk 8. (4a2h2 � 8a3h � 3a4) (2a2)

2j � 3k 2h2 � 4ah �

9. (n2 � 7n � 10) (n � 5) 10. (d2 � 4d � 3) (d � 1)

n � 2 d � 3

11. (2s2 � 13s � 15) (s � 5) 12. (6y2 � y � 2)(2y � 1)�1

2s � 3 3y � 2

13. (4g2 � 9) (2g � 3) 14. (2x2 � 5x � 4) (x � 3)

2g � 3 2x � 1 �

15. 16.

u � 8 � 2x � 1 �

17. (3v2 � 7v � 10)(v � 4)�1 18. (3t4 � 4t3 � 32t2 � 5t � 20)(t � 4)�1

3v � 5 � 3t3 � 8t2 � 5

19. 20.

y2 � 3y � 6 � 2x2 � 5x � 4 �

21. (4p3 � 3p2 � 2p) ( p � 1) 22. (3c4 � 6c3 � 2c � 4)(c � 2)�1

4p2 � p � 3 � 3c3 � 2 �

23. GEOMETRY The area of a rectangle is x3 � 8x2 � 13x � 12 square units. The width ofthe rectangle is x � 4 units. What is the length of the rectangle? x2 � 4x � 3 units

8�c � 2

3�p � 1

3�x � 3

18�y � 2

2x3 � x2 � 19x � 15��x � 3

y3 � y2 � 6��y � 2

10�v � 4

1�x � 3

12�u � 3

2x2 � 5x � 4��x � 3

u2 � 5u � 12��u � 3

1�x � 3

3a2�

2

3f 2�

21

�q2

2�x

12x2 � 4x � 8��4x

15y3 � 6y2 � 3y��3y

12x � 20��4

10c � 6�2


Simplify.

1. 3r6 � r4 � 2. � 6k2 �

3. (�30x3y � 12x2y2 � 18x2y) (�6x2y) 4. (�6w3z4 � 3w2z5 � 4w � 5z) (2w2z)

5x � 2y � 3 �3wz3 � � �

5. (4a3 � 8a2 � a2)(4a)�1 6. (28d3k2 � d2k2 � 4dk2)(4dk2)�1

a2 � 2a � �a4

� 7d2 � �d4

� � 1

7. f � 5 8. 2x � 7

9. (a3 � 64) (a � 4) a2 � 4a � 16 10. (b3 � 27) (b � 3) b2 � 3b � 9

11. 2x2 � 8x � 38 12. 2x2 � 6x � 22 �

13. (3w3 � 7w2 � 4w � 3) (w � 3) 14. (6y4 � 15y3 � 28y � 6) (y � 2)

3w2 � 2w � 2 � �w �

33

� 6y3 � 3y2 � 6y � 16 � �y

2�6

2�

15. (x4 � 3x3 � 11x2 � 3x � 10) (x � 5) 16. (3m5 � m � 1) (m � 1)

x3 � 2x2 � x � 2 3m4 � 3m3 � 3m2 � 3m � 4 �

17. (x4 � 3x3 � 5x � 6)(x � 2)�1 18. (6y2 � 5y � 15)(2y � 3)�1

x3 � 5x2 � 10x � 15 � �x

2�4

2� 3y � 7 � �

2y6� 3�

19. 20.

2x � 2 � �2x

1�2

3� 2x � 1 � �

3x6� 1�

21. (2r3 � 5r2 � 2r � 15) (2r � 3) 22. (6t3 � 5t2 � 2t � 1) (3t � 1)

r2 � 4r � 5 2t2 � t � 1 � �3t

2� 1�

23. 24.

2p3 � 3p2 � 4p � 1 2h2 � h � 3

25. GEOMETRY The area of a rectangle is 2x2 � 11x � 15 square feet. The length of therectangle is 2x � 5 feet. What is the width of the rectangle? x � 3 ft

26. GEOMETRY The area of a triangle is 15x4 � 3x3 � 4x2 � x � 3 square meters. Thelength of the base of the triangle is 6x2 � 2 meters. What is the height of the triangle?5x2 � x � 3 m

2h4 � h3 � h2 � h � 3��

h2 � 14p4 � 17p2 � 14p � 3��2p � 3

6x2 � x � 7��3x � 1

4x2 � 2x � 6��2x � 3

5�m � 1

72�x � 3

2x3 � 4x � 6��x � 3

2x3 � 6x � 152��x � 4

2x2 � 3x � 14��x � 2

f2 � 7f � 10��f � 2

5�2w2

2�wz

3z4�

2

9m�2k

3k�m

6k2m � 12k3m2 � 9m3��

2km28�r2

15r10 � 5r8 � 40r2��

5r4

Practice (Average)

Dividing Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3

Reading to Learn MathematicsDividing Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3


Less

on

5-3

Pre-Activity How can you use division of polynomials in manufacturing?


Using the division symbol (), write the division problem that you woulduse to answer the question asked in the introduction. (Do not actuallydivide.) (32x2 � x) � (8x)

Reading the Lesson

1. a. Explain in words how to divide a polynomial by a monomial. Divide each term ofthe polynomial by the monomial.

b. If you divide a trinomial by a monomial and get a polynomial, what kind ofpolynomial will the quotient be? trinomial

2. Look at the following division example that uses the division algorithm for polynomials.

2x � 4x � 4��2x2 � 4x � 7

2x2 � 8x4x � 74x � 16

23Which of the following is the correct way to write the quotient? CA. 2x � 4 B. x � 4 C. 2x � 4 � D.

3. If you use synthetic division to divide x3 � 3x2 � 5x � 8 by x � 2, the division will looklike this:

2 1 3 �5 �82 10 10

1 5 5 2

Which of the following is the answer for this division problem? BA. x2 � 5x � 5 B. x2 � 5x � 5 �

C. x3 � 5x2 � 5x � D. x3 � 5x2 � 5x � 2


4. When you translate the numbers in the last row of a synthetic division into the quotientand remainder, what is an easy way to remember which exponents to use in writing theterms of the quotient? Sample answer: Start with the power that is one lessthan the degree of the dividend. Decrease the power by one for eachterm after the first. The final number will be the remainder. Drop any termthat is represented by a 0.

2�x � 2

2�x � 2

23�x � 4

23�x � 4


Oblique AsymptotesThe graph of y � ax � b, where a � 0, is called an oblique asymptote of y � f(x) if the graph of f comes closer and closer to the line as x → ∞ or x → �∞. ∞ is themathematical symbol for infinity, which means endless.

For f(x) � 3x � 4 � �2x�, y � 3x � 4 is an oblique asymptote because

f(x) � 3x � 4 � �2x�, and �

2x� → 0 as x → ∞ or �∞. In other words, as |x |

increases, the value of �2x� gets smaller and smaller approaching 0.

Find the oblique asymptote for f(x) � �x2 �

x8�

x2� 15

�.

�2 1 8 15 Use synthetic division.

�2 �121 6 3

y � �x2 �

x8�x

2� 15� � x � 6 � �x �

32�

As |x | increases, the value of �x �3

2� gets smaller. In other words, since

�x �3

2� → 0 as x → ∞ or x → �∞, y � x � 6 is an oblique asymptote.

Use synthetic division to find the oblique asymptote for each function.

1. y � �8x2 �

x �4x

5� 11

�

2. y � �x2 �

x3�x

2� 15�

3. y � �x2 �

x2�x

3� 18�

4. y � �ax2

x�

�bx

d� c

�

5. y � �ax2

x�

�bx

d� c

�

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3

ExampleExample

Study Guide and InterventionFactoring Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4


Less

on

5-4

Factor Polynomials

For any number of terms, check for:greatest common factor

For two terms, check for:Difference of two squares

a2 � b2 � (a � b)(a � b)Sum of two cubes

a3 � b3 � (a � b)(a2 � ab � b2)Difference of two cubes

a3 � b3 � (a � b)(a2 � ab � b2)

Techniques for Factoring Polynomials For three terms, check for:Perfect square trinomials

a2 � 2ab � b2 � (a � b)2

a2 � 2ab � b2 � (a � b)2

General trinomialsacx2 � (ad � bc)x � bd � (ax � b)(cx � d )

For four terms, check for:Groupingax � bx � ay � by � x(a � b) � y(a � b)

� (a � b)(x � y)

Factor 24x2 � 42x � 45.

First factor out the GCF to get 24x2 � 42x � 45 � 3(8x2 � 14x � 15). To find the coefficientsof the x terms, you must find two numbers whose product is 8 � (�15) � �120 and whosesum is �14. The two coefficients must be �20 and 6. Rewrite the expression using �20x and6x and factor by grouping.

8x2 � 14x � 15 � 8x2 � 20x � 6x � 15 Group to find a GCF.

� 4x(2x � 5) � 3(2x � 5) Factor the GCF of each binomial.

� (4x � 3)(2x � 5) Distributive Property

Thus, 24x2 � 42x � 45 � 3(4x � 3)(2x � 5).

Factor completely. If the polynomial is not factorable, write prime.

1. 14x2y2 � 42xy3 2. 6mn � 18m � n � 3 3. 2x2 � 18x � 16

14xy2(x � 3y) (6m � 1)(n � 3) 2(x � 8)(x � 1)

4. x4 � 1 5. 35x3y4 � 60x4y 6. 2r3 � 250

(x2 � 1)(x � 1)(x � 1) 5x3y(7y3 � 12x) 2(r � 5)(r2 � 5r � 25)

7. 100m8 � 9 8. x2 � x � 1 9. c4 � c3 � c2 � c

(10m4 � 3)(10m4 � 3) prime c(c � 1)2(c � 1)

ExampleExample

ExercisesExercises


Simplify Quotients In the last lesson you learned how to simplify the quotient of twopolynomials by using long division or synthetic division. Some quotients can be simplified byusing factoring.

Simplify .

� Factor the numerator and denominator.

� Divide. Assume x � 8, � .

Simplify. Assume that no denominator is equal to 0.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

9x2 � 6x � 4��

3x � 2y2 � 4y � 16��

3y � 52x � 3

��4x2 � 2x � 1

27x3 � 8��9x2 � 4

y3 � 64��3y2 � 17y � 20

4x2 � 4x � 3��8x3 � 1

2x � 3�x � 8

7x � 3�x � 2

x � 9�2x � 5

4x2 � 12x � 9��2x2 � 13x � 24

7x2 � 11x � 6��x2 � 4

x2 � 81��2x2 � 23x � 45

x � 7�x � 5

3x � 5�2x � 3

2x � 5�x � 1

x2 � 14x � 49��x2 � 2x � 35

3x2 � 4x � 15��2x2 � 3x � 9

4x2 � 16x � 15��2x2 � x � 3

x � 2�3x � 7

6x � 1�x � 2

2x � 1�x � 2

x2 � 7x � 10��3x2 � 8x � 35

6x2 � 25x � 4��x2 � 6x � 8

4x2 � 4x � 3��2x2 � x � 6

5x � 1�x � 3

2x � 1�4x � 1

x � 3�x � 5

5x2 � 9x � 2��x2 � 5x � 6

2x2 � 5x � 3��4x2 � 11x � 3

x2 � x � 6��x2 � 7x � 10

x � 5�x � 1

x � 5�2x� 3

x � 4�x � 2

x2 � 11x � 30��x2 � 5x � 6

x2 � 6x � 5��2x2 � x � 3

x2 � 7x � 12��x2 � x � 6

3�2

x � 4�x � 8

(2x � 3)( x � 4)��(x � 8)(2x � 3)

8x2 � 11x � 12��2x2 � 13x � 24

8x2 � 11x � 12��2x2 � 13x � 24


Factoring Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4

ExampleExample

ExercisesExercises

Skills PracticeFactoring Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4


Less

on

5-4


1. 7x2 � 14x 2. 19x3 � 38x2

7x(x � 2) 19x2(x � 2)

3. 21x3 � 18x2y � 24xy2 4. 8j3k � 4jk3 � 73x(7x2 � 6xy � 8y2) prime

5. a2 � 7a � 18 6. 2ak � 6a � k � 3(a � 9)(a � 2) (2a � 1)(k � 3)

7. b2 � 8b � 7 8. z2 � 8z � 10(b � 7)(b � 1) prime

9. m2 � 7m � 18 10. 2x2 � 3x � 5(m � 2)(m � 9) (2x � 5)(x � 1)

11. 4z2 � 4z � 15 12. 4p2 � 4p � 24(2z � 5)(2z � 3) 4(p � 2)(p � 3)

13. 3y2 � 21y � 36 14. c2 � 1003(y � 4)(y � 3) (c � 10)(c � 10)

15. 4f2 � 64 16. d2 � 12d � 364(f � 4)(f � 4) (d � 6)2

17. 9x2 � 25 18. y2 � 18y � 81prime (y � 9)2

19. n3 � 125 20. m4 � 1(n � 5)(n2 � 5n � 25) (m2 � 1)(m � 1)(m � 1)


21. 22.

23. 24.x � 1�x � 7

x2 � 6x � 7��

x2 � 49

x � 5�

xx2 � 10x � 25��

x2 � 5x

x � 1�x � 3

x2 � 4x � 3��x2 � 6x � 9

x � 2�x � 5

x2 � 7x � 18��x2 � 4x � 45



1. 15a2b � 10ab2 2. 3st2 � 9s3t � 6s2t2 3. 3x3y2 � 2x2y � 5xy5ab(3a � 2b) 3st(t � 3s2 � 2st) xy(3x2y � 2x � 5)

4. 2x3y � x2y � 5xy2 � xy3 5. 21 � 7t � 3r � rt 6. x2 � xy � 2x � 2yxy(2x2 � x � 5y � y2) (7 � r )(3 � t) (x � 2)(x � y)

7. y2 � 20y � 96 8. 4ab � 2a � 6b � 3 9. 6n2 � 11n � 2(y � 8)(y � 12) (2a � 3)(2b � 1) (6n � 1)(n � 2)

10. 6x2 � 7x � 3 11. x2 � 8x � 8 12. 6p2 � 17p � 45(3x � 1)(2x � 3) prime (2p � 9)(3p � 5)

13. r3 � 3r2 � 54r 14. 8a2 � 2a � 6 15. c2 � 49r (r � 9)(r � 6) 2(4a � 3)(a � 1) (c � 7)(c � 7)

16. x3 � 8 17. 16r2 � 169 18. b4 � 81(x � 2)(x2 � 2x � 4) (4r � 13)(4r � 13) (b2 � 9)(b � 3)(b � 3)

19. 8m3 � 25 prime 20. 2t3 � 32t2 � 128t 2t(t � 8)2

21. 5y5 � 135y2 5y2(y � 3)(y2 � 3y � 9) 22. 81x4 � 16 (9x2 � 4)(3x � 2)(3x � 2)


23. 24. 25.

26. DESIGN Bobbi Jo is using a software package to create a drawing of a cross section of a brace as shown at the right.Write a simplified, factored expression that represents the area of the cross section of the brace. x(20.2 � x) cm2

27. COMBUSTION ENGINES In an internal combustion engine, the up and down motion of the pistons is converted into the rotary motion of the crankshaft, which drives the flywheel. Let r1 represent the radius of the flywheel at the right and let r2 represent the radius of thecrankshaft passing through it. If the formula for the area of a circle is A � �r2, write a simplified, factored expression for the area of the cross section of the flywheel outside the crankshaft. � (r1 � r2)(r1 � r2)

r1

r2

x cm

x cm

8.2 cm

12 c

m

3(x � 3)��x2 � 3x � 9

3x2 � 27��x3 � 27

x � 8�x � 9

x2 � 16x � 64��

x2 � x � 72

x � 4�x � 5

x2 � 16��x2 � x � 20

Practice (Average)

Factoring Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4

Reading to Learn MathematicsFactoring Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4


Less

on

5-4

Pre-Activity How does factoring apply to geometry?


If a trinomial that represents the area of a rectangle is factored into twobinomials, what might the two binomials represent? the length andwidth of the rectangle

Reading the Lesson

1. Name three types of binomials that it is always possible to factor. difference of twosquares, sum of two cubes, difference of two cubes

2. Name a type of trinomial that it is always possible to factor. perfect squaretrinomial

3. Complete: Since x2 � y2 cannot be factored, it is an example of a polynomial.

4. On an algebra quiz, Marlene needed to factor 2x2 � 4x � 70. She wrote the followinganswer: (x � 5)(2x � 14). When she got her quiz back, Marlene found that she did notget full credit for her answer. She thought she should have gotten full credit because shechecked her work by multiplication and showed that (x � 5)(2x � 14) � 2x2 � 4x � 70.

a. If you were Marlene’s teacher, how would you explain to her that her answer was notentirely correct? Sample answer: When you are asked to factor apolynomial, you must factor it completely. The factorization was notcomplete, because 2x � 14 can be factored further as 2(x � 7).

b. What advice could Marlene’s teacher give her to avoid making the same kind of errorin factoring in the future? Sample answer: Always look for a commonfactor first. If there is a common factor, factor it out first, and then seeif you can factor further.


5. Some students have trouble remembering the correct signs in the formulas for the sumand difference of two cubes. What is an easy way to remember the correct signs?Sample answer: In the binomial factor, the operation sign is the same asin the expression that is being factored. In the trinomial factor, theoperation sign before the middle term is the opposite of the sign in theexpression that is being factored. The sign before the last term is alwaysa plus.

prime


Using Patterns to FactorStudy the patterns below for factoring the sum and the difference of cubes.

a3 � b3 � (a � b)(a2 � ab � b2)a3 � b3 � (a � b)(a2 � ab � b2)

This pattern can be extended to other odd powers. Study these examples.

Factor a5 � b5.Extend the first pattern to obtain a5 � b5 � (a � b)(a4 � a3b � a2b2 � ab3 � b4).Check: (a � b)(a4 � a3b � a2b2 � ab3 � b4) � a5 � a4b � a3b2 � a2b3 � ab4

� a4b � a3b2 � a2b3 � ab4 � b5

� a5 � b5

Factor a5 � b5.Extend the second pattern to obtain a5 � b5 � (a � b)(a4 � a3b � a2b2 � ab3 � b4).Check: (a � b)(a4 � a3b � a2b2 � ab3 � b4) � a5 � a4b � a3b2 � a2b3 � ab4

� a4b � a3b2 � a2b3 � ab4 � b5

� a5 � b5

In general, if n is an odd integer, when you factor an � bn or an � bn, one factor will beeither (a � b) or (a � b), depending on the sign of the original expression. The other factorwill have the following properties:

• The first term will be an � 1 and the last term will be bn � 1.• The exponents of a will decrease by 1 as you go from left to right.• The exponents of b will increase by 1 as you go from left to right.• The degree of each term will be n � 1.• If the original expression was an � bn, the terms will alternately have � and � signs.• If the original expression was an � bn, the terms will all have � signs.

Use the patterns above to factor each expression.

1. a7 � b7

2. c9 � d9

3. e11 � f 11

To factor x10 � y10, change it to (x5 � y5)(x5 � y5) and factor each binomial. Usethis approach to factor each expression.

4. x10 � y10

5. a14 � b14

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4

Example 1Example 1

Example 2Example 2

Study Guide and InterventionRoots of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5


Less

on

5-5

Simplify Radicals

Square Root For any real numbers a and b, if a2 � b, then a is a square root of b.

nth RootFor any real numbers a and b, and any positive integer n, if an � b, then a is an n throot of b.

1. If n is even and b � 0, then b has one positive root and one negative root.Real nth Roots of b, 2. If n is odd and b � 0, then b has one positive root.n

�b�, �n

�b� 3. If n is even and b 0, then b has no real roots.4. If n is odd and b 0, then b has one negative root.

Simplify �49z8�.

�49z8� � �(7z4)2� � 7z4

z4 must be positive, so there is no need totake the absolute value.

Simplify �3�(2a �� 1)6�

�3�(2a �� 1)6� � �

3�[(2a �� 1)2]3� � (2a � 1)2

Example 1Example 1 Example 2Example 2

ExercisesExercises

Simplify.

1. �81� 2.3��343� 3. �144p6�

9 �7 12|p3|

4. �4a10� 5.5�243p1�0� 6. �

3�m6n9��2a5 3p2 �m2n3

7.3��b12� 8. �16a10�b8� 9. �121x6�

�b4 4|a5|b4 11|x3|

10. �(4k)4� 11. �169r4� 12. �3��27p�6�

16k2 �13r 2 3p2

13. ��625y2�z4� 14. �36q34� 15. �100x2�y4z6��25|y |z2 6|q17| 10|x |y2|z3|

16.3��0.02�7� 17. ��0.36� 18. �0.64p�10�

�0.3 not a real number 0.8|p5|

19.4�(2x)8� 20. �(11y2)�4� 21.

3�(5a2b)�6�4x2 121y4 25a4b2

22. �(3x �� 1)2� 23.3�(m ��5)6� 24. �36x2 �� 12x �� 1�

|3x � 1| (m � 5)2 |6x � 1|


Approximate Radicals with a Calculator

Irrational Number a number that cannot be expressed as a terminating or a repeating decimal

Radicals such as �2� and �3� are examples of irrational numbers. Decimal approximationsfor irrational numbers are often used in applications. These approximations can be easilyfound with a calculator.

Approximate 5

�18.2� with a calculator.5�18.2� � 1.787

Use a calculator to approximate each value to three decimal places.

1. �62� 2. �1050� 3.3�0.054�

7.874 32.404 0.378

4. �4�5.45� 5. �5280� 6. �18,60�0�

�1.528 72.664 136.382

7. �0.095� 8.3��15� 9.

5�100�

0.308 �2.466 2.512

10.6�856� 11. �3200� 12. �0.05�

3.081 56.569 0.224

13. �12,50�0� 14. �0.60� 15. �4�500�

111.803 0.775 �4.729

16.3�0.15� 17.

6�4200� 18. �75�

0.531 4.017 8.660

19. LAW ENFORCEMENT The formula r � 2�5L� is used by police to estimate the speed rin miles per hour of a car if the length L of the car’s skid mark is measures in feet.Estimate to the nearest tenth of a mile per hour the speed of a car that leaves a skidmark 300 feet long. 77.5 mi/h

20. SPACE TRAVEL The distance to the horizon d miles from a satellite orbiting h milesabove Earth can be approximated by d � �8000h� � h2�. What is the distance to thehorizon if a satellite is orbiting 150 miles above Earth? about 1100 ft


Roots of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

ExampleExample

ExercisesExercises

Skills PracticeRoots of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5


Less

on

5-5


1. �230� 15.166 2. �38� 6.164

3. ��152� �12.329 4. �5.6� 2.366

5.3�88� 4.448 6.

3��222� �6.055

7. �4�0.34� �0.764 8.

5�500� 3.466

Simplify.

9. �81� �9 10. �144� 12

11. �(�5)2� 5 12. ��52� not a real number

13. �0.36� 0.6 14. ��

15.3��8� �2 16. �

3�27� �3

17.3�0.064� 0.4 18.

5�32� 2

19.4�81� 3 20. �y2� |y |

21.3�125s3� 5s 22. �64x6� 8|x3|

23.3��27a�6� �3a2 24. �m8n4� m4n2

25. ��100p4�q2� �10p2|q | 26.4�16w4v�8� 2|w |v2

27. �(�3c)4� 9c2 28. �(a � b�)2� |a � b |

2�3

4�9



1. �7.8� 2. ��89� 3.3�25� 4.

3��4�2.793 �9.434 2.924 �1.587

5.4�1.1� 6.

5��0.1� 7.6�5555� 8.

4�(0.94)�2�1.024 �0.631 4.208 0.970

Simplify.

9. �0.81� 10. ��324� 11. �4�256� 12.

6�64�

0.9 �18 �4 2

13.3��64� 14.

3�0.512� 15.5��243� 16. �

4�1296�

�4 0.8 �3 �6

17. 5�� 18.5�243x1�0� 19. �(14a)2� 20. ��(14a�)2� not a

��43

� 3x2 14|a| real number

21. �49m2t�8� 22. �� 23.3��64r6�w15� 24. �(2x)8�

7|m |t4 �4r2w5 16x4

25. �4�625s8� 26.

3�216p3�q9� 27. �676x4�y6� 28.3��27x9�y12�

�5s2 6pq3 26x2|y3| �3x3y4

29. ��144m�8n6� 30.5��32x5�y10� 31.

6�(m ��4)6� 32.3�(2x �� 1)3�

�12m4|n3| �2xy2 |m � 4| 2x � 1

33. ��49a10�b16� 34.4�(x � 5�)8� 35.

3�343d6� 36. �x2 � 1�0x � 2�5��7|a5|b8 (x � 5)2 7d2 |x � 5|

37. RADIANT TEMPERATURE Thermal sensors measure an object’s radiant temperature,which is the amount of energy radiated by the object. The internal temperature of an object is called its kinetic temperature. The formula Tr � Tk

4�e� relates an object’s radianttemperature Tr to its kinetic temperature Tk. The variable e in the formula is a measureof how well the object radiates energy. If an object’s kinetic temperature is 30°C and e � 0.94, what is the object’s radiant temperature to the nearest tenth of a degree?29.5C

38. HERO’S FORMULA Salvatore is buying fertilizer for his triangular garden. He knowsthe lengths of all three sides, so he is using Hero’s formula to find the area. Hero’sformula states that the area of a triangle is �s(s ��a)(s �� b)(s �� c)�, where a, b, and c arethe lengths of the sides of the triangle and s is half the perimeter of the triangle. If thelengths of the sides of Salvatore’s garden are 15 feet, 17 feet, and 20 feet, what is thearea of the garden? Round your answer to the nearest whole number. 124 ft2

4|m|�

5

16m2�25

�1024�243

Practice (Average)

Roots of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

Reading to Learn MathematicsRoots of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5


Less

on

5-5

Pre-Activity How do square roots apply to oceanography?


Suppose the length of a wave is 5 feet. Explain how you would estimate thespeed of the wave to the nearest tenth of a knot using a calculator. (Do notactually calculate the speed.) Sample answer: Using a calculator,find the positive square root of 5. Multiply this number by 1.34.Then round the answer to the nearest tenth.

Reading the Lesson

1. For each radical below, identify the radicand and the index.

a.3�23� radicand: index:

b. �15x2� radicand: index:

c.5��343� radicand: index:

2. Complete the following table. (Do not actually find any of the indicated roots.)

NumberNumber of Positive Number of Negative Number of Positive Number of Negative

Square Roots Square Roots Cube Roots Cube Roots

27 1 1 1 0

�16 0 0 0 1

3. State whether each of the following is true or false.

a. A negative number has no real fourth roots. true

b. �121� represents both square roots of 121. true

c. When you take the fifth root of x5, you must take the absolute value of x to identifythe principal fifth root. false


4. What is an easy way to remember that a negative number has no real square roots buthas one real cube root? Sample answer: The square of a positive or negativenumber is positive, so there is no real number whose square is negative.However, the cube of a negative number is negative, so a negativenumber has one real cube root, which is a negative number.

5�343

215x2

323


Approximating Square RootsConsider the following expansion.

(a � �2ba�)2

� a2 � �22aab

� � �4ba

2

2�

� a2 � b � �4ba

2

2�

Think what happens if a is very great in comparison to b. The term �4ba

2

2� is very small and can be disregarded in an approximation.

(a � �2ba�)2

a2 � b

a � �2ba� �a2 � b�

Suppose a number can be expressed as a2 � b, a � b. Then an approximate value

of the square root is a � �2ba�. You should also see that a � �2

ba� �a2 � b�.

Use the formula �a2 ��b� � a � �2ba� to approximate �101� and �622�.

a. �101� � �100 �� 1� � �102 �� 1� b. �622� � �625 �� 3� � �252 �� 3�

Let a � 10 and b � 1. Let a � 25 and b � 3.

�101� 10 � �2(110)� �622� 25 � �2(

325)�

10.05 24.94

Use the formula to find an approximation for each square root to the nearest hundredth. Check your work with a calculator.

1. �626� 2. �99� 3. �402�

4. �1604� 5. �223� 6. �80�

7. �4890� 8. �2505� 9. �3575�

10. �1,441�,100� 11. �290� 12. �260�

13. Show that a � �2ba� �a2 � b� for a � b. �

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

ExampleExample

Study Guide and InterventionRadical Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6


Less

on

5-6

Simplify Radical Expressions

For any real numbers a and b, and any integer n � 1:

Product Property of Radicals 1. if n is even and a and b are both nonnegative, then n�ab� �

n�a� �n�b�.

2. if n is odd, then n�ab� �

n�a� �n�b�.

To simplify a square root, follow these steps:1. Factor the radicand into as many squares as possible.2. Use the Product Property to isolate the perfect squares.3. Simplify each radical.

Quotient Property of RadicalsFor any real numbers a and b � 0, and any integer n � 1,

n�� , if all roots are defined.

To eliminate radicals from a denominator or fractions from a radicand, multiply thenumerator and denominator by a quantity so that the radicand has an exact root.

n�a��n�b�

a�b

Simplify 3��16a�5b7�.

3��16a�5b7� �3�(�2)3�� 2 � a�3 � a2�� (b2)3� � b�

� �2ab2 3�2a2b�

Simplify ��.

�� Quotient Property

� Factor into squares.

� Product Property

� Simplify.

� � Rationalize thedenominator.

� Simplify.2|x|�10xy��

15y3

�5y��5y�

2|x|�2x��3y2�5y�

2|x|�2x��3y2�5y�

�(2x)2� � �2x��(3y2)2� � �5y�

�(2x)2 �� 2x��(3y2)2� � 5y�

�8x3��45y5�

8x3�45y5

8x3�45y5


ExercisesExercises

Simplify.

1. 5�54� 15�6� 2.4�32a9b�20� 2a2|b5| 4�2a� 3. �75x4y�7� 5x2y3�5y�

4. �� 5. �� 6. 3�� pq3�5p2�

��10

p5q3�40

|a3|b�2b��

14a6b3�98

6�5��

2536�125


Operations with Radicals When you add expressions containing radicals, you canadd only like terms or like radical expressions. Two radical expressions are called likeradical expressions if both the indices and the radicands are alike.

To multiply radicals, use the Product and Quotient Properties. For products of the form (a�b� � c�d�) � (e�f� � g�h�), use the FOIL method. To rationalize denominators, use conjugates. Numbers of the form a�b� � c�d� and a�b� � c�d�, where a, b, c, and d arerational numbers, are called conjugates. The product of conjugates is always a rationalnumber.

Simplify 2�50� � 4�500� � 6�125�.

2�50� � 4�500� � 6�125� � 2�52 � 2� � 4�102 � 5� � 6�52 � 5� Factor using squares.

� 2 � 5 � �2� � 4 � 10 � �5� � 6 � 5 � �5� Simplify square roots.

� 10�2� � 40�5� � 30�5� Multiply.

� 10�2� � 10�5� Combine like radicals.


Radical Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6

Example 1Example 1

Simplify (2�3� � 4�2�)(�3� � 2�2�).(2�3� � 4�2�)(�3� � 2�2�)

� 2�3� � �3� � 2�3� � 2�2� � 4�2� � �3� � 4�2� � 2�2�� 6 � 4�6� � 4�6� � 16� �10

Simplify .

� �

�

�

�11 � 5�5��4

6 � 5�5� � 5��9 � 5

6 � 2�5� � 3�5� � (�5� )2��

32 � (�5�)2

3 � �5��3 � �5�

2 � �5��3 � �5�

2 � �5��3 � �5�

2 � �5��3 � �5�


ExercisesExercises

Simplify.

1. 3�2� � �50� � 4�8� 2. �20� � �125� � �45� 3. �300� � �27� � �75�

0 4�5� 2�3�

4.3�81� �

3�24� 5.3�2�( 3�4� �

3�12�) 6. 2�3�(�15� � �60� )

63�9� 2 � 2

3�3� 18�5�

7. (2 � 3�7�)(4 � �7�) 8. (6�3� � 4�2�)(3�3� � �2�) 9. (4�2� � 3�5�)(2�20 ��5�)

29 � 14�7� 46 � 6�6� 40�2� � 30�5�

10. 5 11. 5 � 3�2� 12. 13�3� � 23��

115 � 3�3��1 � 2�3�

4 � �2��2 � �2�

5�48� � �75��

5�3�

Skills PracticeRadical Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6


Less

on

5-6

Simplify.

1. �24� 2�6� 2. �75� 5�3�

3.3�16� 2

3�2� 4. �4�48� �2

4�3�

5. 4�50x5� 20x2�2x� 6.4�64a4b�4� 2|ab | 4�4�

7. 3�� d2f5 � f3�d 2f 2� 8. ��s2t |s |�t�

9. �� 10. 3��

11. ��25gz3� 12. (3�3�)(5�3�) 45

13. (4�12� )(3�20� ) 48�15� 14. �2� � �8� � �50� 8�2�

15. �12� � 2�3� � �108� 6�3� 16. 8�5� � �45� � �80� �5�

17. 2�48� � �75� � �12� �3� 18. (2 � �3�)(6 � �2�) 12 � 2�2� � 6�3� � �6�

19. (1 � �5�)(1 � �5�) �4 20. (3 � �7�)(5 � �2�) 15 � 3�2� � 5�7� � �14�

21. (�2� � �6�)2 8 � 4�3� 22.

23. 24. 40 � 5�6��

585

�8 � �6�

12 � 4�2��

74

�3 � �2�

21 � 3�2��

473

�7 � �2�

g�10gz��

5z

3�6��

32�9

�21��

73�7

5�6

25�36

1�2

1�8


Simplify.

1. �540� 6�15� 2.3��432� �6

3�2� 3.3�128� 4

3�2�

4. �4�405� �3

4�5� 5.3��500�0� �10

3�5� 6.5��121�5� �3

5�5�

7.3�125t6w�2� 5t 2 3�w2� 8.

4�48v8z�13� 2v2z3 4�3z� 9.3�8g3k8� 2gk2 3�k2�

10. �45x3y�8� 3xy4�5x� 11. �� 12. 3�� 3�9�

13. ��c4d7 c2d 3�2d� 14. �� 15. 4��16. (3�15� )(�4�45� ) 17. (2�24� )(7�18� ) 18. �810� � �240� � �250�

�180�3� 168�3� 4�10� � 4�15�

19. 6�20� � 8�5� � 5�45� 20. 8�48� � 6�75� � 7�80� 21. (3�2� � 2�3�)2

5�5� 2�3� � 28�5� 30 � 12�6�

22. (3 � �7�)2 23. (�5� � �6�)(�5� � �2�) 24. (�2� � �10� )(�2� � �10� )16 � 6�7� 5 � �10� � �30� � 2�3� �8

25. (1 � �6�)(5 � �7�) 26. (�3� � 4�7�)2 27. (�108� � 6�3�)2

5 � �7� � 5�6� � �42� 115 � 8�21� 0

28. �15� � 2�3� 29. 6�2� � 6 30.

31. 32. 27 � 11�6� 33.

34. BRAKING The formula s � 2�5�� estimates the speed s in miles per hour of a car whenit leaves skid marks � feet long. Use the formula to write a simplified expression for s if � � 85. Then evaluate s to the nearest mile per hour. 10�17�; 41 mi/h

35. PYTHAGOREAN THEOREM The measures of the legs of a right triangle can berepresented by the expressions 6x2y and 9x2y. Use the Pythagorean Theorem to find asimplified expression for the measure of the hypotenuse. 3x2 |y |�13�

6 � 5�x� � x��

4 � x3 � �x��2 � �x�

3 � �6��5 � �24�

8 � 5�2��

23 � �2��2 � �2�

17 � �3��

135 � �3��4 � �3�

6��2� � 1

�3��5� � 2

4�72a��

3a8

�9a33a2�a��

8b29a5�64b4

1�16

1�128

216�24

�11��

311�9

Practice (Average)

Radical Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6

Reading to Learn MathematicsRadical Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6


Less

on

5-6

Pre-Activity How do radical expressions apply to falling objects?


Describe how you could use the formula given in your textbook and acalculator to find the time, to the nearest tenth of a second, that it wouldtake for the water balloons to drop 22 feet. (Do not actually calculate thetime.) Sample answer: Multiply 22 by 2 (giving 44) and divideby 32. Use the calculator to find the square root of the result.Round this square root to the nearest tenth.

Reading the Lesson

1. Complete the conditions that must be met for a radical expression to be in simplified form.

• The n is as as possible.

• The contains no (other than 1) that are nth

powers of a(n) or polynomial.

• The radicand contains no .

• No appear in the .

2. a. What are conjugates of radical expressions used for? to rationalize binomialdenominators

b. How would you use a conjugate to simplify the radical expression ?

Multiply numerator and denominator by 3 � �2�.

c. In order to simplify the radical expression in part b, two multiplications are

necessary. The multiplication in the numerator would be done by themethod, and the multiplication in the denominator would be done by finding the

of .


3. One way to remember something is to explain it to another person. When rationalizing the

denominator in the expression , many students think they should multiply numerator

and denominator by . How would you explain to a classmate why this is incorrect

and what he should do instead. Sample answer: Because you are working withcube roots, not square roots, you need to make the radicand in thedenominator a perfect cube, not a perfect square. Multiply numerator and

denominator by to make the denominator 3�8�, which equals 2.3

�4��3�4�

3�2��3�2�

1�3�2�

squarestwodifference

FOIL

1 � �2��3 � �2�

denominatorradicals

fractions

integer

factorsradicand

smallindex


Special Products with RadicalsNotice that (�3�)(�3�) � 3, or (�3�)2

� 3.

In general, (�x�)2� x when x � 0.

Also, notice that (�9�)(�4�) � �36�.

In general, (�x�)(�y�) � �xy� when x and y are not negative.

You can use these ideas to find the special products below.

(�a� � �b�)(�a� � �b�) � (�a�)2� (�b�)2

� a � b

(�a� � �b�)2� (�a�)2

� 2�ab� � (�b�)2� a � 2�ab� � b

(�a� � �b�)2� (�a�)2

� 2�ab� � (�b�)2� a � 2�ab� � b

Find the product: (�2� � �5�)(�2� � �5�).(�2� � �5�)(�2� � �5�) � (�2�)2

� (�5�)2� 2 � 5 � �3

Evaluate (�2� � �8�)2.

(�2� � �8�)2� (�2�)2

� 2�2��8� � (�8�)2

� 2 � 2�16� � 8 � 2 � 2(4) � 8 � 2 � 8 � 8 � 18

Multiply.

1. (�3� � �7�)(�3� � �7�) 2. (�10� � �2�)(�10� � �2�)

3. (�2x� � �6�)(�2x� � �6�) 4. (�3� � 27)2

5. (�1000� � �10� )2 6. (�y� � �5�)(�y� � �5�)

7. (�50� � �x�)2 8. (�x� � 20)2

You can extend these ideas to patterns for sums and differences of cubes.Study the pattern below.

( 3�8� � 3�x�)( 3�82� �

3�8x� �3�x2�) �

3�83� � 3�x3� � 8 � x

Multiply.

9. ( 3�2� �3�5�)( 3�22� �

3�10� �3�52�)

10. ( 3�y� �3�w�)( 3�y2� �

3�yw� �3�w2� )

11. ( 3�7� �3�20� )( 3�72� �

3�140� �3�202� )

12. ( 3�11� �3�8�)( 3�112� �

3�88� �3�82�)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6

Example 1Example 1

Example 2Example 2

Study Guide and InterventionRational Exponents

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7


Less

on

5-7

Rational Exponents and Radicals

Definition of b�n1� For any real number b and any positive integer n,

b�1n

��

n�b�, except when b 0 and n is even.

Definition of b�mn� For any nonzero real number b, and any integers m and n, with n � 1,

b�mn�

�n�bm� � ( n�b�)m, except when b 0 and n is even.

Write 28�12

�in radical form.

Notice that 28 � 0.

28�12

�� 28�� 22 � 7�� 22� � �7�� 2�7�

Evaluate � ��13

�

.

Notice that �8 0, �125 0, and 3 is odd.

� ��13

��

�

�2�5

�2��5

3��8��3��125�

�8��125

�8��125


ExercisesExercises

Write each expression in radical form.

1. 11�17

�

2. 15�13

�

3. 300�32

�

7�11� 3�15� �3003�

Write each radical using rational exponents.

4. �47� 5.3�3a5b2� 6.

4�162p5�

47�12

�3

�13

�a

�53

�b

�23

�3 2

�14

� p

�54

�

Evaluate each expression.

7. �27�23

�

8. 9. (0.0004)�12

�

9 0.02

10. 8�23

�

� 4�32

�

11. 12.

32 1�2

1�4

16��

12

�

�

(0.25)�12

�

144��

12

�

�

27��

13

�

1�10

5��

12

�

�2�5�


Simplify Expressions All the properties of powers from Lesson 5-1 apply to rationalexponents. When you simplify expressions with rational exponents, leave the exponent inrational form, and write the expression with all positive exponents. Any exponents in thedenominator must be positive integers

When you simplify radical expressions, you may use rational exponents to simplify, but youranswer should be in radical form. Use the smallest index possible.


Rational Exponents

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7

Simplify y�23

� y

�38

�.

y�23

�� y

�38

�� y

�23

� � �38

�� y

�2254�

Simplify 4�144x6�.

4�144x6� � (144x6)�14

�

� (24 � 32 � x6)�14

�

� (24)�14

�� (32)

�14

�� (x6)

�14

�

� 2 � 3�12

�� x

�32

�� 2x � (3x)

�12

�� 2x�3x�


ExercisesExercises

Simplify each expression.

1. x�45

�� x

�65

�2. �y�

23

�� 34

�3. p

�45

�� p

�170�

x2 y �12

� p�32

�

4. �m��65

�� 25

� 5. x��38

�� x

�43

�6. �s��

16

��43

�

x�2234�

s�29

�

7. 8. �a�23

�� 65

�� a�

25

��3 9.

p�23

� a2

10.6�128� 11.

4�49� 12.5�288�

2�2� �7� 2 5�9�

13. �32� � 3�16� 14.3�25� � �125� 15.

6�16�

48�2� 25 6�5� 3�4�

16. 17. ��3�48� 18.

6�48� �a�6�b5�

�b

x�3� �6�35�

��6

a3�b4�

��ab3�

x �3

�3��

�12�

x�56

�

�x

x��12

�

�x��

13

�

p�

p�13

�

1�m3

Skills PracticeRational Exponents

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7


Less

on

5-7


1. 3�16

� 6�3� 2. 8�15

� 5�8�

3. 12�23

� 3�122� or ( 3�12�)24. (s3)�

35

� s5�s4�


5. �51� 51�12

�6.

3�37� 37�13

�

7.4�153� 15

�34

�8.

3�6xy2� 6�13

�x

�13

�y

�23

�


9. 32�15

� 2 10. 81�14

� 3

11. 27��13

� 12. 4��12

�

13. 16�32

� 64 14. (�243)�45

� 81

15. 27�13

�� 27

�53

� 729 16. � ��32

�


17. c�152�

� c�35

� c3 18. m�29

�� m

�196� m2

19. �q�12

��3q

�32

�20. p��

15

�

21. x��161� 22. x

�152�

23. 24.

25.12�64� �2� 26.

8�49a8b�2� |a | 4�7b�

n�23

�

�n

n�13

�

�

n�16

�� n

�12

�

y�14

�

�y

y��12

�

�

y�14

�

x�23

�

�

x�14

�

x�151�

�x

p�45

�

�p

8�27

4�9

1�2

1�3



1. 5�13

�2. 6

�25

�3. m

�47

�4. (n3)�

25

�

3�5� 5�62� or ( 5�6�)2 7�m4� or ( 7�m�)4 n5�n�


5. �79� 6.4�153� 7.

3�27m6n�4� 8. 5�2a10b�

79�12

�153�

14

� 3m2n�43

�5 2

�12

� |a5 |b�12

�


9. 81�14

� 3 10. 1024��15

�11. 8��

53

�

12. �256��34

�� 13. (�64)��

23

�14. 27

�13

�� 27

�43

� 243

15. � ��23

� 16. 17. �25�12

��64��13

��


18. g�47

�� g

�37

� g 19. s�34

�� s

�143� s4 20. �u��

13

��45

� u�145�

21. y��12

�

22. b��35

�23. q

�15

�24. 25.

26.10�85� 2�2� 27. �12� �

5�123� 28.4�6� � 3

4�6� 29.

1210�12� 3�6�

30. ELECTRICITY The amount of current in amperes I that an appliance uses can be

calculated using the formula I � � ��12

�, where P is the power in watts and R is the

resistance in ohms. How much current does an appliance use if P � 500 watts and R � 10 ohms? Round your answer to the nearest tenth. 7.1 amps

31. BUSINESS A company that produces DVDs uses the formula C � 88n�13

�� 330 to

calculate the cost C in dollars of producing n DVDs per day. What is the company’s costto produce 150 DVDs per day? Round your answer to the nearest dollar. $798

P�R

a�3b��

3ba

��3b�

2z � 2z�12

�

��z � 1

2z�12

�

�

z�12

�� 1

t�11

12�

�5

t�23

�

�

5t�12

�� t��

34

�

q�35

�

�

q�25

�

b�25

�

�b

y�12

�

�y

5�4

16�49

64�23

�

�

343�23

�

25�36

125�216

1�16

1�64

1�32

1�4

Practice (Average)

Rational Exponents

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7

Reading to Learn MathematicsRational Exponents

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7


Less

on

5-7

Pre-Activity How do rational exponents apply to astronomy?


The formula in the introduction contains the exponent . What do you think

it might mean to raise a number to the power?

Sample answer: Take the fifth root of the number and square it.

Reading the Lesson

1. Complete the following definitions of rational exponents.

• For any real number b and for any positive integer n, b�n1

�� except

when b and n is .

• For any nonzero real number b, and any integers m and n, with n ,

b�mn

�� , except when b and

n is .

2. Complete the conditions that must be met in order for an expression with rationalexponents to be simplified.

• It has no exponents.

• It has no exponents in the .

• It is not a fraction.

• The of any remaining is the number possible.

3. Margarita and Pierre were working together on their algebra homework. One exercise

asked them to evaluate the expression 27�43

�. Margarita thought that they should raise

27 to the fourth power first and then take the cube root of the result. Pierre thought thatthey should take the cube root of 27 first and then raise the result to the fourth power.Whose method is correct? Both methods are correct.


4. Some students have trouble remembering which part of the fraction in a rationalexponent gives the power and which part gives the root. How can your knowledge ofinteger exponents help you to keep this straight? Sample answer: An integer exponent can be written as a rational exponent. For example, 23 � 2

�31

�.

You know that this means that 2 is raised to the third power, so thenumerator must give the power, and, therefore, the denominator mustgive the root.

leastradicalindex

complex

denominatorfractional

negative

even

� 0( n�b�)mn�bm�

� 1

even� 0

n�b�

2�5

2�5


Lesser-Known Geometric FormulasMany geometric formulas involve radical expressions.

Make a drawing to illustrate each of the formulas given on this page.Then evaluate the formula for the given value of the variable. Round answers to the nearest hundredth.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7

1. The area of an isosceles triangle. Twosides have length a; the other side haslength c. Find A when a � 6 and c � 7.

A � �4c

��4a2 �� c2�A � 17.06

3. The area of a regular pentagon with aside of length a. Find A when a � 4.

A � �a4

2��25 ��10�5��

A � 27.53

5. The volume of a regular tetrahedronwith an edge of length a. Find V when a � 2.

V � �1a2

3��2�

V � 0.94

7. Heron’s Formula for the area of a triangle uses the semi-perimeter s,

where s � �a �

2b � c�. The sides of the

triangle have lengths a, b, and c. Find Awhen a � 3, b � 4, and c � 5.

A � �s(s ��a)(s �� b)(s �� c)�A � 6

2. The area of an equilateral triangle witha side of length a. Find A when a � 8.

A � �a4

2��3�

A � 27.71

4. The area of a regular hexagon with aside of length a. Find A when a � 9.

A � �32a2��3�

A � 210.44

6. The area of the curved surface of a rightcone with an altitude of h and radius ofbase r. Find S when r � 3 and h � 6.

S � �r�r2 � h�2�S � 63.22

8. The radius of a circle inscribed in a giventriangle also uses the semi-perimeter.Find r when a � 6, b � 7, and c � 9.

r �

r � 1.91

�s(s ��a)(s �� b)(s �� c)��s

Study Guide and InterventionRadical Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8


Less

on

5-8

Solve Radical Equations The following steps are used in solving equations that havevariables in the radicand. Some algebraic procedures may be needed before you use thesesteps.

Step 1 Isolate the radical on one side of the equation.Step 2 To eliminate the radical, raise each side of the equation to a power equal to the index of the radical.Step 3 Solve the resulting equation.Step 4 Check your solution in the original equation to make sure that you have not obtained any extraneous roots.

Solve 2�4x ��8� � 4 � 8.

2�4x � 8� � 4 � 8 Original equation

2�4x � 8� � 12 Add 4 to each side.

�4x � 8� � 6 Isolate the radical.

4x � 8 � 36 Square each side.

4x � 28 Subtract 8 from each side.

x � 7 Divide each side by 4.

Check2�4(7) �� 8� � 4 � 8

2�36� � 4 � 8

2(6) � 4 � 8

8 � 8The solution x � 7 checks.

Solve �3x ��1� � �5x� � 1.

�3x � 1� � �5x� � 1 Original equation

3x � 1 � 5x � 2�5x� � 1 Square each side.

2�5x� � 2x Simplify.

�5x� � x Isolate the radical.

5x � x2 Square each side.

x2 � 5x � 0 Subtract 5x from each side.

x(x � 5) � 0 Factor.

x � 0 or x � 5Check�3(0) �� 1� � 1, but �5(0)� � 1 � �1, so 0 isnot a solution.�3(5) �� 1� � 4, and �5(5)� � 1 � 4, so thesolution is x � 5.


ExercisesExercises

Solve each equation.

1. 3 � 2x�3� � 5 2. 2�3x � 4� � 1 � 15 3. 8 � �x � 1� � 2

15 no solution

4. �5 � x� � 4 � 6 5. 12 � �2x � 1� � 4 6. �12 ��x� � 0

�95 no solution 12

7. �21� � �5x � 4� � 0 8. 10 � �2x� � 5 9. �x2 � 7�x� � �7x � 9�

5 12.5 no solution

10. 43�2x � 1�1� � 2 � 10 11. 2�x � 11� � �x � 2� � �3x � 6� 12. �9x � 1�1� � x � 1

8 14 3, 4

�3��

3


Solve Radical Inequalities A radical inequality is an inequality that has a variablein a radicand. Use the following steps to solve radical inequalities.

Step 1 If the index of the root is even, identify the values of the variable for which the radicand is nonnegative.Step 2 Solve the inequality algebraically.Step 3 Test values to check your solution.

Solve 5 � �20x �� 4� �3.


Radical Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8

ExampleExample

Since the radicand of a square rootmust be greater than or equal tozero, first solve20x � 4 � 0.20x � 4 � 0

20x � �4

x � �1�5

Now solve 5 � �20x �� 4� � � 3.

5 � �20x �� 4� � �3 Original inequality

�20x �� 4� � 8 Isolate the radical.

20x � 4 � 64 Eliminate the radical by squaring each side.

20x � 60 Subtract 4 from each side.

x � 3 Divide each side by 20.

It appears that � � x � 3 is the solution. Test some values.

x � �1 x � 0 x � 4

�20(�1�) � 4� is not a real 5 � �20(0)�� 4� � 3, so the 5 � �20(4)�� 4� � �4.2, sonumber, so the inequality is inequality is satisfied. the inequality is notnot satisfied. satisfied

Therefore the solution � � x � 3 checks.

Solve each inequality.

1. �c � 2� � 4 � 7 2. 3�2x � 1� � 6 15 3. �10x �� 9� � 2 � 5

c 11 � x � 5 x � 4

4. 53�x � 2� � 8 2 5. 8 � �3x � 4� � 3 6. �2x � 8� � 4 � 2

x � 6 � � x � 7 x � 14

7. 9 � �6x � 3� � 6 8. � 4 9. 2�5x � 6� � 1 5

� � x � 1 x 8 � x � 3

10. �2x � 1�2� � 4 � 12 11. �2d ��1� � �d� � 5 12. 4�b � 3� � �b � 2� � 10

x 26 0 � d � 4 b 6

6�5

1�2

20��3x � 1�

4�3

1�2

1�5

1�5

ExercisesExercises

Skills PracticeRadical Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8


Less

on

5-8

Solve each equation or inequality.

1. �x� � 5 25 2. �x� � 3 � 7 16

3. 5�j� � 1 4. v�12

�� 1 � 0 no solution

5. 18 � 3y�12

�� 25 no solution 6.

3�2w� � 4 32

7. �b � 5� � 4 21 8. �3n ��1� � 5 8

9.3�3r � 6� � 3 11 10. 2 � �3p ��7� � 6 3

11. �k � 4� � 1 � 5 40 12. (2d � 3)�13

�� 2

13. (t � 3)�13

�� 2 11 14. 4 � (1 � 7u)

�13

�� 0 �9

15. �3z � 2� � �z � 4� no solution 16. �g � 1� � �2g ��7� 8

17. �x � 1� � 4�x � 1� no solution 18. 5 � �s � 3� � 6 3 � s � 4

19. �2 � �3x � 3� 7 �1 � x � 26 20. ��2a ��4� � �6 �2 � a � 16

21. 2�4r � 3� � 10 r � 7 22. 4 � �3x � 1� � 3 � � x � 0

23. �y � 4� � 3 � 3 y 32 24. �3�11r �� 3� � �15 � � r � 23�11

1�3

5�2

1�25


Solve each equation or inequality.

1. �x� � 8 64 2. 4 � �x� � 3 1

3. �2p� � 3 � 10 4. 4�3h� � 2 � 0

5. c�12

�� 6 � 9 9 6. 18 � 7h

�12

�� 12 no solution

7.3�d � 2� � 7 341 8.

5�w � 7� � 1 8

9. 6 �3�q � 4� � 9 31 10.

4�y � 9� � 4 � 0 no solution

11. �2m �� 6� � 16 � 0 131 12.3�4m �� 1� � 2 � 2

13. �8n ��5� � 1 � 2 14. �1 � 4�t� � 8 � �6 �

15. �2t � 5� � 3 � 3 16. (7v � 2)�14

�� 12 � 7 no solution

17. (3g � 1)�12

�� 6 � 4 33 18. (6u � 5)

�13

�� 2 � �3 �20

19. �2d ��5� � �d � 1� 4 20. �4r � 6� � �r� 2

21. �6x � 4� � �2x � 1�0� 22. �2x � 5� � �2x � 1� no solution

23. 3�a� � 12 a 16 24. �z � 5� � 4 � 13 �5 � z � 76

25. 8 � �2q� � 5 no solution 26. �2a ��3� 5 � a � 14

27. 9 � �c � 4� � 6 c 5 28.3�x � 1� �2 x � �7

29. STATISTICS Statisticians use the formula � � �v� to calculate a standard deviation �,where v is the variance of a data set. Find the variance when the standard deviation is 15. 225

30. GRAVITATION Helena drops a ball from 25 feet above a lake. The formula

t � �25 ��h� describes the time t in seconds that the ball is h feet above the water.

How many feet above the water will the ball be after 1 second? 9 ft

1�4

3�2

7�2

41�2

3�4

7�4

63�4

1�12

49�2

Practice (Average)

Radical Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8

Reading to Learn MathematicsRadical Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8


Less

on

5-8

Pre-Activity How do radical equations apply to manufacturing?


Explain how you would use the formula in your textbook to find the cost ofproducing 125,000 computer chips. (Describe the steps of the calculation in theorder in which you would perform them, but do not actually do the calculation.)

Sample answer: Raise 125,000 to the power by taking the

cube root of 125,000 and squaring the result (or raise 125,000

to the power by entering 125,000 ^ (2/3) on a calculator).

Multiply the number you get by 10 and then add 1500.

Reading the Lesson

1. a. What is an extraneous solution of a radical equation? Sample answer: a numberthat satisfies an equation obtained by raising both sides of the originalequation to a higher power but does not satisfy the original equation

b. Describe two ways you can check the proposed solutions of a radical equation in orderto determine whether any of them are extraneous solutions. Sample answer: Oneway is to check each proposed solution by substituting it into theoriginal equation. Another way is to use a graphing calculator to graphboth sides of the original equation. See where the graphs intersect.This can help you identify solutions that may be extraneous.

2. Complete the steps that should be followed in order to solve a radical inequality.

Step 1 If the of the root is , identify the values of

the variable for which the is .

Step 2 Solve the algebraically.

Step 3 Test to check your solution.


3. One way to remember something is to explain it to another person. Suppose that yourfriend Leora thinks that she does not need to check her solutions to radical equations bysubstitution because she knows she is very careful and seldom makes mistakes in herwork. How can you explain to her that she should nevertheless check every proposedsolution in the original equation? Sample answer: Squaring both sides of anequation can produce an equation that is not equivalent to the originalone. For example, the only solution of x � 5 is 5, but the squaredequation x2 � 25 has two solutions, 5 and �5.

values

inequality

nonnegativeradicand

evenindex

2�3

2�3


Truth TablesIn mathematics, the basic operations are addition, subtraction, multiplication,division, finding a root, and raising to a power. In logic, the basic operations are the following: not (), and (�), or (�), and implies (→).

If P and Q are statements, then P means not P; Q means not Q; P � Qmeans P and Q; P � Q means P or Q; and P → Q means P implies Q. The operations are defined by truth tables. On the left below is the truth table for the statement P. Notice that there are two possible conditions for P, true (T) or false (F). If P is true, P is false; if P is false, P is true. Also shown are the truth tables for P � Q, P � Q, and P → Q.

P P P Q P � Q P Q P � Q P Q P → QT F T T T T T T T T TF T T F F T F T T F F

F T F F T T F T TF F F F F F F F T

You can use this information to find out under what conditions a complex statement is true.

Under what conditions is P Q true?

Create the truth table for the statement. Use the information from the truth table above for P Q to complete the last column.

P Q P P � QT T F TT F F FF T T TF F T T

The truth table indicates that P � Q is true in all cases except where P is true and Q is false.

Use truth tables to determine the conditions under which each statement is true.

1. P � Q 2. P → (P → Q)

3. (P � Q) � (P � Q) 4. (P → Q) � (Q → P)

5. (P → Q) � (Q → P) 6. (P � Q) → (P � Q)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8

ExampleExample

To solve a quadratic equation that does not have real solutions, you can use the fact thati2 � �1 to find complex solutions.

Solve 2x2 � 24 � 0.

2x2 � 24 � 0 Original equation

2x2 � �24 Subtract 24 from each side.

x2 � �12 Divide each side by 2.

x � ��12� Take the square root of each side.

x � 2i�3� ��12� � �4� � ��1� � �3�

Simplify.

1. (�4 � 2i) � (6 � 3i) 2. (5 � i) � (3 � 2i) 3. (6 � 3i) � (4 � 2i)2 � i 2 � i 10 � 5i

4. (�11 � 4i) � (1 � 5i) 5. (8 � 4i) � (8 � 4i) 6. (5 � 2i) � (�6 � 3i)�12 � 9i 16 11 � 5i

7. (12 � 5i) � (4 � 3i) 8. (9 � 2i) � (�2 � 5i) 9. (15 � 12i) � (11 � 13i)8 � 8i 7 � 7i 26 � 25i

10. i4 11. i6 12. i15

1 �1 �i


13. 5x2 � 45 � 0 14. 4x2 � 24 � 0 15. �9x2 � 9�3i �i�6� �i

Study Guide and InterventionComplex Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9


Less

on

5-9

Add and Subtract Complex Numbers

A complex number is any number that can be written in the form a � bi, Complex Number where a and b are real numbers and i is the imaginary unit (i 2 � �1).

a is called the real part, and b is called the imaginary part.

Addition and Combine like terms.Subtraction of (a � bi) � (c � di) � (a � c) � (b � d )iComplex Numbers (a � bi) � (c � di) � (a � c) � (b � d )i

Simplify (6 � i) � (4 � 5i).

(6 � i) � (4 � 5i)� (6 � 4) � (1 � 5)i� 10 � 4i

Simplify (8 � 3i) � (6 � 2i).

(8 � 3i) � (6 � 2i)� (8 � 6) � [3 � (�2)]i� 2 � 5i


Example 3Example 3

ExercisesExercises


Multiply and Divide Complex Numbers

Multiplication of Complex NumbersUse the definition of i2 and the FOIL method:(a � bi)(c � di) � (ac � bd ) � (ad � bc)i

To divide by a complex number, first multiply the dividend and divisor by the complexconjugate of the divisor.

Complex Conjugatea � bi and a � bi are complex conjugates. The product of complex conjugates is always a real number.

Simplify (2 � 5i) (�4 � 2i).

(2 � 5i) � (�4 � 2i)� 2(�4) � 2(2i) � (�5i)(�4) � (�5i)(2i) FOIL

� �8 � 4i � 20i � 10i2 Multiply.

� �8 � 24i � 10(�1) Simplify.

� 2 � 24i Standard form

Simplify .

� � Use the complex conjugate of the divisor.

� Multiply.

� i2 � �1

� � i Standard form

Simplify.

1. (2 � i)(3 � i) 7 � i 2. (5 � 2i)(4 � i) 18 � 13i 3. (4 � 2i)(1 � 2i) �10i

4. (4 � 6i)(2 � 3i) 26 5. (2 � i)(5 � i) 11 � 3i 6. (5 � 3i)(�1 � i) �8 � 2i

7. (1 � i)(2 � 2i)(3 � 3i) 8. (4 � i)(3 � 2i)(2 � i) 9. (5 � 2i)(1 � i)(3 � i)12 � 12i 31 � 12i 16 � 18i

10. � i 11. � � i 12. � � 2i

13. 1 � i 14. � � 2i 15. � � i

16. � 17. �1 � 2i�2� 18. � 2i�6��

3�3��

3�6� � i�3��

�2� � i4 � i�2��

i�2�3i�5��

72�7

3 � i�5��3 � i�5�

31�41

8�41

3 � 4i�4 � 5i

1�2

�5 � 3i�2 � 2i

4 � 2i�3 � i

5�3

6 � 5i�3i

7�2

13�2

7 � 13i�2i

1�2

3�2

5�3 � i

11�13

3�13

3 � 11i�13

6 � 9i � 2i � 3i2��

4 � 9i2

2 � 3i�2 � 3i

3 � i�2 � 3i

3 � i�2 � 3i

3 � i�2 � 3i


Complex Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeComplex Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9


Less

on

5-9

Simplify.

1. ��36� 6i 2. ��196� 14i

3. ��81x6� 9|x3|i 4. ��23� � ��46� �23�2�

5. (3i)(�2i)(5i) 30i 6. i11 �i

7. i65 i 8. (7 � 8i) � (�12 � 4i) �5 � 12i

9. (�3 � 5i) � (18 � 7i) 15 � 2i 10. (10 � 4i) � (7 � 3i) 3 � 7i

11. (2 � i)(2 � 3i) 1 � 8i 12. (2 � i)(3 � 5i) 11 � 7i

13. (7 � 6i)(2 � 3i) �4 � 33i 14. (3 � 4i)(3 � 4i) 25

15. 16.


17. 3x2 � 3 � 0 �i 18. 5x2 � 125 � 0 �5i

19. 4x2 � 20 � 0 �i�5� 20. �x2 � 16 � 0 �4i

21. x2 � 18 � 0 �3i�2� 22. 8x2 � 96 � 0 �2i�3�

Find the values of m and n that make each equation true.

23. 20 � 12i � 5m � 4ni 4, �3 24. m � 16i � 3 � 2ni 3, 8

25. (4 � m) � 2ni � 9 � 14i 5, 7 26. (3 � n) � (7m � 14)i � 1 � 7i 3, 2

3 � 6i�

103i

�4 � 2i�6 � 8i�

38 � 6i�3i


Simplify.

1. ��49� 7i 2. 6��12� 12i�3� 3. ��121�s8� 11s4i

4. ��36a�3b4� 5. ��8� � ��32� 6. ��15� � ��25�6|a|b2i�a� �16 �5�15�

7. (�3i)(4i)(�5i) 8. (7i)2(6i) 9. i42

�60i �294i �1

10. i55 11. i89 12. (5 � 2i) � (�13 � 8i)�i i �8 � 10i

13. (7 � 6i) � (9 � 11i) 14. (�12 � 48i) � (15 � 21i) 15. (10 � 15i) � (48 � 30i)16 � 5i 3 � 69i �38 � 45i

16. (28 � 4i) � (10 � 30i) 17. (6 � 4i)(6 � 4i) 18. (8 � 11i)(8 � 11i)18 � 26i 52 �57 � 176i

19. (4 � 3i)(2 � 5i) 20. (7 � 2i)(9 � 6i) 21.

23 � 14i 75 � 24i

22. 23. 24. �1 � i


25. 5n2 � 35 � 0 �i�7� 26. 2m2 � 10 � 0 �i�5�

27. 4m2 � 76 � 0 �i�19� 28. �2m2 � 6 � 0 �i�3�

29. �5m2 � 65 � 0 �i�13� 30. x2 � 12 � 0 �4i

Find the values of m and n that make each equation true.

31. 15 � 28i � 3m � 4ni 5, �7 32. (6 � m) � 3ni � �12 � 27i 18, 9

33. (3m � 4) � (3 � n)i � 16 � 3i 4, 6 34. (7 � n) � (4m � 10)i � 3 � 6i 1, �4

35. ELECTRICITY The impedance in one part of a series circuit is 1 � 3j ohms and theimpedance in another part of the circuit is 7 � 5j ohms. Add these complex numbers tofind the total impedance in the circuit. 8 � 2 j ohms

36. ELECTRICITY Using the formula E � IZ, find the voltage E in a circuit when thecurrent I is 3 � j amps and the impedance Z is 3 � 2j ohms. 11 � 3j volts

3�4

2 � 4i�1 � 3i

7 � i�

53 � i�2 � i

14 � 16i��

1132

�7 � 8i

�5 � 6i�

26 � 5i�

�2i

Practice (Average)

Complex Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9

Reading to Learn MathematicsComplex Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9


Less

on

5-9

Pre-Activity How do complex numbers apply to polynomial equations?


Suppose the number i is defined such that i2 � �1. Complete each equation.

2i2 � (2i)2 � i4 �

Reading the Lesson

1. Complete each statement.

a. The form a � bi is called the of a complex number.

b. In the complex number 4 � 5i, the real part is and the imaginary part is .

This is an example of a complex number that is also a(n) number.

c. In the complex number 3, the real part is and the imaginary part is .

This is example of complex number that is also a(n) number.

d. In the complex number 7i, the real part is and the imaginary part is .

This is an example of a complex number that is also a(n) number.

2. Give the complex conjugate of each number.

a. 3 � 7i

b. 2 � i

3. Why are complex conjugates used in dividing complex numbers? The product ofcomplex conjugates is always a real number.

4. Explain how you would use complex conjugates to find (3 � 7i) (2 � i). Write thedivision in fraction form. Then multiply numerator and denominator by 2 � i .


5. How can you use what you know about simplifying an expression such as to

help you remember how to simplify fractions with imaginary numbers in thedenominator? Sample answer: In both cases, you can multiply thenumerator and denominator by the conjugate of the denominator.

1 � �3��2 � �5�

2 � i

3 � 7i

pure imaginary

70

real

03

imaginary

54

standard form

1�4�2


Conjugates and Absolute ValueWhen studying complex numbers, it is often convenient to represent a complex number by a single variable. For example, we might let z � x � yi. We denote the conjugate of z by z�. Thus, z� � x � yi.

We can define the absolute value of a complex number as follows.

z � x � yi � �x2 � y�2�

There are many important relationships involving conjugates and absolute values of complex numbers.

Show �z �2 � zz� for any complex number z.

Let z � x � yi. Then,z � (x � yi)(x � yi)

� x2 � y2

� �(x2 � y2�)2�� z 2

Show is the multiplicative inverse for any nonzero

complex number z.

We know z 2 � zz�. If z � 0, then we have z� � � 1.

Thus, is the multiplicative inverse of z.

For each of the following complex numbers, find the absolute value andmultiplicative inverse.

1. 2i 2. �4 � 3i 3. 12 � 5i

4. 5 � 12i 5. 1 � i 6. �3� � i

7. � i 8. � i 9. �12� � i�3�

�2�2��2

�2��2

�3��3

�3��3

z�� z 2

z�� z 2

z��z �2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9

Example 1Example 1

Example 2Example 2

Chapter 5 Test, Form 1

NAME DATE PERIOD

SCORE


Ass

essm

ent

55

1 �2 11

1 �4 17

�2 14

2

1 �6 19

1 �4 17

�2 12

�2

1 �2 �3

1 �4 �7

�2 �4

2

1 �8 1�9

1 �4 1�7

�2 �16

�2

Write the letter for the correct answer in the blank at the right of each question.

1. Simplify (3x0)2(2x4).A. x4 B. 12x4 C. 18x6 D. 18x4 1.

2. Simplify �135y2

yz5�. Assume that no variable equals 0.

A. �5zy3� B. �

y53z� C. 5y3z D. �

y57z� 2.

3. Scientists have determined that the speed at which light travels is approximately 300,000,000 meters per second. Express this speed in scientific notation.A. 3 � 107 m/s B. 3 � 108 m/sC. 3 � 10�8 m/s D. 30 � 10�7m/s 3.

4. Simplify (5m � 9) � (4m � 2).A. 9m � 11 B. m � 11 C. 9m � 7 D. 20m2 � 18 4.

5. Simplify 3x(2x2 � y).A. 5x3 � 3xy B. 12x � y C. 6x2 � 3y D. 6x3 � 3xy 5.

6. Simplify (x2 � 2x � 35) � (x � 5).A. x2 � x � 30 B. x � 7C. x � 5 D. x3 � 3x2 � 45x � 175 6.

7. Which represents the correct synthetic division of (x2 � 4x � 7) � (x � 2)?

A. B.

C. D. 7.

8. Factor m2 � 9m � 14 completely.A. m(m � 23) B. (m � 14)(m � 1)C. (m � 7)(m � 2) D. m(m � 9) � 14 8.

9. Simplify �t2t2

��7tt

��

610�. Assume that the denominator is not equal to 0.

A. �tt

��

35� B. �

tt

��

25� C. �

tt

��

35� D. �

tt

��

35� 9.


Chapter 5 Test, Form 1 (continued)

10. Simplify �121�.A. 11 B. �11 C. �11 D. �11� 10.

11. Use a calculator to approximate �224� to three decimal places.A. 15.0 B. 14.97 C. 14.966 D. 14.967 11.

12. Simplify �48�.A. 16�3� B. 4�3� C. 6 D. 4�6� 12.

13. Simplify (2 � �5�)(3 � �5�).A. 1 � �5� B. 1 � �5� C. �1 � �5� D. �1 � �5� 13.

14. Simplify �75� � �12�.A. 21 B. �87� C. 10�3� D. 7�3� 14.

15. Write the expression 5�17�

in radical form.A. �

751� B. 35 C. �

75� D. �

57� 15.

16. Simplify the expression m�25�

� m�15�.

A. m�53�

B. m�35�

C. m�225� D. m

�25�

16.

17. Solve �3x � 4� � 5.A. �7 B. 7 C. 21 D. �

235� 17.

18. Solve 2 � �5x � 1� 5.A. x 5 B. x �2 C. x 2 D. x 2 18.

19. Simplify (5 � 2i)(1 � 3i).A. 5 � 6i B. �1 C. �1 � 17i D. 11 � 17i 19.

20. ELECTRICITY The total impedance of a series circuit is the sum of the impedances of all parts of the circuit. A technician determined that the impedance of the first part of a particular circuit was 2 � 5j ohms. The impedance of the remaining part of the circuit was 3 � 2j ohms. What was the total impedance of the circuit?A. 5 � 3j ohms B. 5 � 7j ohmsC. �1 � 7j ohms D. 16 � 11j ohms 20.

Bonus Find the value of k so that x � 3 divides 2x3 � 11x2 � 19x � k with no remainder. B:

NAME DATE PERIOD

55

Chapter 5 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Simplify (3a0b2)(2a�3b2)2.

A. �1a2b

6

6� B. �

3a6b

6

8� C. 6b8 D. �

12ab6� 1.

2. Simplify �124aa2

4

bb�

2

5cc3�. Assume that no variable equals 0.

A. �a82

cb2

7� B. �

a32

cb2

3� C. �

a32

bc3

2� D. �

a32

cb2

7� 2.

3. Neptune is approximately 4.5 � 109 kilometers from the Sun. If light travels at approximately 3.0 � 105 kilometers per second, how long does it take light from the Sun to reach Neptune? Express your answer in scientific notation.A. 15 � 104 s B. 1.5 � 104 s C. 1.5 � 103 s D. 1500 s 3.

4. Simplify (3a3 � 7a2 � a) � (6a3 � 4a2 � 8).A. �3a6 � 3a4 � a � 8 B. �3a3 � 11a2 � a � 8C. �3a6 � 11a4 � a � 8 D. �3a3 � 3a2 � a � 8 4.

5. Simplify (7m � 8)2.A. 49m2 � 64 B. 49m2 � 64C. 49m2 � 112m � 64 D. 49m2 � 30m � 64 5.

6. Simplify (4x3 � 2x2 � 8x � 8) � (2x � 1).

A. 2x2 � 2x � 5 � �2x3� 1� B. 2x2 � 4 � �2x

9� 1�

C. 2x2 � 4 � �2x1�2

1� D. x2 � 4x � 6 � �2x1�4

1� 6.

7. Which represents the correct synthetic division of (2x3 � 5x � 40) � (x � 3)?

A. B.

C. D. 7.

8. Factor y3 � 64 completely.A. (y � 4)3 B. (y � 4)(y2 � 4y � 16)C. (y � 4)(y � 4)2 D. (y � 4)(y2 � 4y � 16) 8.

55

2 �5 40

�6 33

3�3

2 �5 40

�6 13

3

2 �11 73

�3

0 �5 40

6 18 39

2 �1 43

2

6 13 792

2 �5 �40

18 �39

�0

�6

2 13 �41�6


Chapter 5 Test, Form 2A (continued)

9. Simplify �xx2

2��

39xx

��


A. �xx

��

72� B. �

xx

��

42� C. �

xx

��

47� D. �

xx

��

42� 9.

10. Simplify �64n6w�4�.A. 8� n3 �w2 B. 8n3w2 C. �8n3w2 D. 32� n3 �w2 10.

11. Use a calculator to approximate �3

257� to three decimal places.A. 6.357 B. 4.004 C. 16.031 D. 6.358 11.

12. Simplify �3625x5�.

A. �25�3

x� B. 25x2 C. 5x�35x2� D. �5x�3

5x� 12.

13. Simplify �5� � �20� � �27� � �147�.A. 5�3� � 6 B. 3�5� � 4�3� C. 3�5� � 10�3� D. 2�5� � 3�3� 13.

14. Simplify �4 �6�2��.

A. �12 �76�2�� B. �

4 �2�2�� C. �

4 �3�2�� D. �

12 �73�2�� 14.

15. Write the radical �6y4� using rational exponents.

A. y�16�

B. y�32�

C. y�23�

D. y24 15.

16. Simplify the expression .

A. m�175� B. m

��12�

C. m�175�

D. m�38�

16.

17. A correct step in the solution of the equation (2m � 1)�14�

� 2 � 1 is _____.A. (2m � 1) � 16 � 1 B. 2m � 1 � 81

C. (2m � 1)�14�

� 1 D. 2m � 1 � 3�14�

17.

18. Solve �2x � 4� � 1 � 5.A. x � 0 B. x � �2 C. �2 � x � 6 D. x � 6 18.

19. Simplify (4 � 12i) � (�8 � 4i).A. 12 � 8 B. 28 C. 12 � 16i D. 12 � 16i 19.

20. Simplify �47��

23ii�.

A. �2191� � �

1239�

i B. �2191� � �

1249�

i C. �1239�

� �1279�

i D. �1279�

� �1239�

i 20.

Bonus Find the value of k so that (x3 � 2x2 � kx � 6) � (x � 2) has remainder 8. B:

m�23�

�

m�15�

NAME DATE PERIOD

55

Chapter 5 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Simplify (3x0y�4)(2x2y)3.

A. �24

yx6� B. �

21y69x6� C. 24x5 D. �

6yx6� 1.

2. Simplify �28x2

xy6

5

yzz�

3

4�. Assume that no variable equals 0.

A. �4yx

4

4z� B. �6xy4

4

z7� C. �4xy4

4

z7� D. �4y4

xz4� 2.

3. Saturn is approximately 1.4 � 109 kilometers from the Sun. If light travels at approximately 3.0 � 105 kilometers per second, how long does it take light from the Sun to reach Saturn? Express your answer in scientific notation.A. 4.7 � 104 s B. 4700 s C. 4.7 � 103 s D. 47 � 103 s 3.

4. Simplify (7x3 � 2x2 � 3) � (x2 � x � 5).A. 7x3 � 2x2 � x � 2 B. 7x3 � 3x2 � 2C. 8x5 � 3x3 � 2 D. 7x3 � x2 � x � 2 4.

5. Simplify (5x � 4)2.A. 25x2 � 16 B. 25x2 � 20x � 16C. 25x2 � 40x � 16 D. 25x2 � 18x � 16 5.

6. Simplify (6x3 � 16x2 � 11x � 5) � (3x � 2).

A. 6x2 � 12x � 3 � �3x9� 2� B. 2x2 � 4x � 1 � �3x

3� 2�

C. 2x2 � 4x � 1 � �3x1� 2� D. x2 � 8x � 3 � �3x

9� 2� 6.

7. Which represents the correct synthetic division of (3x3 � 2x � 5) � (x � 2)?

A. B.

C. D. 7.

8. Factor 27x3 � 1 completely.A. (3x � 1)(9x2 � 3x � 1) B. (3x � 1)(9x2 � 3x � 1)C. (3x � 1)3 D. (3x � 1)(9x2 � 3x � 1) 8.

55

3 �6 10 �15

3 �0 �2

�6 12

�25

�20

�2

3 �8 21

3 �2 15

�6 16

3 6 10 25

2

�2

3 �4 13

3 0 �2

6 12

15

20

3 �2 15

�6 18

2


Chapter 5 Test, Form 2B (continued)

9. Simplify �xx2

2��

38xx

��


A. �xx

��

65� B. �

xx

��

65� C. �

xx

��

35� D. �

xx

��

63� 9.

10. Simplify �25p4q�2�.A. 5� p2 �q B. 5p2q C. �5p2q D. 5p2� q � 10.


160� to three decimal places.A. 3.556 B. 12.649 C. 3.557 D. 5.429 11.

12. Simplify �3256t4�.

A. 4t�3

4t� B. 16t�3

t� C. �4t�3

4t� D. 4t�4t� 12.

13. Simplify �32� � �18� � �54� � �150�.

A. 7�2� � 2�6� B. 7�2� � 8�6� C. 3�2� � 3�6� D. �2� � 8�6� 13.

14. Simplify �2 �5�3��.

A. 10 � 5�3� B. 10 � 5�3� C. �10 � 5�3� D. �10 � 5�3� 14.

15. Write the radical �5m3� using rational exponents.

A. m2 B. m�53�

C. m�35�

D. m15 15.

16. Simplify the expression .

A. t�2 B. t�2101�

C. t�12

90� D. t

�230� 16.

17. A correct step in the solution of the equation (5z � 1)�13�

� 3 � 1 is _____.

A. 5z � 1 � 4�13�

B. (5z � 1) � 27 � 1C. (5z � 1) � 9 � 3 D. 5z � 1 � 64 17.

18. Solve �3x � 6� � 1 � 5.A. x � 0 B. �2 � x � 10 C. x � 10 D. x � �2 18.

19. Simplify (15 � 13i) � (�1 � 17i).A. 16 � 30i B. 16 � 4i C. 16 � 30i D. 46 19.


23ii�.

A. �87� � �

17�i B. �

87� � i C. �4 � 7i D. ��1

43�

� �173�

i 20.

Bonus Factor x2z2 � 36y2 � 4y2z2 � 9x2 completely. B:

t�34�

�

t�15�

NAME DATE PERIOD

55

Chapter 5 Test, Form 2C



1. (5r2t)2(3r0t�4) 1.

2. �182aa2

4

bb7cc5

�1� 2.

For Questions 3–12, simplify.

3. (4c2 � 12c � 7) � (c2 � 2c � 5) 3.

4. (9p2 � 7p) � (5p2 � 4p � 12) 4.

5. (3x � 4)(2x � 5) 5.

6. ��449�� 6.

7. �49x6y�4� 7.

8. �3 24a6b�5� 8.

9. 5�72� � �75� � �288� 9.

10. (5 � �6�)(4 � �6�) 10.

11. (7 � 12i) � (15 � 7i) 11.

12. (2 � 3i)(6 � i) 12.

13. Evaluate (8 � 10�4)(3.5 � 109). Express the result in 13.scientific notation.

14. Use long division to find (10y3 � 9y2 � 6y � 10) � (2y � 3). 14.

15. Use synthetic division to find (x3 � 4x2 � 17x � 50) � (x � 3). 15.

16. Factor 2xz � 3yz � 8x � 12y completely. If the polynomial 16.is not factorable, write prime.

NAME DATE PERIOD

SCORE 55

Ass

essm

ent


Chapter 5 Test, Form 2C (continued)

17. Simplify �x2 �x2

2�x

3�6

24�. Assume that the denominator 17.

is not equal to 0.

18. TREES The diameter of a tree d (in inches) is related to its 18.

basal area BA (in square feet) by the formula d � ��576

(B�A)��.

If the basal area of a tree is 12.4 square feet, what is the diameter of the tree? Use a calculator to approximate your answer to three decimal places.

19. Write the radical �5

32m3� using rational exponents. 19.

20. Simplify the expression . 20.

21. Solve �3

3m �� 1� � 4. 21.

22. Solve 4 � �5y � 1�0� � �1. 22.

23. POPULATION In 2000, the population of New York City 23.was approximately 8,000,000. Its total area is about 300 square miles. What was the population density (number of people per square mile) of New York City in 2000? Express your answer in scientific notation.

24. ELECTRICITY The total impedance of a series circuit is 24.the sum of the impedances of all parts of the circuit.Suppose that the first part of a circuit has an impedance of 6 � 5j ohms and that the total impedance of the circuit is 12 � 7j ohms. What is the impedance of the remainder of the circuit?

25. ELECTRICITY In an AC circuit, the voltage E (in volts), 25.current I (in amps), and impedance Z (in ohms) are related by the formula E � I � Z. Find the current in a circuit with voltage 10 � 3j volts and impedance 4 � j ohms.

Bonus Simplify �23�i

i� � �23�i

i�. B:

x�

x�12�

� x�13�

NAME DATE PERIOD

55

Chapter 5 Test, Form 2D



1. (2c2d0)3(5c�7d2) 1.

2. �4182aa6

2

bb3

4

cc�

5

3� 2.


3. (3f 2 � 5f � 9) � (4f 2 � 7f � 12) 3.

4. (6g3 � 2g � 1) � (3g2 � 5g � 7) 4.

5. (5m � 6)(2m � 1) 5.

6. ��295�� 6.

7. �4

16x4y�8� 7.

8. �3

�64a�6b7� 8.

9. 2�50� � �45� � �18� 9.

10. (2 � �3�)(4 � �3�) 10.

11. (7 � 6i) � (�3 � 2i) 11.

12. (4 � i)(1 � 7i) 12

13. Evaluate(6 � 10�4)(2.5 � 106). Express the result in 13.scientific notation.

14. Use long division to find (8x3 � 10x2 � 9x � 10) � (2x � 1). 14.


16. Factor 20x2 � 8x � 5xy � 2y completely. If the polynomial 16.is not factorable, write prime.

NAME DATE PERIOD

SCORE 55

Ass

essm

ent


Chapter 5 Test, Form 2D (continued)

17. Simplify �x2

x�2 �

x �25

20�. Assume that the denominator is 17.

not equal to 0.

18. TREES The diameter of a tree d (in inches) is related to its 18.

basal area BA (in square feet) by the formula d � ��576

(B�A)��.

If the basal area of a tree is 8.9 square feet, what is the diameter of the tree? Use a calculator to approximate your answer to three decimal places.

19. Write the radical �3�125�x2� using rational exponents. 19.


21. Solve �410s �� 1� � 3. 21.

22. Solve 2 � �3t � 6� 5. 22.

23. POPULATION In 2000, the population of Hong Kong 23.was approximately 6.8 million. Its total area is about 1000 square kilometers. What was the population density (number of people per square kilometer) of Hong Kong in 2000? Express your answer in scientific notation.

24. ELECTRICITY The total impedance of a series circuit is 24.the sum of the impedances of all parts of the circuit.Suppose that the first part of a circuit has an impedance of 7 � 4j ohms and that the total impedance of the circuit is 16 � 2j ohms. What is the impedance of the remainder of the circuit?

25. ELECTRICITY In an AC circuit, the voltage E (in volts), 25.current I (in amps), and impedance Z (in ohms) are related by the formula E � I � Z. Find the impedance in a circuit with voltage 12 � 2j volts and current 3 � 5j amps.

Bonus Simplify �34�i

i� � �34�i

i�. B:

x�85�

�

x � x�12�

NAME DATE PERIOD

55

Chapter 5 Test, Form 3



1. �(�2

4aa

2

2)�2

� 1.

2. �2x5�

(2

�y0

2(x5yx2y)2)2

� 2.


3. �12p2 � �65�r2 � �

43�pr� � (3pr � 2r2) 3.

4. (m � 2n)2 4.

5. �4x2 �� 20x �� 25� 5.

6. �3

�27x6�y3� 6.

7. �3

x5y7� 7.

8. 2�15� � �60� � 3�45� 8.

9. ��xx��

�93� 9.

10. (3 � 2i)(�1 � 4i) 10.

11. (5 � i) � (2 � 4i) � (3 � i) 11.

12. Evaluate �33..90

��

1100

�

�

4

1�. Express the result in scientific notation. 12.

13. Use long division to find �x4

x�

2x�

2

3�x

2�x

1� 7

�. 13.

14. Use synthetic division to find �2x3

x�

�x2

1� 1

�. 14.

NAME DATE PERIOD

SCORE 55

Ass

essm

ent


Chapter 5 Test, Form 3 (continued)

For Questions 15 and 16, factor completely. If the polynomial is not factorable, write prime.

15. 162w4 � 2n4 15.

16. x6 � 8y6 16.

17. Simplify . Assume that the 17.

denominator is not equal to 0.

18. GEOMETRY The volume V of a sphere and the length of 18.

its radius r are the related by the formula r � �3 �34V ��. Use the

formula to find radius of a sphere with volume 800 cubic meters. Approximate your answer to three decimal places.

19. Write the expression �4

16x9y�4� using rational exponents. 19.


21. Solve �x � 11� � 10 � 14. 21.

22. Solve �x � 2� 5 � �2x � 5�. 22.


ii��

5�5��. 23.

24. Solve �37�x2 � 5 � 0 24.

25. STATISTICS During fiscal year 1998, total New York state 25.expenditures were approximately $87.3 billion dollars. The population of New York in 1998 was approximately 18 million. Find New York’s 1998 per capita (per person) expenditures. Express your answer in scientific notation.

Bonus For a circuit with two impedances in parallel, the total impedance of the circuit Zt is given by the equation

Zt � �ZZ

1

1�

ZZ2

2�, where Z1 and Z2 are the parallel

impedances. Find the total impedance of a circuit with Z1 � 3 � 4j ohms and Z2 � 6 � 2j ohms. B:

3�12�

� 1�

2 � 3�12�

m � 1��(m2 � 4m � 5)(m � 5)�2

NAME DATE PERIOD

55

Chapter 5 Open-Ended Assessment


Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solution in more thanone way or investigate beyond the requirements of the problem.

1. Jorge works for the A-Glide Sled Company. This company estimates itsmonthly profit for the sale of x sleds, in hundreds of dollars, is given bythe expression �3x � 1�9�. Tia works for a competing sled manufacturer,SnowFun. Tia’s company estimates that its monthly profit for the sale ofx sleds, in hundreds of dollars, is given by the expression 3 � �2x�.Mark has been offered a job at both companies and decides he will workfor the company that has the greatest monthly profit. Before he makeshis decision, however, he asks Jorge and Tia the average number of sledssold each month by each of their companies.a. Why is the number of sleds sold important to Mark?b. Assume both companies make the same number of sleds in a certain

month. Determine the number of sleds that would make Mark wantto work for SnowFun, and give the profit, to the nearest dollar,earned by each company during that month.

c. After talking to Jorge and Tia, Mark decided to work for A-Glide.Assume that both companies average the same number of sleds soldper month. Write and solve an inequality to determine the possibleresponses Mark might have heard from Jorge and Tia. What doesthis tell you about the number of sleds sold each month?

2. You are given an unlimited number of tiles with the given dimensions.length: x units length: x units length: 1 unit width: x units width: 1 unit width: 1 unit area: x2 units2 area: x units2 area: 1 unit2

The polynomial 2x2 � 3x � 1 can be represented by the figure at the right.

These tiles can be arranged to form the rectangle shown.Notice that the area of the rectangle is 2x2 � 3x � 1 units2.a. Find the length and width of the rectangle.b. Explain how to find the perimeter of the rectangle.

Then find the perimeter.c. Select a value for x and substitute that value into each of the

expressions above. For your value of x, state the length, width,perimeter, and area of the rectangle. Discuss any restrictions on yourchoice of x.

d. Factor the polynomial 2x2 � 3x � 1.e. Compare your answers to parts a and d.f. Draw a tile model for a different polynomial. Then write your

polynomial and its factors. Explain how your model relates to thefactors of your polynomial.

x

x 1

x 2

x

x 2

NAME DATE PERIOD

SCORE 55

Ass

essm

ent

x x x1

x 2x 2


Chapter 5 Vocabulary Test/Review

Underline or circle the correct word or phrase to complete each sentence.

1. A polynomial with two terms is called a (monomial, binomial, trinomial).

2. 4 � 5i is a(n) (pure imaginary number, imaginary unit, complex number).

3. (Scientific notation, Synthetic division, Absolute value) is used to write very large and very small numbers without having to write many zeros.

4. If you square both sides of a radical equation and obtain a solution that does not satisfy the original equation, you have found a(n) (coefficient, complex number, extraneous solution).

5. �3x � 5� 0 and �2x � 1� � 0 are (radical equations, radical inequalities,like radical expressions).

6. The expressions 7 � �5� and 7 � �5� are (complex conjugates, conjugates,like terms).

7. The monomials that make up a polynomial are called (conjugates, terms,principal roots).

8. A monomial that contains no variables is called a (power, constant, degree).

9. The expression 106 is a (trinomial, nth root, power).

10. One of the steps that may be necessary to simplify a radical expression is (synthetic division, scientific notation, rationalizing the denominator).

In your own words–Define each term.

11. square root

12. pure imaginary number

absolute valuebinomialcoefficientcomplex conjugatescomplex numberconjugatesconstantdegree

dimensional analysisextraneous solutionFOIL methodimaginary unitlike radical expressionslike termsmonomialnth root

polynomialpowerprincipal rootpure imaginary numberradical equationradical inequalityrationalizing the

denominator

scientific notationsimplifysquare rootsynthetic divisiontermstrinomial

NAME DATE PERIOD

SCORE 55

Chapter 5 Quiz (Lessons 5–1 through 5–3)

55


For Questions 1 and 2, simplify. Assume that no variable equals 0.

1. (4n5y2)(�6n2y�5) 2. �126((xxy3

0y))4

2� 2.

3. Express 0.00000068 in scientific notation. 3.

4. Evaluate (3.8 � 10�2)(4 � 105). Express the result in 4.scientific notation.

Simplify.

5. (3p � 5q) � (6p � 4q) 5.

6. (2x � 3) � (5x � 6) 6.

7. (4x � 5)(2x � 7) 7.

8. Standardized Test Practice Which expression is equal to(30a2 � 11a � 15)(5a � 6)�1?

A. 6a � 5 � �5a4�5

6� B. 6a � 5

C. 6a � 5 � �5a4�5

6� D. �6a � 5 � �5a4�5

6� 8.

Simplify.

9. (m2 � m � 6) � (m � 4) 9.

10. (a3 � 6a2 � 10a � 3) � (a � 3) 10.

NAME DATE PERIOD

SCORE

Chapter 5 Quiz (Lessons 5–4 and 5–5)

For Questions 1 and 2, factor completely. If the polynomial is not factorable, write prime.

1. 2c2 � 98 1.

2. 6a2 � 3a � 18 2.

3. Simplify �x2 �

x27�x

1�6

12�. Assume that the denominator is not 3.

equal to 0.

4. Simplify �3

�27w�9y6�. 4.


�56� to three decimal 5.places.

NAME DATE PERIOD

SCORE 55

Ass

essm

ent

1.



1. ��25x�� 2. �18m5n�6�

3. 4�12� � �18� � �108� � 7�72�

4. (�5� � �7�)2

5. (7 � �5�)(3 � 2�5�) 6. �24

��

��

6�6��

7. Write the expression x�58�

in radical form.


32z3� using rational exponents.

9. Evaluate 16��32�. 10. Simplify 6t�

23�

� t�43�.


Solve each equation

1. �5y � 3� � �7y � 9�

2. �3

2v � 7� � �2

For Questions 3 and 4, solve each inequality.

3. 2 � �5x � 1� � 4

4. �2x � 5� � 1 4

5. Solve 5x2 � 100 � 0.

Simplify.

6. ��80� 7. ��6� � ��12�

8. (6 � 9i) � (17 � 12i) 9. (7 � 3i)(8 � 4i)

10. �23

��

ii�

NAME DATE PERIOD

SCORE


55

NAME DATE PERIOD

SCORE

55

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Chapter 5 Mid-Chapter Test (Lessons 5–1 through 5–5)



Simplify.

1. (5x3y)2(�2x5y�1)

A. �50x10y B. ��5

y0x11� C. �50x11y D. �10x8y 1.

2. �31x2�

x2

yy2

4

zz3

0�

A. �4xy3

2

z3� B. �4yx

2

z3� C. �9xy2

2

z3� D. �x43

zy2

2� 2.

3. (x2 � 2x � 5) � (3x2 � 4x � 7)A. 2x2 � 2x � 12 B. �2x2 � 6x � 12 C. 4x2 � 2x � 2 D. 4x2 � 6x � 2 3.

4. (s � 3)(s � 4)A. s2 � 7s � 12 B. s � 1 C. s2 � 7s � 12 D. s2 � s � 12 4.

5. �x2x2

��141x

x��

350� (Assume that the denominator is not equal to 0.)

A. �xx

��

16� B. �1 C. �

xx

��

16� D. �

xx

��

16� 5.

6. �3

216x9�A. 6x6 B. 6� x3 � C. �6x3 D. 6x3 6.

7. �4x2y2z�4�A. 2xyz2 B. 2� xy �z2 C. �2xyz2 D. 2x2y2z4 7.

8. Use long division to find (2x3 � 7x2 � 7x � 2) � (x � 2). 8.



287� to three decimal 10.places.

11. Some computer chips can perform a floating point 11.operation in less than 0.00000000125 second. Express this value in scientific notation.

12. Evaluate (4 � 106)(2.8 � 10�3). Express your answer in 12.scientific notation.

13. Factor ax2 � 4a � 5x2 � 20 completely. If the polynomial is 13.not factorable, write prime.

14. Simplify (5x4 � 22x2 � 7x � 6) � (x � 2). 14.

Part II

Part I

NAME DATE PERIOD

SCORE 55

Ass

essm

ent

NAME DATE PERIOD

SCORE


Chapter 5 Cumulative Review (Chapters 1–5)

1. Write an algebraic expression to represent the verbal 1.expression the square of the sum of a number and three.(Lesson 1-3)

2. If f(x) � x2 � 3x, find f(2 � a). (Lesson 2-1) 2.

3. Write an equation in slope-intercept form of the line 3.through (1, 3) and (–3, 7). (Lesson 2-4)

4. Solve the system of equations 2x � 5y � 16 4.4x � 3y � 6 by using elimination. (Lesson 3-2)

5. STORES The floor area of a furniture storeroom is 5.500 square yards. One sofa requires 3 square yards and one dining table requires 4 square yards of space. The room can hold a maximum of 150 pieces of furniture. Let s represent the number of sofas and t represent the number of tables.Write a system of inequalities to represent the number of pieces of furniture that can be placed in the storeroom.(Lesson 3-4)

6. State the dimensions of matrix A if A � � . (Lesson 4-1) 6.

7. Find the product � � � , if possible. (Lesson 4-3) 7.

8. Write a matrix equation for the system of equations 8.3m � 2n � 16 4m � 5n � 9. (Lesson 4-8)

9. Evaluate �12.6.4

��

1100�

9

2�. Express the result in scientific notation. 9.

(Lesson 5-1)

10. Use long division to find (6x3 � x2 � x) � (2x � 1). (Lesson 5-3) 10.

11. Simplify �49x2y�4�. (Lesson 5-5) 11.


25z6� using rational exponents. (Lesson 5-7) 12.


24ii�. (Lesson 5-8) 13.

15

�2

3 6 40 �5 2

0 310 70 �4

NAME DATE PERIOD

55

Standardized Test Practice (Chapters 1–5)


1. In the figure, circle P represents all prime numbers, circle Q represents all numbers whose square roots are not integers, and circle R represents all multiples of 4.In which region does 24 belong?A. a B. bC. c D. d 1.

2. Find the reciprocal of �3x� � �

25�.

E. �2x

5�x

15� F. �

56x� G. �2x

5�x

15� H. x 2.

3. Suppose a set of data contains just two data items. If the median is w, the mean is x, and the mode is y, which of the following must be equal?A. w, x, and y B. x and y C. w and x D. w and y 3.

4. What is the value of cd in the equation 32cd � 11cd � 42?

E. ��12� F. �

12� G. 2 H. –2 4.

5. Hoshiko owns one-fifth of a business. She sells her share for $15,000. What is the total value of the business?A. $3000 B. $75,000 C. $100,000 D. $150,000 5.

6. If r is an odd integer and m � 8r, then �m2� will always be _____.

E. odd F. even G. positive H. negative 6.

7. 13% of 160 is 16% of _____.A. 13 B. 130 C. 1300 D. 13,000 7.

8. If m2 � 3, then what is the value of 5m6?E. 15 F. 30 G. 45 H. 135 8.

9. Which is the equation of a line that passes through a point with coordinates (7, –1) and is perpendicular to the graph of y � 2x � 1?

A. y � 2x � 15 B. y � �2x � 13 C. y � �12�x � 2�

12� D. y � �

12�x � 4�

12� 9.

10. If mn � 16 and m2 � n2 � 68, then (m � n)2 � _____.E. 68 F. 84G. 100 H. cannot be determined 10. HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

55

Ass

essm

ent

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

P Q

R

a

cb d


Standardized Test Practice (continued)

11. What is the value of � a � b � � � b � a � if 11. 12.

a � b � �13�?

12. The volume of a cube with a surface area of 384 in.2 is _____ in3.

13. The circle is divided into eight 13. 14.sectors of equal area. In two consecutive spins, what is the probability of spinning a “50”and then spinning an odd-numbered region?

14. For all positive integers m, m � 2m2 � 1.What is the value of x if x � 45,001?

Column A Column B

15. 15.

16. x 0 16.

17. x � k 17.

18. 2x�1 � 128 18.

7x

DCBA

2(x � k)�2(x � k)

DCBA

80% of x�180x�

DCBA

702y

DCBA

y˚

3y˚

(y � 30)˚

Part 3: Quantitative Comparison

Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

NAME DATE PERIOD

55

NAME DATE PERIOD

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

45 5

30 15

50 35

25 40

A

D

C

B

Standardized Test PracticeStudent Record Sheet (Use with pages 282–283 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

NAME DATE PERIOD

55

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5 8

3 6 9

Solve the problem and write your answer in the blank.

Also enter your answer by writing each number or symbol in a box. Then fill inthe corresponding oval for that number or symbol.

10 12 14 16

11 13 15 17

Select the best answer from the choices given and fill in the corresponding oval.

18 20 22

19 21 DCBADCBA

DCBADCBADCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

DCBADCBADCBA

DCBADCBADCBA

DCBADCBADCBA

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison


Answers (Lesson 5-1)

Stu

dy G

uid

e a

nd I

nte

rven

tion

Mo

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-1

5-1

©G

lenc

oe/M

cGra

w-H

ill23

9G

lenc

oe A

lgeb

ra 2

Lesson 5-1

Mo

no

mia

lsA

mon

omia

lis

a n

um

ber,

a va

riab

le,o

r th

e pr

odu

ct o

f a

nu

mbe

r an

d on

e or

mor

e va

riab

les.

Con

stan

ts a

re m

onom

ials

th

at c

onta

in n

o va

riab

les.

Neg

ativ

e E

xpo

nen

ta�

n�

and

�an

for

any

real

num

ber

a�

0 an

d an

y in

tege

r n.

Wh

en y

ou s

imp

lify

an

exp

ress

ion

,you

rew

rite

it

wit

hou

t pa

ren

thes

es o

r n

egat

ive

expo

nen

ts.T

he

foll

owin

g pr

oper

ties

are

use

ful

wh

en s

impl

ifyi

ng

expr

essi

ons.

Pro

du

ct o

f P

ow

ers

am�

an�

am�

nfo

r an

y re

al n

umbe

r a

and

inte

gers

man

d n.

Qu

oti

ent

of

Po

wer

s�

am�

nfo

r an

y re

al n

umbe

r a

�0

and

inte

gers

man

d n.

For

a,

bre

al n

umbe

rs a

nd m

, n

inte

gers

:(a

m)n

�am

n

Pro

per

ties

of

Po

wer

s(a

b)m

�am

bm

��n

�,

b�

0

��

n�

��n

or

, a

�0,

b�

0

Sim

pli

fy.A

ssu

me

that

no

vari

able

eq

ual

s 0.

bn� an

b � aa � b

an� bn

a � bam� an

1� a�

n1 � an

Exam

ple

Exam

ple

a.(3

m4 n

�2 )

(�5m

n)2

(3m

4 n�

2 )(�

5mn

)2�

3m4 n

�2

�25

m2 n

2

�75

m4 m

2 n�

2 n2

�75

m4

�2 n

�2

�2

�75

m6

b.

� ��

m12

�4m

4

��

4m16

�m

12� � 4m

14

�

(�m

4 )3

� (2m

2 )�

2

(�m

4 )3

��

(2m

2 )�

2

Exer

cises

Exer

cises

Sim

pli

fy.A

ssu

me

that

no

vari

able

eq

ual

s 0.

1.c1

2�

c�4

�c6

c14

2.b

63.

(a4 )

5a

20

4.5.

��

16.

��2

7.(�

5a2 b

3 )2 (

abc)

25a

6 b8 c

28.

m7

�m

8m

159.

10.

211

.4j(

2j�

2 k2 )

(3j3

k�7 )

12.

m2

3 � 22m

n2 (

3m2 n

)2�

�12

m3 n

424

j2�

k5

23 c4 t

2� 22 c

4 t2

2m2

�n

8m3 n

2� 4m

n3

1 � 5

x2� y

4x2 y� xy

3b � a

5a2 b� a�

3 b2

y2

� x6

x�2y

� x4 y�

1

b8� b2

©G

lenc

oe/M

cGra

w-H

ill24

0G

lenc

oe A

lgeb

ra 2

Scie

nti

fic

No

tati

on

Sci

enti

fic

no

tati

on

Anu

mbe

r ex

pres

sed

in t

he f

orm

a�

10n ,

whe

re 1

�a

10

and

nis

an

inte

ger

Exp

ress

46,

000,

000

in s

cien

tifi

c n

otat

ion

.

46,0

00,0

00 �

4.6

�10

,000

,000

1 �

4.6

10

�4.

6 �

107

Writ

e 10

,000

,000

as

a po

wer

of

ten.

Eva

luat

e .E

xpre

ss t

he

resu

lt i

n s

cien

tifi

c n

otat

ion

.

��

�0.

7 �

106

�7

�10

5

Exp

ress

eac

h n

um

ber

in

sci

enti

fic

not

atio

n.

1.24

,300

2.0.

0009

93.

4,86

0,00

02.

43 �

104

9.9

�10

�4

4.86

�10

6

4.52

5,00

0,00

05.

0.00

0003

86.

221,

000

5.25

�10

83.

8 �

10�

62.

21 �

105

7.0.

0000

0006

48.

16,7

509.

0.00

0369

6.4

�10

�8

1.67

5 �

104

3.69

�10

�4

Eva

luat

e.E

xpre

ss t

he

resu

lt i

n s

cien

tifi

c n

otat

ion

.

10.(

3.6

�10

4 )(5

�10

3 )11

.(1.

4 �

10�

8 )(8

�10

12)

12.(

4.2

�10

�3 )

(3 �

10�

2 )1.

8 �

108

1.12

�10

51.

26 �

10�

4

13.

14.

15.

2.5

�10

99

�10

�8

7.4

�10

3

16.(

3.2

�10

�3 )

217

.(4.

5 �

107 )

218

.(6.

8 �

10�

5 )2

1.02

4 �

10�

52.

025

�10

154.

624

�10

�9

19.A

STR

ON

OM

YP

luto

is

3,67

4.5

mil

lion

mil

es f

rom

th

e su

n.W

rite

th

is n

um

ber

insc

ien

tifi

c n

otat

ion

.So

urce

:New

Yor

k Tim

es A

lman

ac3.

6745

�10

9m

iles

20.C

HEM

ISTR

YT

he

boil

ing

poin

t of

th

e m

etal

tu

ngs

ten

is

10,2

20°F

.Wri

te t

his

tem

pera

ture

in

sci

enti

fic

not

atio

n.

Sour

ce:N

ew Y

ork

Times

Alm

anac

1.02

2 �

104

21.B

IOLO

GY

The

hum

an b

ody

cont

ains

0.0

004%

iodi

ne b

y w

eigh

t.H

ow m

any

poun

ds o

fio

dine

are

the

re i

n a

120

-pou

nd

teen

ager

? E

xpre

ss y

our

answ

er i

n s

cien

tifi

c n

otat

ion

.So

urce

:Uni

versa

l Alm

anac

4.8

�10

�4

lb

4.81

�10

8�

�6.

5 �

104

1.62

�10

�2

��

1.8

�10

59.

5 �

107

��

3.8

�10

�2

104

� 10�

23.

5�5

3.5

�10

4�

�5

�10

�2

3.5

�10

4�

�5

�10

�2

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Mo

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-1

5-1

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Mo

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-1

5-1

©G

lenc

oe/M

cGra

w-H

ill24

1G

lenc

oe A

lgeb

ra 2

Lesson 5-1

Sim

pli

fy.A

ssu

me

that

no

vari

able

eq

ual

s 0.

1.b4

�b3

b7

2.c5

�c2

�c2

c9

3.a�

4�

a�3

4.x5

�x�

4�

xx

2

5.(g

4 )2

g8

6.(3

u)3

27u

3

7.(�

x)4

x4

8.�

5(2z

)3�

40z3

9.�

(�3d

)4�

81d

410

.(�

2t2 )

3�

8t6

11.(

�r7

)3�

r21

12.

s3

13.

14.(

�3f

3 g)3

�27

f9g

3

15.(

2x)2

(4y)

264

x2 y

216

.�2g

h(g

3 h5 )

�2g

4 h6

17.1

0x2 y

3 (10

xy8 )

100x

3 y11

18.

19.

�20

.

Exp

ress

eac

h n

um

ber

in

sci

enti

fic

not

atio

n.

21.5

3,00

05.

3 �

104

22.0

.000

248

2.48

�10

�4

23.4

10,1

00,0

004.

101

�10

824

.0.0

0000

805

8.05

�10

�6

Eva

luat

e.E

xpre

ss t

he

resu

lt i

n s

cien

tifi

c n

otat

ion

.

25.(

4 �

103 )

(1.6

�10

�6 )

6.4

�10

�3

26.

6.4

�10

109.

6 �

107

��

1.5

�10

�32q

2� p

2�

10pq

4 r�

��

5p3 q

2 rc7

� 6a3 b

�6a

4 bc8

��

36a7 b

2 c

8z2

� w2

24w

z7� 3w

3 z5

1 � kk9� k10

s15� s12

1 � a7

©G

lenc

oe/M

cGra

w-H

ill24

2G

lenc

oe A

lgeb

ra 2

Sim

pli

fy.A

ssu

me

that

no

vari

able

eq

ual

s 0.

1.n

5�

n2

n7

2.y7

�y3

�y2

y12

3.t9

�t�

8t

4.x�

4�

x�4

�x4

5.(2

f4)6

64f2

46.

(�2b

�2 c

3 )3

�

7.(4

d2 t

5 v�

4 )(�

5dt�

3 v�

1 )�

8.8u

(2z)

364

uz3

9.�

10.

�

11.

12. �

�2

13.�

(4w

�3 z

�5 )

(8w

)2�

14.(

m4 n

6 )4 (

m3 n

2 p5 )

6m

34n

36p

30

15. �

d2 f

4 �4 ��d

5 f�3

�12

d23

f19

16. �

��2

17.

18.

�

Exp

ress

eac

h n

um

ber

in

sci

enti

fic

not

atio

n.

19.8

96,0

0020

.0.0

0005

621

.433

.7 �

108

8.96

�10

55.

6 �

10�

54.

337

�10

10

Eva

luat

e.E

xpre

ss t

he

resu

lt i

n s

cien

tifi

c n

otat

ion

.

22.(

4.8

�10

2 )(6

.9 �

104 )

23.(

3.7

�10

9 )(8

.7 �

102 )

24.

3.31

2 �

107

3.21

9 �

1012

3 �

10�

5

25.C

OM

PUTI

NG

Th

e te

rm b

it,s

hor

t fo

r bi

nar

y d

igit

,was

fir

st u

sed

in 1

946

by J

ohn

Tu

key.

A s

ingl

e bi

t h

olds

a z

ero

or a

on

e.S

ome

com

pute

rs u

se 3

2-bi

t n

um

bers

,or

stri

ngs

of

32 c

onse

cuti

ve b

its,

to i

den

tify

eac

h a

ddre

ss i

n t

hei

r m

emor

ies.

Eac

h 3

2-bi

t n

um

ber

corr

espo

nds

to

a n

um

ber

in o

ur

base

-ten

sys

tem

.Th

e la

rges

t 32

-bit

nu

mbe

r is

nea

rly

4,29

5,00

0,00

0.W

rite

th

is n

um

ber

in s

cien

tifi

c n

otat

ion

.4.

295

�10

9

26.L

IGH

TW

hen

lig

ht

pass

es t

hro

ugh

wat

er,i

ts v

eloc

ity

is r

edu

ced

by 2

5%.I

f th

e sp

eed

ofli

ght

in a

vac

uu

m i

s 1.

86 �

105

mil

es p

er s

econ

d,at

wh

at v

eloc

ity

does

it

trav

el t

hro

ugh

wat

er?

Wri

te y

our

answ

er i

n s

cien

tifi

c n

otat

ion

.1.

395

�10

5m

i/s

27.T

REE

SD

ecid

uou

s an

d co

nif

erou

s tr

ees

are

har

d to

dis

tin

guis

h i

n a

bla

ck-a

nd-

wh

ite

phot

o.B

ut

beca

use

dec

idu

ous

tree

s re

flec

t in

frar

ed e

ner

gy b

ette

r th

an c

onif

erou

s tr

ees,

the

two

type

s of

tre

es a

re m

ore

dist

ingu

ish

able

in

an

in

frar

ed p

hot

o.If

an

in

frar

edw

avel

engt

h m

easu

res

abou

t 8

�10

�7

met

ers

and

a bl

ue

wav

elen

gth

mea

sure

s ab

out

4.5

�10

�7

met

ers,

abou

t h

ow m

any

tim

es l

onge

r is

th

e in

frar

ed w

avel

engt

h t

han

th

ebl

ue

wav

elen

gth

? ab

ou

t 1.

8 ti

mes

2.7

�10

6�

�9

�10

10

4v2

� m2

�20

(m2 v

)(�

v)3

��

5(�

v)2 (

�m

4 )15

x11�

y3

(3x�

2 y3 )

(5xy

�8 )

��

(x�

3 )4 y

�2

y6

� 4x2

2x3 y

2� �

x2 y5

4 � 33 � 2

256

� wz5

4� 9r

4 s6 z

122

� 3r2 s

3 z6

27x

6�

16�

27x3 (

�x7 )

��

16x4

s4� 3x

4�

6s5 x

3� 18

sx7

4m7 y

2�

312

m8y6

� �9m

y4

20d

3 t2

�v

5

8c9

� b6

1 � x4

Pra

ctic

e (

Ave

rag

e)

Mo

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-1

5-1



Readin

g t

o L

earn

Math

em

ati

csM

on

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-1

5-1

©G

lenc

oe/M

cGra

w-H

ill24

3G

lenc

oe A

lgeb

ra 2

Lesson 5-1

Pre-

Act

ivit

yW

hy

is s

cien

tifi

c n

otat

ion

use

ful

in e

con

omic

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

5-1

at

the

top

of p

age

222

in y

our

text

book

.

You

r te

xtbo

ok g

ives

th

e U

.S.p

ubl

ic d

ebt

as a

n e

xam

ple

from

eco

nom

ics

that

invo

lves

lar

ge n

um

bers

th

at a

re d

iffi

cult

to

wor

k w

ith

wh

en w

ritt

en i

nst

anda

rd n

otat

ion

.Giv

e an

exa

mpl

e fr

om s

cien

ce t

hat

in

volv

es v

ery

larg

en

um

bers

an

d on

e th

at i

nvo

lves

ver

y sm

all

nu

mbe

rs.

Sam

ple

an

swer

:d

ista

nce

s b

etw

een

Ear

th a

nd

th

e st

ars,

size

s o

f m

ole

cule

s an

d a

tom

s

Rea

din

g t

he

Less

on

1.T

ell

wh

eth

er e

ach

exp

ress

ion

is

a m

onom

ial

or n

ot a

mon

omia

l.If

it

is a

mon

omia

l,te

llw

het

her

it

is a

con

stan

t or

not

a c

onst

ant.

a.3x

2m

on

om

ial;

no

t a

con

stan

tb

.y2

�5y

�6

no

t a

mo

no

mia

l

c.�

73m

on

om

ial;

con

stan

td

.n

ot

a m

on

om

ial

2.C

ompl

ete

the

foll

owin

g de

fin

itio

ns

of a

neg

ativ

e ex

pon

ent

and

a ze

ro e

xpon

ent.

For

an

y re

al n

um

ber

a�

0 an

d an

y in

tege

r n

,a�

n�

.

For

an

y re

al n

um

ber

a�

0,a0

�.

3.N

ame

the

prop

erty

or

prop

erti

es o

f ex

pon

ents

th

at y

ou w

ould

use

to

sim

plif

y ea

chex

pres

sion

.(D

o n

ot a

ctu

ally

sim

plif

y.)

a.q

uo

tien

t o

f p

ow

ers

b.

y6�

y9p

rod

uct

of

po

wer

s

c.(3

r2s)

4p

ow

er o

f a

pro

du

ct a

nd

po

wer

of

a p

ow

er

Hel

pin

g Y

ou

Rem

emb

er

4.W

hen

wri

tin

g a

nu

mbe

r in

sci

enti

fic

not

atio

n,s

ome

stu

den

ts h

ave

trou

ble

rem

embe

rin

gw

hen

to

use

pos

itiv

e ex

pon

ents

an

d w

hen

to

use

neg

ativ

e on

es.W

hat

is

an e

asy

way

to

rem

embe

r th

is?

Sam

ple

an

swer

:U

se a

po

siti

ve e

xpo

nen

t if

th

e n

um

ber

is10

or

gre

ater

.Use

a n

egat

ive

nu

mb

er if

th

e n

um

ber

is le

ss t

han

1.

m8

� m3

1

� a1 n�

1 � z

©G

lenc

oe/M

cGra

w-H

ill24

4G

lenc

oe A

lgeb

ra 2

Pro

per

ties

of

Exp

on

ents

Th

e ru

les

abou

t po

wer

s an

d ex

pon

ents

are

usu

ally

giv

en w

ith

let

ters

su

ch a

s m

,n,

and

kto

rep

rese

nt

expo

nen

ts.F

or e

xam

ple,

one

rule

sta

tes

that

am

�an

�am

�n.

In p

ract

ice,

such

exp

onen

ts a

re h

andl

ed a

s al

gebr

aic

expr

essi

ons

and

the

rule

s of

al

gebr

a ap

ply.

Sim

pli

fy 2

a2 (

an

�1

�a

4n).

2a2 (

an�

1�

a4n)

�2a

2�

an�

1�

2a2

�a4

nU

se t

he D

istr

ibut

ive

Law

.

�2a

2 �

n�

1�

2a2

�4n

Rec

all a

m�

an�

am�

n .

�2a

n�

3�

2a2

�4n

Sim

plify

the

exp

onen

t 2

�n

�1

as n

�3.

It i

s im

port

ant

alw

ays

to c

olle

ct l

ike

term

s on

ly.

Sim

pli

fy (

an

�bm

)2.

(an

�bm

)2�

(an

�bm

)(an

�bm

)F

OI

L�

an�

an�

an�

bm�

an�

bm�

bm�

bmT

he s

econ

d an

d th

irdte

rms

are

like

term

s.

� a

2n�

2anbm

�b2

m

Sim

pli

fy e

ach

exp

ress

ion

by

per

form

ing

the

ind

icat

ed o

per

atio

ns.

1.23

2m23

�m

2.(a

3 )n

a3n

3.(4

nb2

)k4k

nb

2k

4.(x

3 aj )

mx3

ma

jm5.

(�ay

n)3

�a

3 y3n

6.(�

bkx)

2�

b2k

x2

7.(c

2 )h

kc

2hk

8.(�

2dn)5

�32

d5n

9.(a

2 b)(

anb2

) a

2 �

nb

3

10.(

xnym

)(xm

yn)

11.

12.

xn

�m

yn

�m

an

�2

3x3

�n

13.(

ab2

�a2

b)(3

an�

4bn)

3an

�1 b

2�

4ab

n�

2�

3an

�2 b

�4a

2 bn

�1

14.a

b2(2

a2bn

�1

�4a

bn�

6bn

�1 )

2a3 b

n �

1�

4a2 b

n �

2�

6ab

n �

3

12x3

� 4xn

an� a2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-1

5-1

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-2

5-2

©G

lenc

oe/M

cGra

w-H

ill24

5G

lenc

oe A

lgeb

ra 2

Lesson 5-2

Ad

d a

nd

Su

btr

act

Poly

no

mia

ls

Po

lyn

om

ial

a m

onom

ial o

r a

sum

of

mon

omia

ls

Lik

e Te

rms

term

s th

at h

ave

the

sam

e va

riabl

e(s)

rai

sed

to t

he s

ame

pow

er(s

)

To

add

or s

ubt

ract

pol

ynom

ials

,per

form

th

e in

dica

ted

oper

atio

ns

and

com

bin

e li

ke t

erm

s.

Sim

pli

fy �

6rs

�18

r2�

5s2

� 1

4r2

�8r

s �

6s2 .

�6r

s�

18r2

�5s

2�

14r2

�8r

s�6s

2

�(1

8r2

�14

r2)

�(�

6rs

�8r

s) �

(�5s

2�

6s2 )

Gro

up li

ke t

erm

s.

�4r

2�

2rs

�11

s2C

ombi

ne li

ke t

erm

s.

Sim

pli

fy 4

xy2

�12

xy�

7x2 y

�(2

0xy

�5x

y2�

8x2 y

).

4xy2

�12

xy�

7x2 y

�(2

0xy

�5x

y2�

8x2 y

)�

4xy2

�12

xy�

7x2 y

�20

xy�

5xy2

�8x

2 yD

istr

ibut

e th

e m

inus

sig

n.

� (

�7x

2 y�

8x2 y

) �

(4xy

2�

5xy2

) �

(12x

y�

20xy

)G

roup

like

ter

ms.

�x2

y�

xy2

�8x

yC

ombi

ne li

ke t

erm

s.

Sim

pli

fy.

1.(6

x2�

3x�

2) �

(4x2

�x

�3)

2.(7

y2�

12xy

�5x

2 ) �

(6xy

�4y

2�

3x2 )

2x2

�4x

�5

3y2 �

18xy

�8x

2

3.(�

4m2

�6m

) �

(6m

�4m

2 )4.

27x2

�5y

2�

12y2

�14

x2

�8m

2�

12m

13x

2�

7y2

5.(1

8p2

�11

pq�

6q2 )

�(1

5p2

�3p

q�

4q2 )

6.17

j2�

12k2

�3j

2�

15j2

�14

k2

3p2

�14

pq

�10

q2

5j2

�2k

2

7.(8

m2

�7n

2 ) �

(n2

�12

m2 )

8.14

bc�

6b�

4c�

8b�

8c�

8bc

20m

2�

8n2

14b

�22

bc

�12

c

9.6r

2 s�

11rs

2�

3r2 s

�7r

s2�

15r2

s�

9rs2

10.�

9xy

�11

x2�

14y2

�(6

y2�

5xy

�3x

2 )24

r2s

�5r

s2

14x

2�

4xy

�20

y2

11.(

12xy

�8x

�3y

) �

(15x

�7y

�8x

y)12

.10.

8b2

�5.

7b�

7.2

�(2

.9b2

�4.

6b�

3.1)

7x�

4xy

�4y

7.9b

2�

1.1b

�10

.3

13.(

3bc

�9b

2�

6c2 )

�(4

c2�

b2�

5bc)

14.1

1x2

�4y

2�

6xy

�3y

2�

5xy

�10

x2

�10

b2

�8b

c�

2c2

x2

�xy

�7y

2

15.

x2�

xy�

y2�

xy�

y2�

x216

.24p

3�

15p2

�3p

�15

p3�

13p2

�7p

�x

2�

xy�

y2

9p3

�2p

2�

4p3 � 4

7 � 81 � 8

3 � 81 � 4

1 � 21 � 2

3 � 81 � 4Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill24

6G

lenc

oe A

lgeb

ra 2

Mu

ltip

ly P

oly

no

mia

lsYo

u u

se t

he

dist

ribu

tive

pro

pert

y w

hen

you

mu

ltip

lypo

lyn

omia

ls.W

hen

mu

ltip

lyin

g bi

nom

ials

,th

e F

OIL

patt

ern

is

hel

pfu

l.

To m

ultip

ly t

wo

bino

mia

ls,

add

the

prod

ucts

of

Fth

e fir

stte

rms,

FO

IL P

atte

rnO

the

oute

rte

rms,

Ith

e in

ner

term

s, a

ndL

the

last

term

s.

Fin

d 4

y(6

�2y

�5y

2 ).

4y(6

�2y

�5y

2 ) �

4y(6

) �

4y(�

2y)

�4y

(5y2

)D

istr

ibut

ive

Pro

pert

y

�24

y�

8y2

�20

y3M

ultip

ly t

he m

onom

ials

.

Fin

d (

6x�

5)(2

x�

1).

(6x

�5)

(2x

�1)

�6x

�2x

�6x

�1

�(�

5) �

2x�

(�5)

�1

Firs

t te

rms

Out

er t

erm

sIn

ner

term

sLa

st t

erm

s

�12

x2�

6x�

10x

�5

Mul

tiply

mon

omia

ls.

�12

x2�

4x�

5A

dd li

ke t

erm

s.

Fin

d e

ach

pro

du

ct.

1.2x

(3x2

�5)

2.

7a(6

�2a

�a2

)3.

�5y

2 (y2

�2y

�3)

6x3

�10

x42

a�

14a

2�

7a3

�5y

4�

10y

3�

15y

2

4.(x

�2)

(x�

7)

5.(5

�4x

)(3

�2x

)6.

(2x

�1)

(3x

�5)

x2

�5x

�14

15 �

22x

�8x

26x

2�

7x�

5

7.(4

x�

3)(x

�8)

8.(7

x�

2)(2

x�

7)

9.(3

x�

2)(x

�10

)4x

2�

35x

�24

14x

2�

53x

�14

3x2

�28

x�

20

10.3

(2a

�5c

) �

2(4a

�6c

)11

.2(a

�6)

(2a

�7)

12

.2x(

x�

5) �

x2(3

�x)

�2a

�27

c4a

2�

10a

�84

x3

�x

2�

10x

13.(

3t2

�8)

(t2

�5)

14

.(2r

�7)

215

.(c

�7)

(c�

3)3t

4�

7t2

�40

4r2

�28

r�

49c

2�

4c�

21

16.(

5a�

7)(5

a�

7)17

.(3x

2�

1)(2

x2�

5x)

25a

2�

496x

4�

15x

3�

2x2

�5x

18.(

x2�

2)(x

2�

5)19

.(x

�1)

(2x2

�3x

�1)

x4

�7x

2�

102x

3�

x2

�2x

�1

20.(

2n2

�3)

(n2

�5n

�1)

21.(

x�

1)(x

2�

3x�

4)2n

4�

10n

3�

5n2

�15

n�

3x

3�

4x2

�7x

�4

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-2

5-2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-2

5-2

©G

lenc

oe/M

cGra

w-H

ill24

7G

lenc

oe A

lgeb

ra 2

Lesson 5-2

Det

erm

ine

wh

eth

er e

ach

exp

ress

ion

is

a p

olyn

omia

l.If

it

is a

pol

ynom

ial,

stat

e th

ed

egre

e of

th

e p

olyn

omia

l.

1.x2

�2x

�2

yes;

22.

no

3.8x

z�

yye

s;2

Sim

pli

fy.

4.(g

�5)

�(2

g�

7)5.

(5d

�5)

�(d

�1)

3g�

124d

�4

6.(x

2�

3x�

3) �

(2x2

�7x

�2)

7.(�

2f2

�3f

�5)

�(�

2f2

�3f

�8)

3x2

�4x

�5

�4f

2�

6f�

3

8.(4

r2�

6r�

2) �

(�r2

�3r

�5)

9.(2

x2�

3xy)

�(3

x2�

6xy

�4y

2 )5r

2�

9r�

3�

x2

�3x

y�

4y2

10.(

5t�

7) �

(2t2

�3t

�12

)11

.(u

�4)

�(6

�3u

2�

4u)

2t2

�8t

�5

�3u

2�

5u�

10

12.�

5(2c

2�

d2 )

13.x

2 (2x

�9)

�10

c2

�5d

22x

3�

9x2

14.2

q(3p

q�

4q4 )

15.8

w(h

k2�

10h

3 m4

�6k

5 w3 )

6pq

2�

8q5

8hk

2 w�

80h

3 m4 w

�48

k5 w

4

16.m

2 n3 (

�4m

2 n2

�2m

np

�7)

17.�

3s2 y

(�2s

4 y2

�3s

y3�

4)�

4m4 n

5�

2m3 n

4 p�

7m2 n

36s

6 y3

�9s

3 y4

�12

s2 y

18.(

c�

2)(c

�8)

19.(

z�

7)(z

�4)

c2

�10

c�

16z

2�

3z�

28

20.(

a�

5)2

21.(

2x�

3)(3

x�

5)a2

�10

a�

256x

2�

19x

�15

22.(

r�

2s)(

r�

2s)

23.(

3y�

4)(2

y�

3)r2

�4s

26y

2�

y�

12

24.(

3 �

2b)(

3 �

2b)

25.(

3w�

1)2

9 �

4b2

9w2

�6w

�1

1 � 2b2 c� d

4

©G

lenc

oe/M

cGra

w-H

ill24

8G

lenc

oe A

lgeb

ra 2

Det

erm

ine

wh

eth

er e

ach

exp

ress

ion

is

a p

olyn

omia

l.If

it

is a

pol

ynom

ial,

stat

e th

ed

egre

e of

th

e p

olyn

omia

l.

1.5x

3�

2xy4

�6x

yye

s;5

2.�

ac�

a5d

3ye

s;8

3.n

o

4.25

x3z

�x�

78�ye

s;4

5.6c

�2

�c

�1

no

6.�

no

Sim

pli

fy.

7.(3

n2

�1)

�(8

n2

�8)

8.(6

w�

11w

2 ) �

(4 �

7w2 )

11n

2�

7�

18w

2�

6w�

4

9.(�

6n�

13n

2 ) �

(�3n

�9n

2 )10

.(8x

2�

3x)

�(4

x2�

5x�

3)�

9n�

4n2

4x2

�8x

�3

11.(

5m2

�2m

p�

6p2 )

�(�

3m2

�5m

p�

p2)

12.(

2x2

�xy

�y2

) �

(�3x

2�

4xy

�3y

2 )8m

2�

7mp

�7p

2�

x2

�3x

y�

4y2

13.(

5t�

7) �

(2t2

�3t

�12

)14

.(u

�4)

�(6

�3u

2�

4u)

2t2

�8t

�5

�3u

2�

5u�

10

15.�

9(y2

�7w

)16

.�9r

4 y2 (

�3r

y7�

2r3 y

4�

8r10

)�

9y2

�63

w27

r5y

9�

18r7

y6

�72

r14 y

2

17.�

6a2 w

(a3 w

�aw

4 )18

.5a2

w3 (

a2w

6�

3a4 w

2�

9aw

6 )�

6a5 w

2�

6a3 w

55a

4 w9

�15

a6 w

5�

45a

3 w9

19.2

x2(x

2�

xy�

2y2 )

20.�

ab3 d

2 (�

5ab2

d5

�5a

b)

2x4

�2x

3 y�

4x2 y

23a

2 b5 d

7�

3a2 b

4 d2

21.(

v2�

6)(v

2�

4)22

.(7a

�9y

)(2a

�y)

v4

�2v

2�

2414

a2

�11

ay�

9y2

23.(

y�

8)2

24.(

x2�

5y)2

y2

�16

y�

64x

4�

10x

2 y�

25y

2

25.(

5x�

4w)(

5x�

4w)

26.(

2n4

�3)

(2n

4�

3)25

x2

�16

w2

4n8

�9

27.(

w�

2s)(

w2

�2w

s�

4s2 )

28.(

x�

y)(x

2�

3xy

�2y

2 )w

3�

8s3

x3

�2x

2 y�

xy2

�2y

3

29.B

AN

KIN

GT

erry

in

vest

s $1

500

in t

wo

mu

tual

fu

nds

.Th

e fi

rst

year

,on

e fu

nd

grow

s 3.

8%an

d th

e ot

her

gro

ws

6%.W

rite

a p

olyn

omia

l to

rep

rese

nt

the

amou

nt

Ter

ry’s

$15

00gr

ows

to i

n t

hat

yea

r if

xre

pres

ents

th

e am

oun

t h

e in

vest

ed i

n t

he

fun

d w

ith

th

e le

sser

grow

th r

ate.

�0.

022x

�15

90

30.G

EOM

ETRY

Th

e ar

ea o

f th

e ba

se o

f a

rect

angu

lar

box

mea

sure

s 2x

2�

4x�

3 sq

uar

eu

nit

s.T

he

hei

ght

of t

he

box

mea

sure

s x

un

its.

Fin

d a

poly

nom

ial

expr

essi

on f

or t

he

volu

me

of t

he

box.

2x3

�4x

2�

3xu

nit

s3

3 � 5

6 � s5 � r12

m8 n

9�

�(m

�n

)24 � 3

Pra

ctic

e (

Ave

rag

e)

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-2

5-2


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csP

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-2

5-2

©G

lenc

oe/M

cGra

w-H

ill24

9G

lenc

oe A

lgeb

ra 2

Lesson 5-2

Pre-

Act

ivit

yH

ow c

an p

olyn

omia

ls b

e ap

pli

ed t

o fi

nan

cial

sit

uat

ion

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

5-2

at

the

top

of p

age

229

in y

our

text

book

.

Su

ppos

e th

at S

hen

equ

a de

cide

s to

en

roll

in

a f

ive-

year

en

gin

eeri

ng

prog

ram

rath

er t

han

a f

our-

year

pro

gram

.Usi

ng

the

mod

el g

iven

in

you

r te

xtbo

ok,

how

cou

ld s

he

esti

mat

e th

e tu

itio

n f

or t

he

fift

h y

ear

of h

er p

rogr

am?

(Do

not

act

ual

ly c

alcu

late

,bu

t de

scri

be t

he

calc

ula

tion

th

at w

ould

be

nec

essa

ry.)

Mu

ltip

ly $

15,6

04 b

y 1.

04.

Rea

din

g t

he

Less

on

1.S

tate

wh

eth

er t

he

term

s in

eac

h o

f th

e fo

llow

ing

pair

s ar

e li

ke t

erm

sor

un

like

ter

ms.

a.3x

2 ,3y

2u

nlik

e te

rms

b.

�m

4 ,5m

4lik

e te

rms

c.8r

3 ,8s

3u

nlik

e te

rms

d.

�6,

6lik

e te

rms

2.S

tate

wh

eth

er e

ach

of

the

foll

owin

g ex

pres

sion

s is

a m

onom

ial,

bin

omia

l,tr

inom

ial,

orn

ot a

pol

ynom

ial.

If t

he

expr

essi

on i

s a

poly

nom

ial,

give

its

deg

ree.

a.4r

4�

2r�

1tr

ino

mia

l;d

egre

e 4

b.

�3x�

no

t a

po

lyn

om

ial

c.5x

�4y

bin

om

ial;

deg

ree

1d

.2a

b�

4ab2

�6a

b3tr

ino

mia

l;d

egre

e 4

3.a.

Wh

at i

s th

e F

OIL

met

hod

use

d fo

r in

alg

ebra

?to

mu

ltip

ly b

ino

mia

lsb

.T

he

FO

IL m

eth

od i

s an

app

lica

tion

of

wh

at p

rope

rty

of r

eal

nu

mbe

rs?

Dis

trib

uti

ve P

rop

erty

c.In

th

e F

OIL

met

hod

,wh

at d

o th

e le

tter

s F,

O,I

,an

d L

mea

n?

firs

t,o

ute

r,in

ner

,las

td

.S

upp

ose

you

wan

t to

use

th

e F

OIL

met

hod

to

mu

ltip

ly (

2x�

3)(4

x�

1).S

how

th

ete

rms

you

wou

ld m

ult

iply

,bu

t do

not

act

ual

ly m

ult

iply

th

em.

F(2

x)(

4x)

O(2

x)(

1)

I(3

)(4x

)

L(3

)(1)

Hel

pin

g Y

ou

Rem

emb

er

4.Yo

u c

an r

emem

ber

the

diff

eren

ce b

etw

een

mon

omia

ls,b

inom

ials

,an

d tr

inom

ials

byth

inki

ng

of c

omm

on E

ngl

ish

wor

ds t

hat

beg

in w

ith

th

e sa

me

pref

ixes

.Giv

e tw

o w

ords

un

rela

ted

to m

ath

emat

ics

that

sta

rt w

ith

mon

o-,t

wo

that

beg

in w

ith

bi-

,an

d tw

o th

atbe

gin

wit

h t

ri-.

Sam

ple

an

swer

:m

on

oto

no

us,

mo

no

gra

m;

bic

ycle

,bifo

cal;

tric

ycle

,tri

po

d

©G

lenc

oe/M

cGra

w-H

ill25

0G

lenc

oe A

lgeb

ra 2

Po

lyn

om

ials

wit

h F

ract

ion

al C

oef

fici

ents

Pol

ynom

ials

may

hav

e fr

acti

onal

coe

ffic

ien

ts a

s lo

ng

as t

her

e ar

e n

o va

riab

les

in t

he

den

omin

ator

s.C

ompu

tin

g w

ith

fra

ctio

nal

coe

ffic

ien

ts i

s pe

rfor

med

in

th

e sa

me

way

as

com

puti

ng

wit

h w

hol

e-n

um

ber

coef

fici

ents

.

Sim

pli

ply

.Wri

te a

ll c

oeff

icie

nts

as

frac

tion

s.

1.��3 5� m

��2 7� p

��1 3� n

��7 3� p

��5 2� m

��3 4� n

��3 11 0�

m�

� 15 2�n

��5 25 1�

p

2.��3 2� x

��4 3� y

��5 4� z

��

�1 4� x�

y�

�2 5� z��

��7 8� x

��6 7� y

��1 2� z

��3 8� x�

�2 25 1�y

�� 27 0�

z

3.��1 2� a

2�

�1 3� ab

��1 4� b

2 ��5 6� a

2�

�2 3� ab

��3 4� b

2 ��4 3� a

2�

�1 3� ab

��1 2� b

2

4.��1 2� a

2�

�1 3� ab

��1 4� b

2 ��1 3� a

2�

�1 2� ab

��5 6� b

2 ��1 6� a

2�

�1 6� ab

�� 17 2�

b2

5.��1 2� a

2�

�1 3� ab

��1 4� b

2 ��1 2� a

��2 3� b

��1 4� a3

��1 2� a

2 b�

�2 75 2�ab

2�

�1 6� b3

6.��2 3� a

2�

�1 5� a�

�2 7� ��2 3� a

3�

�1 5� a2

��2 7� a

��4 9� a5

�� 21 5�

a3�

� 34 5�a2

�� 44 9�

a

7.��2 3� x

2�

�3 4� x�

2 ��4 5� x

��1 6� x

2�

�1 2� ��

�1 9� x4

�� 17 29 0�

x3�

�3 5� x2

��4 49 0�

x�

1

8.��1 6�

��1 3� x

��1 6� x

4�

�1 2� x2 ��

��1 6� x3

��1 3�

��1 3� x

�� 31 6�x7

�� 35 6�

x5�

� 37 6�x3

�� 11 8�

x2�

�1 6� x�

� 11 8�

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-2

5-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Div

idin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3

©G

lenc

oe/M

cGra

w-H

ill25

1G

lenc

oe A

lgeb

ra 2

Lesson 5-3

Use

Lo

ng

Div

isio

nT

o di

vide

a p

olyn

omia

l by

a m

onom

ial,

use

th

e pr

oper

ties

of

pow

ers

from

Les

son

5-1

.

To

divi

de a

pol

ynom

ial

by a

pol

ynom

ial,

use

a l

ong

divi

sion

pat

tern

.Rem

embe

r th

at o

nly

like

ter

ms

can

be

adde

d or

su

btra

cted

.

Sim

pli

fy

.

��

�

�p3

�2 t

2�

1 r1

�1

�p2

�2 q

t1�

1 r2

�1

�p3

�2 t

1�

1 r1

�1

�4p

t�

7qr

�3p

Use

lon

g d

ivis

ion

to

fin

d (

x3�

8x2

�4x

�9)

�(x

�4)

.

x2�

4x�

12x

�4 �

x�3 ��

8�x�2 �

��4�

x��

9�(�

)x3

�4x

2

�4x

2�

4x(�

)�4x

2�

16x

�12

x�

9(�

)�12

x�

48�

57T

he

quot

ien

t is

x2

�4x

�12

,an

d th

e re

mai

nde

r is

�57

.

Th

eref

ore

�x2

�4x

�12

�.

Sim

pli

fy.

1.2.

3.

6a2

�10

a�

105a

b�

�7a

4

4.(2

x2�

5x�

3)

(x�

3)5.

(m2

�3m

�7)

(m

�2)

2x�

1m

�5

�

6.(p

3�

6)

(p�

1)7.

(t3

�6t

2�

1)

(t�

2)

p2

�p

�1

�t2

�8t

�16

�

8.(x

5�

1)

(x�

1)9.

(2x3

�5x

2�

4x�

4)

(x �

2)

x4

�x

3�

x2

�x

�1

2x2

�x

�2

31� t

�2

5� p

�1

3� m

�2

4b2

�a

6n3

� m

60a2 b

3�

48b4

�84

a5 b2

��

�12

ab2

24m

n6

�40

m2 n

3�

��

4m2 n

318

a3�

30a2

�� 3a

57� x

�4

x3�

8x2

�4x

�9

��

�x

�4

9 � 321 � 3

12 � 3

9p3 t

r� 3p

2 tr

21p2 q

tr2

��

3p2 t

r12

p3 t2 r

�3p

2 tr

12p3 t

2 r�

21p2 q

tr2

�9p

3 tr

��

��

3p2 t

r

12p

3 t2 r

�21

p2 q

tr2

�9p

3 tr

��

��

3p2 t

rEx

ampl

e1Ex

ampl

e1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill25

2G

lenc

oe A

lgeb

ra 2

Use

Syn

thet

ic D

ivis

ion

Syn

thet

ic d

ivis

ion

a pr

oced

ure

to d

ivid

e a

poly

nom

ial b

y a

bino

mia

l usi

ng c

oeffi

cien

ts o

f th

e di

vide

nd a

nd

the

valu

e of

rin

the

div

isor

x�

r

Use

syn

thet

ic d

ivis

ion

to

fin

d (2

x3�

5x2

�5x

�2)

(x

�1)

.

Ste

p 1

Writ

e th

e te

rms

of t

he d

ivid

end

so t

hat

the

degr

ees

of t

he t

erm

s ar

e in

2x3

�5x

2�

5x�

2 de

scen

ding

ord

er. T

hen

writ

e ju

st t

he c

oeffi

cien

ts.

2 �

5 5

�2

Ste

p 2

Writ

e th

e co

nsta

nt r

of t

he d

ivis

or x

�r

to t

he le

ft, I

n th

is c

ase,

r�

1.

12

�5

5�

2B

ring

dow

n th

e fir

st c

oeffi

cien

t, 2,

as

show

n.2

Ste

p 3

Mul

tiply

the

firs

t co

effic

ient

by

r, 1

�2

�2.

Writ

e th

eir

prod

uct

unde

r th

e1

2�

55

�2

seco

nd c

oeffi

cien

t. T

hen

add

the

prod

uct

and

the

seco

nd c

oeffi

cien

t:2

�5

�2

��

3.2

�3

Ste

p 4

Mul

tiply

the

sum

, �

3, b

y r:

�3

�1

��

3. W

rite

the

prod

uct

unde

r th

e ne

xt1

2�

55

�2

coef

ficie

nt a

nd a

dd:

5 �

(�3)

�2.

2�

32

�3

2

Ste

p 5

Mul

tiply

the

sum

, 2,

by

r: 2

�1

�2.

Writ

e th

e pr

oduc

t un

der

the

next

12

�5

5�

2 co

effic

ient

and

add

: �

2 �

2 �

0. T

he r

emai

nder

is 0

.2

�3

22

�3

20

Th

us,

(2x3

�5x

2�

5x�

2)

(x�

1) �

2x2

�3x

�2.

Sim

pli

fy.

1.(3

x3�

7x2

�9x

�14

)

(x�

2)2.

(5x3

�7x

2�

x�

3)

(x�

1)

3x2

�x

�7

5x2

�2x

�3

3.(2

x3�

3x2

�10

x �

3)

(x�

3)4.

(x3

�8x

2�

19x

�9)

(x

�4)

2x2

�3x

�1

x2

�4x

�3

�

5.(2

x3�

10x2

�9x

�38

)

(x�

5)6.

(3x3

�8x

2�

16x

�1)

(x

�1)

2x2

�9

�3x

2�

5x�

11 �

7.(x

3�

9x2

�17

x�

1)

(x�

2)8.

(4x3

�25

x2�

4x�

20)

(x

�6)

x2

�7x

�3

�4x

2�

x�

2 �

9.(6

x3�

28x2

�7x

�9)

(x

�5)

10.(

x4�

4x3

�x2

�7x

�2)

(x

�2)

6x2

�2x

�3

�x

3�

2x2

�3x

�1

11.(

12x4

�20

x3�

24x2

�20

x�

35)

(3

x�

5)4x

3�

8x�

20 �

�65

� 3x�

5

6� x

�5

8� x

�6

5� x

�2

10� x

�1

7� x

�5

3� x

�4

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Div

idin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Div

idin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3

©G

lenc

oe/M

cGra

w-H

ill25

3G

lenc

oe A

lgeb

ra 2

Lesson 5-3

Sim

pli

fy.

1.5c

�3

2.3x

�5

3.5y

2�

2y�

14.

3x�

1 �

5.(1

5q6

�5q

2 )(5

q4)�

16.

(4f5

�6f

4�

12f3

�8f

2 )(4

f2)�

1

3q2

�f3

��

3f�

2

7.(6

j2k

�9j

k2)

3j

k8.

(4a2

h2

�8a

3 h�

3a4 )

(2

a2)

2j�

3k2h

2�

4ah

�

9.(n

2�

7n�

10)

(n

�5)

10.(

d2

�4d

�3)

(d

�1)

n�

2d

�3

11.(

2s2

�13

s�

15)

(s

�5)

12.(

6y2

�y

�2)

(2y

�1)

�1

2s�

33y

�2

13.(

4g2

�9)

(2

g�

3)14

.(2x

2�

5x�

4)

(x�

3)

2g�

32x

�1

�

15.

16.

u�

8 �

2x�

1 �

17.(

3v2

�7v

�10

)(v

�4)

�1

18.(

3t4

�4t

3�

32t2

�5t

�20

)(t

�4)

�1

3v�

5 �

3t3

�8t

2�

5

19.

20.

y2

�3y

�6

�2x

2�

5x�

4 �

21.(

4p3

�3p

2�

2p)

(p

�1)

22.(

3c4

�6c

3�

2c�

4)(c

�2)

�1

4p2

�p

�3

�3c

3�

2 �

23.G

EOM

ETRY

Th

e ar

ea o

f a

rect

angl

e is

x3

�8x

2�

13x

�12

squ

are

un

its.

Th

e w

idth

of

the

rect

angl

e is

x�

4 u

nit

s.W

hat

is

the

len

gth

of

the

rect

angl

e?x

2�

4x�

3 u

nit

s

8� c

�2

3� p

�1

3� x

�3

18� y

�2

2x3

�x2

�19

x�

15�

��

x�

3y3

�y2

�6

��

y�

2

10� v

�4

1� x

�3

12� u

�3

2x2

�5x

�4

��

x�

3u

2�

5u�

12�

�u

�3

1� x

�33a

2�

2

3f2

�2

1 � q2

2 � x12

x2�

4x�

8�

� 4x15

y3�

6y2

�3y

�� 3y

12x

�20

�� 4

10c

�6

�2

©G

lenc

oe/M

cGra

w-H

ill25

4G

lenc

oe A

lgeb

ra 2

Sim

pli

fy.

1.3r

6�

r4�

2.�

6k2

�

3.(�

30x3

y�

12x2

y2�

18x2

y)

(�6x

2 y)

4.(�

6w3 z

4�

3w2 z

5�

4w�

5z)

(2

w2 z

)

5x�

2y�

3�

3wz3

��

�

5.(4

a3�

8a2

�a2

)(4a

)�1

6.(2

8d3 k

2�

d2 k

2�

4dk2

)(4d

k2)�

1

a2

�2a

��a 4�

7d2

��d 4�

�1

7.f

�5

8.2x

�7

9.(a

3�

64)

(a

�4)

a2

�4a

�16

10.(

b3�

27)

(b

�3)

b2

�3b

�9

11.

2x2

�8x

�38

12.

2x2

�6x

�22

�

13.(

3w3

�7w

2�

4w�

3)

(w�

3)14

.(6y

4�

15y3

�28

y�

6)

(y�

2)

3w2

�2w

�2

�� w

�33

�6y

3�

3y2

�6y

�16

�� y

2 �62

�

15.(

x4�

3x3

�11

x2�

3x�

10)

(x

�5)

16.(

3m5

�m

�1)

(m

�1)

x3�

2x2

�x

�2

3m4

�3m

3�

3m2

�3m

�4

�

17.(

x4�

3x3

�5x

�6)

(x�

2)�

118

.(6y

2�

5y�

15)(

2y�

3)�

1

x3

�5x

2�

10x

�15

�� x

2 �42

�3y

�7

�� 2y

6 �3

�

19.

20.

2x�

2 �

� 2x1 �2

3�

2x�

1 �

� 3x6 �

1�

21.(

2r3

�5r

2�

2r�

15)

(2

r�

3)22

.(6t

3�

5t2

�2t

�1)

(3

t�

1)

r2�

4r�

52t

2�

t�

1 �

� 3t2 �

1�

23.

24.

2p3

�3p

2�

4p�

12h

2�

h�

3

25.G

EOM

ETRY

Th

e ar

ea o

f a

rect

angl

e is

2x2

�11

x�

15 s

quar

e fe

et.T

he

len

gth

of

the

rect

angl

e is

2x

�5

feet

.Wh

at i

s th

e w

idth

of

the

rect

angl

e?x

�3

ft

26.G

EOM

ETRY

Th

e ar

ea o

f a

tria

ngl

e is

15x

4�

3x3

�4x

2�

x�

3 sq

uar

e m

eter

s.T

he

len

gth

of

the

base

of

the

tria

ngl

e is

6x2

�2

met

ers.

Wh

at i

s th

e h

eigh

t of

th

e tr

ian

gle?

5x2

�x

�3

m

2h4

�h

3�

h2

�h

�3

��

�h

2�

14p

4�

17p2

�14

p�

3�

��

2p�

3

6x2

�x

� 7

��

3x�

14x

2�

2x�

6�

�2x

�3

5� m

�1

72� x

�3

2x3

�4x

�6

��

x�

32x

3�

6x�

152

��

x�

4

2x2

�3x

�14

��

x�

2f2

�7f

�10

��

f�

2

5� 2w

22 � wz

3z4

�2

9m � 2k3k � m

6k2 m

�12

k3 m2

�9m

3�

��

2km

28 � r2

15r10

�5r

8�

40r2

��

�5r

4Pra

ctic

e (

Ave

rag

e)

Div

idin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3



Readin

g t

o L

earn

Math

em

ati

csD

ivid

ing

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3

©G

lenc

oe/M

cGra

w-H

ill25

5G

lenc

oe A

lgeb

ra 2

Lesson 5-3

Pre-

Act

ivit

yH

ow c

an y

ou u

se d

ivis

ion

of

pol

ynom

ials

in

man

ufa

ctu

rin

g?

Rea

d th

e in

trod

uct

ion

to

Les

son

5-3

at

the

top

of p

age

233

in y

our

text

book

.

Usi

ng

the

divi

sion

sym

bol

(),

wri

te t

he

divi

sion

pro

blem

th

at y

ou w

ould

use

to

answ

er t

he

ques

tion

ask

ed i

n t

he

intr

odu

ctio

n.(

Do

not

act

ual

lydi

vide

.)(3

2x2

�x)

�(8

x)

Rea

din

g t

he

Less

on

1.a.

Exp

lain

in

wor

ds h

ow t

o di

vide

a p

olyn

omia

l by

a m

onom

ial.

Div

ide

each

ter

m o

fth

e p

oly

no

mia

l by

the

mo

no

mia

l.b

.If

you

div

ide

a tr

inom

ial

by a

mon

omia

l an

d ge

t a

poly

nom

ial,

wh

at k

ind

ofpo

lyn

omia

l w

ill

the

quot

ien

t be

?tr

ino

mia

l

2.L

ook

at t

he

foll

owin

g di

visi

on e

xam

ple

that

use

s th

e di

visi

on a

lgor

ith

m f

or p

olyn

omia

ls.

2x�

4x

�4��

�2x

2�

4x�

72x

2�

8x 4x�

74x

�16 23

Wh

ich

of

the

foll

owin

g is

th

e co

rrec

t w

ay t

o w

rite

th

e qu

otie

nt?

CA

.2x

�4

B.

x�

4C

.2x

�4

�D

.

3.If

you

use

syn

thet

ic d

ivis

ion

to

divi

de x

3�

3x2

�5x

�8

by x

�2,

the

divi

sion

wil

l lo

okli

ke t

his

:

21

3�

5�

82

1010

15

52

Wh

ich

of

the

foll

owin

g is

th

e an

swer

for

th

is d

ivis

ion

pro

blem

?B

A.

x2�

5x�

5B

.x2

�5x

�5

�

C.

x3�

5x2

�5x

�D

.x3

�5x

2�

5x�

2

Hel

pin

g Y

ou

Rem

emb

er

4.W

hen

you

tra

nsl

ate

the

nu

mbe

rs i

n t

he

last

row

of

a sy

nth

etic

div

isio

n i

nto

th

e qu

otie

nt

and

rem

ain

der,

wh

at i

s an

eas

y w

ay t

o re

mem

ber

wh

ich

exp

onen

ts t

o u

se i

n w

riti

ng

the

term

s of

th

e qu

otie

nt?

Sam

ple

an

swer

:S

tart

wit

h t

he

po

wer

th

at is

on

e le

ssth

an t

he

deg

ree

of

the

div

iden

d.D

ecre

ase

the

po

wer

by

on

e fo

r ea

chte

rm a

fter

th

e fi

rst.

Th

e fi

nal

nu

mb

er w

ill b

e th

e re

mai

nd

er.D

rop

any

ter

mth

at is

rep

rese

nte

d b

y a

0.

2� x

�2

2� x

�2

23� x

�4

23� x

�4

©G

lenc

oe/M

cGra

w-H

ill25

6G

lenc

oe A

lgeb

ra 2

Ob

liqu

e A

sym

pto

tes

Th

e gr

aph

of

y�

ax�

b,w

her

e a

� 0

,is

call

ed a

n o

bliq

ue

asym

ptot

e of

y�

f(x)

if

th

e gr

aph

of

fco

mes

clo

ser

and

clos

er t

o th

e li

ne

as x

→ ∞

or x

→ �

∞.∞

is t

he

mat

hem

atic

al s

ymbo

l fo

r in

fin

ity,

wh

ich

mea

ns

end

less

.

For

f(x

) �

3x�

4 �

�2 x� ,y

�3x

�4

is a

n o

bliq

ue

asym

ptot

e be

cau

se

f(x)

�3x

�4

��2 x� ,

and

�2 x�→

0 a

s x

→ ∞

or �

∞.I

n o

ther

wor

ds,a

s |x

|

incr

ease

s,th

e va

lue

of �2 x�

gets

sm

alle

r an

d sm

alle

r ap

proa

chin

g 0.

Fin

d t

he

obli

qu

e as

ymp

tote

for

f(x

) ��x2

� x8 �x

2�15

�.

�2

18

15U

se s

ynth

etic

div

isio

n.

�2

�12

16

3

y��x2

�x

8 �x2�

15�

�x

�6

�� x

�32

�

As

|x|i

ncr

ease

s,th

e va

lue

of � x

�32

�ge

ts s

mal

ler.

In o

ther

wor

ds,s

ince

� x�3

2�

→ 0

as

x →

∞or

x →

�∞

,y�

x�

6 is

an

obl

iqu

e as

ympt

ote.

Use

syn

thet

ic d

ivis

ion

to

fin

d t

he

obli

qu

e as

ymp

tote

for

eac

h f

un

ctio

n.

1.y

��8x

2� x

�4x5�

11�

y�

8x�

44

2.y

��x2

�x

3 �x2�

15�

y�

x�

5

3.y

��x2

�x

2 �x3�

18�

y�

x�

1

4.y

��ax

2

x��bx

d�

c�

y�

ax�

b�

ad

5.y

��ax

2

x��bx

d�

c�

y�

ax�

b�

ad

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3

Exam

ple

Exam

ple


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Fact

ori

ng

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4

5-4

©G

lenc

oe/M

cGra

w-H

ill25

7G

lenc

oe A

lgeb

ra 2

Lesson 5-4

Fact

or

Poly

no

mia

ls

For

any

num

ber

of t

erm

s, c

heck

for

:gr

eate

st c

omm

on f

acto

r

For

tw

o te

rms,

che

ck f

or:

Diff

eren

ce o

f tw

o sq

uare

sa

2�

b2

�(a

�b)

(a�

b)S

um o

f tw

o cu

bes

a3

�b

3�

(a�

b)(a

2�

ab�

b2 )

Diff

eren

ce o

f tw

o cu

bes

a3

�b

3�

(a�

b)(a

2�

ab�

b2 )

Tech

niq

ues

fo

r Fa

cto

rin

g P

oly

no

mia

lsF

or t

hree

ter

ms,

che

ck f

or:

Per

fect

squ

are

trin

omia

lsa

2�

2ab

�b

2�

(a�

b)2

a2

�2a

b�

b2

�(a

�b)

2

Gen

eral

trin

omia

lsac

x2�

(ad

�bc

)x�

bd�

(ax

�b)

(cx

�d

)

For

fou

r te

rms,

che

ck f

or:

Gro

upin

gax

�bx

�ay

�by

�x(

a�

b) �

y(a

�b)

�(a

�b)

(x�

y)

Fac

tor

24x2

�42

x�

45.

Fir

st f

acto

r ou

t th

e G

CF

to

get

24x2

�42

x�

45 �

3(8x

2�

14x

�15

).T

o fi

nd

the

coef

fici

ents

of t

he

xte

rms,

you

mu

st f

ind

two

nu

mbe

rs w

hos

e pr

odu

ct i

s 8

�(�

15)

��

120

and

wh

ose

sum

is

�14

.Th

e tw

o co

effi

cien

ts m

ust

be

�20

an

d 6.

Rew

rite

th

e ex

pres

sion

usi

ng

�20

xan

d6x

and

fact

or b

y gr

oupi

ng.

8x2

�14

x�

15 �

8x2

�20

x�

6x�

15G

roup

to

find

a G

CF.

�4x

(2x

�5)

�3(

2x�

5)F

acto

r th

e G

CF

of

each

bin

omia

l.

�(4

x�

3)(2

x�

5)D

istr

ibut

ive

Pro

pert

y

Th

us,

24x2

�42

x�

45 �

3(4x

�3)

(2x

�5)

.

Fac

tor

com

ple

tely

.If

the

pol

ynom

ial

is n

ot f

acto

rab

le,w

rite

pri

me.

1.14

x2y2

�42

xy3

2.6m

n�

18m

�n

�3

3.2x

2�

18x

�16

14xy

2 (x

�3y

)(6

m�

1)(n

�3)

2(

x�

8)(x

�1)

4.x4

�1

5.35

x3y4

�60

x4y

6.2r

3�

250

(x2

�1)

(x�

1)(x

�1)

5x

3 y(7

y3

�12

x)

2(r

�5)

(r2

�5r

�25

)

7.10

0m8

�9

8.x2

�x

�1

9.c4

�c3

�c2

�c

(10m

4�

3)(1

0m4

�3)

pri

me

c(c

�1)

2(c

�1)

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill25

8G

lenc

oe A

lgeb

ra 2

Sim

plif

y Q

uo

tien

tsIn

th

e la

st l

esso

n y

ou l

earn

ed h

ow t

o si

mpl

ify

the

quot

ien

t of

tw

opo

lyn

omia

ls b

y u

sin

g lo

ng

divi

sion

or

syn

thet

ic d

ivis

ion

.Som

e qu

otie

nts

can

be

sim

plif

ied

byu

sin

g fa

ctor

ing.

Sim

pli

fy

.

�F

acto

r th

e nu

mer

ator

and

den

omin

ator

.

�D

ivid

e. A

ssum

e x

�8,

�.

Sim

pli

fy.A

ssu

me

that

no

den

omin

ator

is

equ

al t

o 0.

1.2.

3.

4.5.

6.

7.8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

9x2

�6x

�4

��

3x�

2y

2�

4y�

16�

�3y

�5

2x�

3�

�4x

2�

2x�

1

27x3

�8

��

9x2

�4

y3�

64�

�3y

2�

17y

�20

4x2

�4x

�3

��

8x3

�1

2x�

3� x

�8

7x�

3� x

�2

x�

9� 2x

�5

4x2

�12

x�

9�

�2x

2�

13x

�24

7x2

�11

x�

6�

�x2

�4

x2�

81�

�2x

2�

23x

�45

x�

7� x

�5

3x�

5� 2x

�3

2x�

5� x

�1

x2�

14x

�49

��

x2�

2x�

353x

2�

4x�

15�

�2x

2�

3x�

94x

2�

16x

�15

��

2x2

�x

�3

x�

2� 3x

�7

6x�

1� x

�2

2x�

1� x

�2

x2�

7x�

10�

�3x

2�

8x�

356x

2�

25x

�4

��

x2�

6x�

84x

2�

4x�

3�

�2x

2�

x�

6

5x�

1� x

�3

2x�

1� 4x

�1

x�

3� x

�5

5x2

�9x

�2

��

x2�

5x�

62x

2�

5x�

3�

�4x

2�

11x

�3

x2�

x�

6�

�x2

�7x

�10

x�

5� x

�1

x�

5� 2x

�3

x�

4� x

�2

x2�

11x

�30

��

x2�

5x�

6x2

�6x

�5

��

2x2

�x

�3

x2�

7x�

12�

�x2

�x

�6

3 � 2x

�4

� x�

8

(2x

�3)

( x

�4)

��

(x�

8)(2

x�

3)8x

2�

11x

�12

��

2x2

�13

x�

24

8x2

�11

x�

12�

�2x

2�

13x

�24

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Fact

ori

ng

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4

5-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Fact

ori

ng

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4

5-4

©G

lenc

oe/M

cGra

w-H

ill25

9G

lenc

oe A

lgeb

ra 2

Lesson 5-4

Fac

tor

com

ple

tely

.If

the

pol

ynom

ial

is n

ot f

acto

rab

le,w

rite

pri

me.

1.7x

2�

14x

2.19

x3�

38x2

7x(x

�2)

19x

2 (x

�2)

3.21

x3�

18x2

y�

24xy

24.

8j3 k

�4j

k3�

73x

(7x2

�6x

y�

8y2 )

pri

me

5.a2

�7a

�18

6.2a

k�

6a�

k�

3(a

�9)

(a�

2)(2

a�

1)(k

�3)

7.b2

�8b

�7

8.z2

�8z

�10

(b�

7)(b

�1)

pri

me

9.m

2�

7m�

1810

.2x2

�3x

�5

(m�

2)(m

�9)

(2x

�5)

(x�

1)

11.4

z2�

4z�

1512

.4p2

�4p

�24

(2z

�5)

(2z

�3)

4(p

�2)

(p�

3)

13.3

y2�

21y

�36

14.c

2�

100

3(y

�4)

(y�

3)(c

�10

)(c

�10

)

15.4

f2�

6416

.d2

�12

d�

364(

f�

4)(f

�4)

(d�

6)2

17.9

x2�

2518

.y2

�18

y�

81p

rim

e(y

�9)

2

19.n

3�

125

20.m

4�

1(n

�5)

(n2

�5n

�25

)(m

2�

1)(m

�1)

(m�

1)

Sim

pli

fy.A

ssu

me

that

no

den

omin

ator

is

equ

al t

o 0.

21.

22.

23.

24.

x�

1� x

�7

x2�

6x�

7�

�x2

�49

x�

5�

xx2

�10

x�

25�

�x2

�5x

x�

1� x

�3

x2�

4x�

3�

�x2

�6x

�9

x�

2� x

�5

x2�

7x�

18�

�x2

�4x

�45

©G

lenc

oe/M

cGra

w-H

ill26

0G

lenc

oe A

lgeb

ra 2

Fac

tor

com

ple

tely

.If

the

pol

ynom

ial

is n

ot f

acto

rab

le,w

rite

pri

me.

1.15

a2b

�10

ab2

2.3s

t2�

9s3 t

�6s

2 t2

3.3x

3 y2

�2x

2 y�

5xy

5ab

(3a

�2b

)3s

t(t

�3s

2�

2st)

xy(3

x2 y

�2x

�5)

4.2x

3 y�

x2y

�5x

y2�

xy3

5.21

�7t

�3r

�rt

6.x2

�xy

�2x

�2y

xy(2

x2

�x

�5y

�y

2 )(7

�r)

(3 �

t)(x

�2)

(x�

y)

7.y2

�20

y�

968.

4ab

�2a

�6b

�3

9.6n

2�

11n

�2

(y�

8)(y

�12

)(2

a�

3)(2

b�

1)(6

n�

1)(n

�2)

10.6

x2�

7x�

311

.x2

�8x

�8

12.6

p2�

17p

�45

(3x

�1)

(2x

�3)

pri

me

(2p

�9)

(3p

�5)

13.r

3�

3r2

�54

r14

.8a2

�2a

�6

15.c

2�

49r(

r�

9)(r

�6)

2(4a

�3)

(a�

1)(c

�7)

(c�

7)

16.x

3�

817

.16r

2�

169

18.b

4�

81(x

�2)

(x2

�2x

�4)

(4r

�13

)(4r

�13

)(b

2�

9)(b

�3)

(b�

3)

19.8

m3

�25

pri

me

20.2

t3�

32t2

�12

8t2t

(t�

8)2

21.5

y5�

135y

25y

2 (y

�3)

(y2

�3y

�9)

22.8

1x4

�16

(9x

2�

4)(3

x�

2)(3

x�

2)

Sim

pli

fy.A

ssu

me

that

no

den

omin

ator

is

equ

al t

o 0.

23.

24.

25.

26.D

ESIG

NB

obbi

Jo

is u

sin

g a

soft

war

e pa

ckag

e to

cre

ate

a dr

awin

g of

a c

ross

sec

tion

of

a br

ace

as s

how

n a

t th

e ri

ght.

Wri

te a

sim

plif

ied,

fact

ored

exp

ress

ion

th

at r

epre

sen

ts t

he

area

of

the

cros

s se

ctio

n o

f th

e br

ace.

x(2

0.2

�x

) cm

2

27.C

OM

BU

STIO

N E

NG

INES

In a

n i

nte

rnal

com

bust

ion

en

gin

e,th

e u

p an

d do

wn

mot

ion

of

the

pist

ons

is c

onve

rted

in

to t

he

rota

ry m

otio

n o

f th

e cr

anks

haf

t,w

hic

h d

rive

s th

e fl

ywh

eel.

Let

r1

repr

esen

t th

e ra

diu

s of

th

e fl

ywh

eel

at t

he

righ

t an

d le

t r 2

repr

esen

t th

e ra

diu

s of

th

ecr

anks

haf

t pa

ssin

g th

rou

gh i

t.If

th

e fo

rmu

la f

or t

he

area

of

a ci

rcle

is

A�

�r2

,wri

te a

sim

plif

ied,

fact

ored

exp

ress

ion

for

th

e ar

ea o

f th

e cr

oss

sect

ion

of

the

flyw

hee

l ou

tsid

e th

e cr

anks

haf

t.�

(r 1�

r 2)(r

1�

r 2)

r 1

r 2

x cm

x cm

8.2

cm

12 cm

3(x

�3)

��

x2

�3x

�9

3x2

�27

��

x3�

27

x�

8� x

�9

x2�

16x

�64

��

x2�

x�

72

x�

4� x

�5

x2�

16�

�x2

�x

�20Pra

ctic

e (

Ave

rag

e)

Fact

ori

ng

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4

5-4


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csFa

cto

rin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4

5-4

©G

lenc

oe/M

cGra

w-H

ill26

1G

lenc

oe A

lgeb

ra 2

Lesson 5-4

Pre-

Act

ivit

yH

ow d

oes

fact

orin

g ap

ply

to

geom

etry

?

Rea

d th

e in

trod

uct

ion

to

Les

son

5-4

at

the

top

of p

age

239

in y

our

text

book

.

If a

tri

nom

ial

that

rep

rese

nts

th

e ar

ea o

f a

rect

angl

e is

fac

tore

d in

to t

wo

bin

omia

ls,w

hat

mig

ht

the

two

bin

omia

ls r

epre

sen

t?th

e le

ng

th a

nd

wid

th o

f th

e re

ctan

gle

Rea

din

g t

he

Less

on

1.N

ame

thre

e ty

pes

of b

inom

ials

th

at i

t is

alw

ays

poss

ible

to

fact

or.

dif

fere

nce

of

two

squ

ares

,su

m o

f tw

o c

ub

es,d

iffe

ren

ce o

f tw

o c

ub

es

2.N

ame

a ty

pe o

f tr

inom

ial

that

it

is a

lway

s po

ssib

le t

o fa

ctor

.p

erfe

ct s

qu

are

trin

om

ial

3.C

ompl

ete:

Sin

ce x

2�

y2ca

nn

ot b

e fa

ctor

ed,i

t is

an

exa

mpl

e of

a

poly

nom

ial.

4.O

n a

n a

lgeb

ra q

uiz

,Mar

len

e n

eede

d to

fac

tor

2x2

�4x

�70

.Sh

e w

rote

th

e fo

llow

ing

answ

er:(

x�

5)(2

x�

14).

Wh

en s

he

got

her

qu

iz b

ack,

Mar

len

e fo

un

d th

at s

he

did

not

get

full

cre

dit

for

her

an

swer

.Sh

e th

ough

t sh

e sh

ould

hav

e go

tten

fu

ll c

redi

t be

cau

se s

he

chec

ked

her

wor

k by

mu

ltip

lica

tion

an

d sh

owed

th

at (

x�

5)(2

x�

14)

�2x

2�

4x�

70.

a.If

you

wer

e M

arle

ne’

s te

ach

er,h

ow w

ould

you

exp

lain

to

her

th

at h

er a

nsw

er w

as n

oten

tire

ly c

orre

ct?

Sam

ple

an

swer

:Wh

en y

ou

are

ask

ed t

o f

acto

r a

po

lyn

om

ial,

you

mu

st f

acto

r it

co

mp

lete

ly.T

he

fact

ori

zati

on

was

no

tco

mp

lete

,bec

ause

2x

�14

can

be

fact

ore

d f

urt

her

as

2(x

�7)

.

b.

Wh

at a

dvic

e co

uld

Mar

len

e’s

teac

her

giv

e h

er t

o av

oid

mak

ing

the

sam

e ki

nd

of e

rror

in f

acto

rin

g in

th

e fu

ture

?S

amp

le a

nsw

er:

Alw

ays

loo

k fo

r a

com

mo

nfa

cto

r fi

rst.

If t

her

e is

a c

om

mo

n f

acto

r,fa

cto

r it

ou

t fi

rst,

and

th

en s

eeif

yo

u c

an f

acto

r fu

rth

er.

Hel

pin

g Y

ou

Rem

emb

er

5.S

ome

stu

den

ts h

ave

trou

ble

rem

embe

rin

g th

e co

rrec

t si

gns

in t

he

form

ula

s fo

r th

e su

man

d di

ffer

ence

of

two

cube

s.W

hat

is

an e

asy

way

to

rem

embe

r th

e co

rrec

t si

gns?

Sam

ple

an

swer

:In

th

e b

ino

mia

l fac

tor,

the

op

erat

ion

sig

n is

th

e sa

me

asin

th

e ex

pre

ssio

n t

hat

is b

ein

g f

acto

red

.In

th

e tr

ino

mia

l fac

tor,

the

op

erat

ion

sig

n b

efo

re t

he

mid

dle

ter

m is

th

e o

pp

osi

te o

f th

e si

gn

in t

he

exp

ress

ion

th

at is

bei

ng

fac

tore

d.T

he

sig

n b

efo

re t

he

last

ter

m is

alw

ays

a p

lus.

pri

me

©G

lenc

oe/M

cGra

w-H

ill26

2G

lenc

oe A

lgeb

ra 2

Usi

ng

Pat

tern

s to

Fac

tor

Stu

dy t

he

patt

ern

s be

low

for

fac

tori

ng

the

sum

an

d th

e di

ffer

ence

of

cube

s.

a3�

b3�

(a�

b)(a

2�

ab�

b2)

a3�

b3�

(a�

b)(a

2�

ab�

b2)

Th

is p

atte

rn c

an b

e ex

ten

ded

to o

ther

odd

pow

ers.

Stu

dy t

hes

e ex

ampl

es.

Fac

tor

a5

�b5

.E

xten

d th

e fi

rst

patt

ern

to

obta

in a

5�

b5�

(a�

b)(a

4�

a3b

�a2

b2�

ab3

�b4

).C

hec

k:(

a�

b)(a

4�

a3b

�a2

b2�

ab3

�b4

) �

a5�

a4b

�a3

b2�

a2b3

�ab

4

�a4

b�

a3b2

�a2

b3�

ab4

�b5

�a5

�b5

Fac

tor

a5

�b5

.E

xten

d th

e se

con

d pa

tter

n t

o ob

tain

a5

�b5

�(a

�b)

(a4

�a3

b�

a2b2

�ab

3�

b4).

Ch

eck

:(a

�b)

(a4

�a3

b�

a2b2

�ab

3�

b4)

�a5

�a4

b �

a3b2

�a2

b3�

ab4

�a4

b�

a3b2

�a2

b3�

ab4

�b5

�a5

�b5

In g

ener

al,i

f n

is a

n o

dd i

nte

ger,

wh

en y

ou f

acto

r an

�bn

or a

n�

bn,o

ne

fact

or w

ill

beei

ther

(a

�b)

or

(a�

b),d

epen

din

g on

th

e si

gn o

f th

e or

igin

al e

xpre

ssio

n.T

he

oth

er f

acto

rw

ill

hav

e th

e fo

llow

ing

prop

erti

es:

•T

he

firs

t te

rm w

ill

be a

n�

1an

d th

e la

st t

erm

wil

l be

bn

�1 .

•T

he

expo

nen

ts o

f a

wil

l de

crea

se b

y 1

as y

ou g

o fr

om l

eft

to r

igh

t.•

Th

e ex

pon

ents

of

bw

ill

incr

ease

by

1 as

you

go

from

lef

t to

rig

ht.

•T

he

degr

ee o

f ea

ch t

erm

wil

l be

n�

1.•

If t

he

orig

inal

exp

ress

ion

was

an

�bn

,th

e te

rms

wil

l al

tern

atel

y h

ave

�an

d �

sign

s.•

If t

he

orig

inal

exp

ress

ion

was

an

�bn

,th

e te

rms

wil

l al

l h

ave

�si

gns.

Use

th

e p

atte

rns

abov

e to

fac

tor

each

exp

ress

ion

.

1.a7

�b7

(a�

b)(

a6

�a

5 b�

a4 b

2�

a3 b

3�

a2 b

4�

ab5

�b

6 )

2.c9

�d

9(c

�d

)(c

8�

c7 d

�c

6 d2

�c

5 d3

�c4

d4

�c

3 d5

�c

2 d6

�cd

7�

d8 )

3.e1

1�

f11

(e�

f)(e

10�

e9 f

�e

8 f2

�e

7 f3

�e

6 f4

�e

5 f5

�e

4 f6

�e

3 f7

�e

2 f8

�ef

9�

f10 )

To

fact

or x

10�

y10 ,

chan

ge i

t to

(x

5�

y5)(

x5

�y5

) an

d f

acto

r ea

ch b

inom

ial.

Use

this

ap

pro

ach

to

fact

or e

ach

exp

ress

ion

.

4.x1

0�

y10

(x�

y)(

x4

�x

3 y�

x2y

2�

xy3

�y

4 )(x

�y

)(x

4�

x3 y

�x

2 y2

�xy

3�

y4 )

5.a1

4�

b14

(a�

b)(

a6�

a5 b

�a

4 b2

�a

3 b3

�a

2 b4

�ab

5�

b6 )

(a�

b)

(a6

�a5

b�

a4b

2�

a3b

3�

a2b

4�

ab5

�b

6 )

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4

5-4

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Ro

ots

of

Rea

l Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-5

5-5

©G

lenc

oe/M

cGra

w-H

ill26

3G

lenc

oe A

lgeb

ra 2

Lesson 5-5

Sim

plif

y R

adic

als

Sq

uar

e R

oo

tF

or a

ny r

eal n

umbe

rs a

and

b, if

a2

�b,

the

n a

is a

squ

are

root

of

b.

nth

Ro

ot

For

any

rea

l num

bers

aan

d b,

and

any

pos

itive

inte

ger

n, if

an

�b,

the

n a

is a

n n

thro

ot o

f b.

1.If

nis

eve

n an

d b

�0,

the

n b

has

one

posi

tive

root

and

one

neg

ativ

e ro

ot.

Rea

l nth

Ro

ots

of

b,

2.If

nis

odd

and

b�

0, t

hen

bha

s on

e po

sitiv

e ro

ot.

n �b�,

�n �

b�3.

If n

is e

ven

and

b

0, t

hen

bha

s no

rea

l roo

ts.

4.If

nis

odd

and

b

0, t

hen

bha

s on

e ne

gativ

e ro

ot.

Sim

pli

fy �

49z8

�.

�49

z8�

��

(7z4

)2�

�7z

4

z4m

ust

be

posi

tive

,so

ther

e is

no

nee

d to

take

th

e ab

solu

te v

alu

e.

Sim

pli

fy �

3 �(2

a�

�1)

6�

�3 �

(2a

��

1)6

��

�3 �

[(2a

��

1)2 ]

3�

�(2

a�

1)2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sim

pli

fy.

1.�

81�2.

3 ��

343

�3.

�14

4p6

�9

�7

12| p

3|

4.

�4a

10�

5.5 �

243p

1�

0 �6.

�3 �

m6 n

9�

�2a

53p

2�

m2 n

3

7.3 �

�b1

2�

8.�

16a1

0�

b8 �9.

�12

1x6

��

b4

4| a5 | b

411

| x3|

10.�

(4k)

4�

11.

�16

9r4

�12

.�3 �

�27

p�

6 �16

k2

�13

r23p

2

13.�

�62

5y2

�z4 �

14.�

36q3

4�

15.�

100x

2�

y4z6

��

25| y

| z2

6| q

17|

10| x

| y2| z

3 |

16.

3 ��

0.02

�7�

17.�

��

0.36

�18

.�0.

64p

�10 �

�0.

3n

ot

a re

al n

um

ber

0.8

| p5 |

19.

4 �(2

x)8

�20

.�(1

1y2 )

�4 �

21.

3 �(5

a2b)

�6 �

4x2

121y

425

a4 b

2

22.�

(3x

��

1)2

�23

.3 �

(m�

�5)

6�

24.�

36x2

��

12x

��

1�| 3

x�

1|(m

�5)

2| 6x

�1|

©G

lenc

oe/M

cGra

w-H

ill26

4G

lenc

oe A

lgeb

ra 2

Ap

pro

xim

ate

Rad

ical

s w

ith

a C

alcu

lato

r

Irra

tio

nal

Nu

mb

era

num

ber

that

can

not

be e

xpre

ssed

as

a te

rmin

atin

g or

a r

epea

ting

deci

mal

Rad

ical

s su

ch a

s �

2�an

d �

3�ar

e ex

ampl

es o

f ir

rati

onal

nu

mbe

rs.D

ecim

al a

ppro

xim

atio

ns

for

irra

tion

al n

um

bers

are

oft

en u

sed

in a

ppli

cati

ons.

Th

ese

appr

oxim

atio

ns

can

be

easi

lyfo

un

d w

ith

a c

alcu

lato

r.

Ap

pro

xim

ate

5 �18

.2�

wit

h a

cal

cula

tor.

5 �18

.2�

�1.

787

Use

a c

alcu

lato

r to

ap

pro

xim

ate

each

val

ue

to t

hre

e d

ecim

al p

lace

s.

1.�

62�2.

�10

50�

3.3 �

0.05

4�

7.87

432

.404

0.37

8

4.�

4 �5.

45�

5.�

5280

�6.

�18

,60

�0�

�1.

528

72.6

6413

6.38

2

7.�

0.09

5�

8.3 �

�15

�9.

5 �10

0�

0.30

8�

2.46

62.

512

10.

6 �85

6�

11.�

3200

�12

.�0.

05�

3.08

156

.569

0.22

4

13.�

12,5

0�

0�14

.�0.

60�

15.�

4 �50

0�

111.

803

0.77

5�

4.72

9

16.

3 �0.

15�

17.

6 �42

00�

18.�

75�

0.53

14.

017

8.66

0

19.L

AW

EN

FOR

CEM

ENT

Th

e fo

rmu

la r

�2�

5L�is

use

d by

pol

ice

to e

stim

ate

the

spee

d r

in m

iles

per

hou

r of

a c

ar i

f th

e le

ngt

h L

of t

he

car’

s sk

id m

ark

is m

easu

res

in f

eet.

Est

imat

e to

th

e n

eare

st t

enth

of

a m

ile

per

hou

r th

e sp

eed

of a

car

th

at l

eave

s a

skid

mar

k 30

0 fe

et l

ong.

77.5

mi/h

20.S

PAC

E TR

AV

ELT

he

dist

ance

to

the

hor

izon

dm

iles

fro

m a

sat

elli

te o

rbit

ing

hm

iles

abov

e E

arth

can

be

appr

oxim

ated

by

d�

�80

00h

��

h2

�.W

hat

is

the

dist

ance

to

the

hor

izon

if

a sa

tell

ite

is o

rbit

ing

150

mil

es a

bove

Ear

th?

abo

ut

1100

ft

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Ro

ots

of

Rea

l Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-5

5-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Ro

ots

of

Rea

l Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-5

5-5

©G

lenc

oe/M

cGra

w-H

ill26

5G

lenc

oe A

lgeb

ra 2

Lesson 5-5

Use

a c

alcu

lato

r to

ap

pro

xim

ate

each

val

ue

to t

hre

e d

ecim

al p

lace

s.

1.�

230

�15

.166

2.�

38�6.

164

3.�

�15

2�

�12

.329

4.�

5.6

�2.

366

5.3 �

88�4.

448

6.3 �

�22

2�

�6.

055

7.�

4 �0.

34�

�0.

764

8.5 �

500

�3.

466

Sim

pli

fy.

9.

�81�

�9

10.�

144

�12

11.�

(�5)

2�

512

.��

52�

no

t a

real

nu

mb

er

13.�

0.36

�0.

614

.��

�

15.

3 ��

8�

�2

16.�

3 �27�

�3

17.

3 �0.

064

�0.

418

.5 �

32�2

19.

4 �81�

320

.�y2 �

| y|

21.

3 �12

5s3

�5s

22.�

64x6

�8| x

3 |

23.

3 ��

27a

�6 �

�3a

224

.�m

8 n4

�m

4 n2

25.�

�10

0p4

�q2 �

�10

p2 | q

|26

.4 �

16w

4 v�

8 �2| w

| v2

27.�

(�3c

)4�

9c2

28.�

(a�

b�

)2 �| a

�b

|

2 � 34 � 9

©G

lenc

oe/M

cGra

w-H

ill26

6G

lenc

oe A

lgeb

ra 2

Use

a c

alcu

lato

r to

ap

pro

xim

ate

each

val

ue

to t

hre

e d

ecim

al p

lace

s.

1.�

7.8

�2.

��

89�3.

3 �25�

4.3 �

�4

�2.

793

�9.

434

2.92

4�

1.58

7

5.4 �

1.1

�6.

5 ��

0.1

�7.

6 �55

55�

8.4 �

(0.9

4)�

2 �1.

024

�0.

631

4.20

80.

970

Sim

pli

fy.

9.�

0.81

�10

.��

324

�11

.�4 �

256

�12

.6 �

64�

0.9

�18

�4

2

13.

3 ��

64�

14.

3 �0.

512

�15

.5 �

�24

3�

16.�

4 �12

96�

�4

0.8

�3

�6

17.

5 ��18

.5 �

243x

1�

0 �19

.�(1

4a)2

�20

.��

(14a

�)2 �

no

t a

��4 3�

3x2

14| a|

real

nu

mb

er

21.�

49m

2 t�

8 �22

. ��

23.

3 ��

64r6

�w

15�

24.�

(2x)

8�

7| m| t4

�4r

2 w5

16x

4

25.�

4 �62

5s8

�26

.3 �

216p

3�

q9 �27

.�67

6x4

�y6 �

28.

3 ��

27x9

�y1

2�

�5s

26p

q3

26x

2 | y3 |

�3x

3 y4

29.�

�14

4m�

8 n6

�30

.5 �

�32

x5�

y10

�31

.6 �

(m�

�4)

6�

32.

3 �(2

x�

�1)

3�

�12

m4 | n

3 |�

2xy

2| m

�4|

2x�

1

33.�

�49

a10

�b1

6�

34.

4 �(x

�5

�)8 �

35.

3 �34

3d6

�36

.�x2

�1

�0x

�2

�5�

�7| a

5| b

8(x

�5)

27d

2| x

�5|

37.R

AD

IAN

T TE

MPE

RA

TUR

ET

her

mal

sen

sors

mea

sure

an

obj

ect’s

rad

ian

tte

mpe

ratu

re,

wh

ich

is

the

amou

nt

of e

ner

gy r

adia

ted

by t

he

obje

ct.T

he

inte

rnal

tem

pera

ture

of

an

obje

ct i

s ca

lled

its

kin

etic

tem

pera

ture

.The

for

mul

a T r

�T k

4 �e�

rela

tes

an o

bjec

t’s r

adia

ntte

mpe

ratu

re T

rto

its

kin

etic

tem

pera

ture

Tk.

Th

e va

riab

le e

in t

he

form

ula

is

a m

easu

reof

how

wel

l th

e ob

ject

rad

iate

s en

ergy

.If

an o

bjec

t’s k

inet

ic t

empe

ratu

re i

s 30

°C a

nd

e�

0.94

,wh

at i

s th

e ob

ject

’s r

adia

nt

tem

pera

ture

to

the

nea

rest

ten

th o

f a

degr

ee?

29.5

C

38.H

ERO

’S F

OR

MU

LAS

alva

tore

is

buyi

ng

fert

iliz

er f

or h

is t

rian

gula

r ga

rden

.He

know

sth

e le

ngt

hs

of a

ll t

hre

e si

des,

so h

e is

usi

ng

Her

o’s

form

ula

to

fin

d th

e ar

ea.H

ero’

sfo

rmu

la s

tate

s th

at t

he

area

of

a tr

ian

gle

is �

s(s

��

a)(s

��

b)(s

��

c)�,w

her

e a,

b,an

d c

are

the

len

gth

s of

th

e si

des

of t

he

tria

ngl

e an

d s

is h

alf

the

peri

met

er o

f th

e tr

ian

gle.

If t

he

len

gth

s of

th

e si

des

of S

alva

tore

’s g

arde

n a

re 1

5 fe

et,1

7 fe

et,a

nd

20 f

eet,

wh

at i

s th

ear

ea o

f th

e ga

rden

? R

oun

d yo

ur

answ

er t

o th

e n

eare

st w

hol

e n

um

ber.

124

ft2

4|m

|�

516m

2�

25

�10

24�

243

Pra

ctic

e (

Ave

rag

e)

Ro

ots

of

Rea

l Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-5

5-5



Readin

g t

o L

earn

Math

em

ati

csR

oo

ts o

f R

eal N

um

ber

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-5

5-5

©G

lenc

oe/M

cGra

w-H

ill26

7G

lenc

oe A

lgeb

ra 2

Lesson 5-5

Pre-

Act

ivit

yH

ow d

o sq

uar

e ro

ots

app

ly t

o oc

ean

ogra

ph

y?

Rea

d th

e in

trod

uct

ion

to

Les

son

5-5

at

the

top

of p

age

245

in y

our

text

book

.

Su

ppos

e th

e le

ngt

h o

f a

wav

e is

5 f

eet.

Exp

lain

how

you

wou

ld e

stim

ate

the

spee

d of

th

e w

ave

to t

he

nea

rest

ten

th o

f a

knot

usi

ng

a ca

lcu

lato

r.(D

o n

otac

tual

ly c

alcu

late

th

e sp

eed.

)S

amp

le a

nsw

er:

Usi

ng

a c

alcu

lato

r,fi

nd

th

e p

osi

tive

sq

uar

e ro

ot

of

5.M

ult

iply

th

is n

um

ber

by

1.34

.T

hen

ro

un

d t

he

answ

er t

o t

he

nea

rest

ten

th.

Rea

din

g t

he

Less

on

1.F

or e

ach

rad

ical

bel

ow,i

den

tify

th

e ra

dica

nd

and

the

inde

x.

a.3 �

23�ra

dica

nd:

inde

x:

b.

�15

x2�

radi

can

d:in

dex:

c.5 �

�34

3�

radi

can

d:in

dex:

2.C

ompl

ete

the

foll

owin

g ta

ble.

(Do

not

act

ual

ly f

ind

any

of t

he

indi

cate

d ro

ots.

)

Nu

mb

erN

um

ber

of

Po

siti

ve

Nu

mb

er o

f N

egat

ive

Nu

mb

er o

f P

osi

tive

N

um

ber

of

Neg

ativ

eS

qu

are

Ro

ots

Sq

uar

e R

oo

tsC

ub

e R

oo

ts

Cu

be

Ro

ots

271

11

0

�16

00

01

3.S

tate

wh

eth

er e

ach

of

the

foll

owin

g is

tru

eor

fal

se.

a.A

neg

ativ

e n

um

ber

has

no

real

fou

rth

roo

ts.

tru

e

b.

�

121

�re

pres

ents

bot

h s

quar

e ro

ots

of 1

21.

tru

e

c.W

hen

you

tak

e th

e fi

fth

roo

t of

x5 ,

you

mu

st t

ake

the

abso

lute

val

ue

of x

to i

den

tify

the

prin

cipa

l fi

fth

roo

t.fa

lse

Hel

pin

g Y

ou

Rem

emb

er

4.W

hat

is

an e

asy

way

to

rem

embe

r th

at a

neg

ativ

e n

um

ber

has

no

real

squ

are

root

s bu

th

as o

ne

real

cu

be r

oot?

Sam

ple

an

swer

:Th

e sq

uar

e o

f a

po

siti

ve o

r n

egat

ive

nu

mb

er is

po

siti

ve,s

o t

her

e is

no

rea

l nu

mb

er w

ho

se s

qu

are

is n

egat

ive.

Ho

wev

er,t

he

cub

e o

f a

neg

ativ

e n

um

ber

is n

egat

ive,

so a

neg

ativ

en

um

ber

has

on

e re

al c

ub

e ro

ot,

wh

ich

is a

neg

ativ

e n

um

ber

.

5�

343

215

x2

323

©G

lenc

oe/M

cGra

w-H

ill26

8G

lenc

oe A

lgeb

ra 2

Ap

pro

xim

atin

g S

qu

are

Ro

ots

Con

side

r th

e fo

llow

ing

expa

nsi

on.

(a�� 2b a�

)2�

a2�

�2 2a ab ��

� 4b a2 2�

�a2

�b

�� 4b a2 2�

Th

ink

wh

at h

appe

ns

if a

is v

ery

grea

t in

com

pari

son

to

b.T

he

term

� 4b a2 2�is

ver

y sm

all

and

can

be

disr

egar

ded

in a

n a

ppro

xim

atio

n.

(a�� 2b a�

)2

a2

�b

a�

� 2b a�

�a2

�b

�

Su

ppos

e a

nu

mbe

r ca

n b

e ex

pres

sed

as a

2�

b,a

� b

.Th

en a

n a

ppro

xim

ate

valu

e

of t

he

squ

are

root

is

a�

� 2b a�.Y

ou s

hou

ld a

lso

see

that

a�

� 2b a�

�a2

�b

�.

Use

th

e fo

rmu

la �

a2

��

b��

a�

� 2b a�to

ap

pro

xim

ate

�10

1�

and

�62

2�

.

a.�

101

��

�10

0�

�1�

��

102

��

1�b

.�

622

��

�62

5 �

�3�

��

252

��

3�

Let

a�

10 a

nd

b�

1.L

et a

�25

an

d b

�3.

�10

1�

1

0 �

� 2(1 10

)�

�62

2�

2

5 �

� 2(3 25

)�

1

0.05

2

4.94

Use

th

e fo

rmu

la t

o fi

nd

an

ap

pro

xim

atio

n f

or e

ach

sq

uar

e ro

ot t

o th

e n

eare

st h

un

dre

dth

.Ch

eck

you

r w

ork

wit

h a

cal

cula

tor.

1.�

626

�25

.02

2.�

99�9.

953.

�40

2�

20.0

5

4.�

1604

�40

.05

5.�

223

�14

.93

6.�

80�8.

94

7.�

4890

�69

.93

8.�

2505

�50

.05

9.�

3575

�59

.79

10.

�1,

441

�,1

00�

1200

.42

11.

�29

0�

17.0

312

.�

260

�16

.12

13.

Sh

ow t

hat

a�

� 2b a�

�a2

�b

�fo

r a

� b

.�a

�� 2b a�

�2�

a2

�b

�� 4b a2 2�

;

dis

reg

ard

� 4b a2 2�;�a

�� 2b a�

�2�

a2

�b

;a

�� 2b a�

��

a2

�b

�

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-5

5-5

Exam

ple

Exam

ple


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Rad

ical

Exp

ress

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-6

5-6

©G

lenc

oe/M

cGra

w-H

ill26

9G

lenc

oe A

lgeb

ra 2

Lesson 5-6

Sim

plif

y R

adic

al E

xpre

ssio

ns

For

any

rea

l num

bers

aan

d b,

and

any

inte

ger

n�

1:

Pro

du

ct P

rop

erty

of

Rad

ical

s1.

if n

is e

ven

and

aan

d b

are

both

non

nega

tive,

the

n n �

ab��

n �a�

�n �

b�.2.

if n

is o

dd,

then

n �

ab��

n �a�

�n �

b�.

To

sim

plif

y a

squ

are

root

,fol

low

th

ese

step

s:1.

Fact

or t

he

radi

can

d in

to a

s m

any

squ

ares

as

poss

ible

.2.

Use

th

e P

rodu

ct P

rope

rty

to i

sola

te t

he

perf

ect

squ

ares

.3.

Sim

plif

y ea

ch r

adic

al.

Qu

oti

ent

Pro

per

ty o

f R

adic

als

For

any

rea

l num

bers

aan

d b

�0,

and

any

inte

ger

n�

1,

n ��

, if

all r

oots

are

def

ined

.

To

elim

inat

e ra

dica

ls f

rom

a d

enom

inat

or o

r fr

acti

ons

from

a r

adic

and,

mu

ltip

ly t

he

nu

mer

ator

an

d de

nom

inat

or b

y a

quan

tity

so

that

th

e ra

dica

nd

has

an

exa

ct r

oot.

n �a�

� n �b�

a � b

Sim

pli

fy

3 ��

16a

�5 b

7�

.3 �

�16

a�

5 b7

��

3 �(�

2)3

��

2 �

a�

3�

a2�

�(b

2 )3

��

b�

��

2ab2

3 �2a

2 b�

Sim

pli

fy ��

.

��

Quo

tient

Pro

pert

y

�F

acto

r in

to s

quar

es.

�P

rodu

ct P

rope

rty

�S

impl

ify.

��

Rat

iona

lize

the

deno

min

ator

.

�S

impl

ify.

2|x|

�10

xy�

��

15y3

�5y�

� �5y�

2|x|

�2x�

� 3y2 �

5y�

2|x|

�2x�

� 3y2 �

5y�

�(2

x)2

��

�2x�

��

�(3

y2 )2

��

�5y�

�(2

x)2

��

2x��

��

(3y2 )

2�

�5y

�

�8x

3�

� �45

y5�

8x3

� 45y5

8x3

� 45y5

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sim

pli

fy.

1.5 �

54�15

�6�

2.4 �

32a9

b�

20 �2a

2 | b5 |

4 �2a�

3.�

75x4

y�

7 �5x

2 y3�

5y�

4.��

5.��

6.3 ��

pq

3 �5p

2�

�� 10

p5 q3

�40

| a3| b

�2b�

�� 14

a6 b3

�98

6�5�

�25

36 � 125

©G

lenc

oe/M

cGra

w-H

ill27

0G

lenc

oe A

lgeb

ra 2

Op

erat

ion

s w

ith

Rad

ical

sW

hen

you

add

exp

ress

ion

s co

nta

inin

g ra

dica

ls,y

ou c

anad

d on

ly l

ike

term

s or

lik

e ra

dic

al e

xpre

ssio

ns.

Tw

o ra

dica

l ex

pres

sion

s ar

e ca

lled

lik

era

dic

al e

xpre

ssio

ns

if b

oth

th

e in

dice

s an

d th

e ra

dica

nds

are

ali

ke.

To

mu

ltip

ly r

adic

als,

use

th

e P

rodu

ct a

nd

Qu

otie

nt

Pro

pert

ies.

For

pro

duct

s of

th

e fo

rm

(a�

b��

c�d�

)�(e

�f�

�g�

h�),u

se t

he

FO

IL m

eth

od.T

o ra

tion

aliz

e de

nom

inat

ors,

use

co

nju

gate

s.N

um

bers

of

the

form

a�

b��

c�d�

and

a�b�

�c�

d�,w

her

e a,

b,c,

and

dar

era

tion

al n

um

bers

,are

cal

led

con

juga

tes.

Th

e pr

odu

ct o

f co

nju

gate

s is

alw

ays

a ra

tion

aln

um

ber.

Sim

pli

fy 2

�50�

�4�

500

��

6�12

5�

.

2�50�

�4�

500

��

6�12

5�

�2�

52�

2�

�4�

102

�5

��

6�52

�5

�F

acto

r us

ing

squa

res.

�2

�5

��

2��

4 �

10 �

�5�

�6

�5

��

5�S

impl

ify s

quar

e ro

ots.

�10

�2�

�40

�5�

�30

�5�

Mul

tiply

.

�10

�2�

�10

�5�

Com

bine

like

rad

ical

s.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Rad

ical

Exp

ress

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-6

5-6

Exam

ple1

Exam

ple1

Sim

pli

fy (2

�3�

�4�

2�)(�

3��

2�2�)

.

(2�

3��

4�2�)

(�3�

�2�

2�)�

2�3�

��

3��

2�3�

�2�

2��

4�2�

��

3��

4�2�

�2�

2��

6 �

4�6�

�4�

6��

16�

�10

Sim

pli

fy

.

��

� � �11

�5�

5��

� 4

6 �

5�5�

�5

��

9 �

5

6 �

2 �5�

�3�

5��

(�5�)

2�

��

32�

(�5� )

2

3 �

�5�

� 3 �

�5�

2 �

�5�

� 3 �

�5�

2 �

�5�

� 3 �

�5�

2 �

�5�

� 3 �

�5�

Exam

ple2

Exam

ple2

Exam

ple3

Exam

ple3

Exer

cises

Exer

cises

Sim

pli

fy.

1.3 �

2��

�50�

�4�

8�2.

�20�

��

125

��

�45�

3.�

300

��

�27�

��

75�

04�

5�2�

3�

4.3 �

81��

3 �24�

5.3 �

2�(3 �

4��

3 �12�

)6.

2�3�(

�15�

��

60�)

63 �

9�2

�2

3 �3�

18�

5�

7.(2

�3�

7�)(4

��

7�)8.

(6�

3��

4�2�)

(3�

3��

�2�)

9.(4

�2�

�3�

5�)(2

�20

��

5�)

29 �

14�

7�46

�6�

6�40

�2�

�30

�5�

10.

511

.5

�3�

2�12

.13

�3�

�23

�� 11

5 �

3�3�

��

1 �

2�3�

4 �

�2�

� 2 �

�2�

5�48�

��

75��

�5�

3�



Skil

ls P

ract

ice

Rad

ical

Exp

ress

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-6

5-6

©G

lenc

oe/M

cGra

w-H

ill27

1G

lenc

oe A

lgeb

ra 2

Lesson 5-6

Sim

pli

fy.

1.�

24�2�

6�2.

�75�

5�3�

3.3 �

16�2

3 �2�

4.�

4 �48�

�2

4 �3�

5.4�

50x5

�20

x2 �

2x�6.

4 �64

a4b

�4 �

2| ab

|4 �4�

7.3 ��

d2 f

5�

f3 �

d2 f

2�

8.��

s2t

| s| �

t�

9.��

�10

.3 ��

11. �

��2 5g z3 �12

.(3 �

3�)(5

�3�)

45

13.(

4 �12�

)(3�

20�)

48�

15�14

.�2�

��

8��

�50�

8�2�

15.�

12��

2�3�

��

108

�6�

3�16

.8�

5��

�45�

��

80��

5�

17.2

�48�

��

75��

�12�

�3�

18.(

2 �

�3�)

(6 �

�2�)

12 �

2�2�

�6�

3��

�6�

19.(

1 �

�5�)

(1 �

�5�)

�4

20.(

3 �

�7�)

(5 �

�2�)

15 �

3�2�

�5�

7��

�14�

21.(

�2�

��

6�)2

8 �

4�3�

22.

23.

24.

40 �

5�6�

�� 58

5� 8

��

6�12

�4�

2��

� 74

� 3 �

�2�

21 �

3�2�

�� 47

3� 7

��

2�

g�

10g

z�

�� 5z

3 �6 �

�3

2 � 9�

21��

73 � 7

5 � 625 � 36

1 � 21 � 8

©G

lenc

oe/M

cGra

w-H

ill27

2G

lenc

oe A

lgeb

ra 2

Sim

pli

fy.

1.�

540

�6�

15�2.

3 ��

432

��

63 �

2�3.

3 �12

8�

43 �

2�

4.�

4 �40

5�

�3

4 �5�

5.3 �

�50

0�

0��

103 �

5�6.

5 ��

121

�5�

�3

5 �5�

7.3 �

125t

6 w�

2 �5t

23 �

w2

�8.

4 �48

v8z

�13 �

2v2 z

34 �

3z�9.

3 �8g

3 k8

�2g

k2

3 �k

2 �

10.�

45x3

y�

8 �3x

y4 �

5x�11

. ��

12.

3 ��3 �

9�

13. ��c

4 d7

c2 d

3 �2d�

14. ��

15.

4 ��16

.(3�

15�)(�

4�45�

)17

.(2�

24�)(7

�18�

)18

.�81

0�

��

240

��

�25

0�

�18

0�3�

168�

3�4�

10��

4�15�

19.6

�20�

�8�

5��

5�45�

20.8

�48�

�6�

75��

7�80�

21.(

3�2�

�2�

3�)2

5�5�

2�3�

�28

�5�

30 �

12�

6�

22.(

3 �

�7�)

223

.(�

5��

�6�)

(�5�

��

2�)24

.(�

2��

�10�

)(�2�

��

10�)

16 �

6�7�

5 �

�10�

��

30��

2�3�

�8

25.(

1 �

�6�)

(5 �

�7�)

26.(

�3�

�4�

7�)2

27.(

�10

8�

�6�

3�)2

5 �

�7�

�5�

6��

�42�

115

�8�

21�0

28.

�15�

�2�

3�29

.6�

2��

630

.

31.

32.

27 �

11�

6�33

.

34.B

RA

KIN

GT

he

form

ula

s�

2�5��

esti

mat

es t

he

spee

d s

in m

iles

per

hou

r of

a c

ar w

hen

it l

eave

s sk

id m

arks

�fe

et l

ong.

Use

th

e fo

rmu

la t

o w

rite

a s

impl

ifie

d ex

pres

sion

for

sif

�

�85

.Th

en e

valu

ate

sto

th

e n

eare

st m

ile

per

hou

r.10

�17�

;41

mi/h

35.P

YTH

AG

OR

EAN

TH

EOR

EMT

he

mea

sure

s of

th

e le

gs o

f a

righ

t tr

ian

gle

can

be

repr

esen

ted

by t

he

expr

essi

ons

6x2 y

and

9x2 y

.Use

th

e P

yth

agor

ean

Th

eore

m t

o fi

nd

asi

mpl

ifie

d ex

pres

sion

for

th

e m

easu

re o

f th

e h

ypot

enu

se.

3x2

| y| �

13�

6 �

5�x�

�x

��

4 �

x3

��

x�� 2

��

x�3

��

6��

�5

��

24�8

�5�

2��

� 23

��

2�� 2

��

2�

17 �

�3�

�� 13

5 �

�3�

� 4 �

�3�

6� �

2��

1�

3��

5��

2

4 �72

a�

�3a

8� 9a

33a

2 �a�

�8b

29a

5� 64

b41 � 16

1� 12

8

216

� 24�

11��

311 � 9

Pra

ctic

e (

Ave

rag

e)

Rad

ical

Exp

ress

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-6

5-6


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csR

adic

al E

xpre

ssio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-6

5-6

©G

lenc

oe/M

cGra

w-H

ill27

3G

lenc

oe A

lgeb

ra 2

Lesson 5-6

Pre-

Act

ivit

yH

ow d

o ra

dic

al e

xpre

ssio

ns

app

ly t

o fa

llin

g ob

ject

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

5-6

at

the

top

of p

age

250

in y

our

text

book

.

Des

crib

e h

ow y

ou c

ould

use

th

e fo

rmu

la g

iven

in

you

r te

xtbo

ok a

nd

aca

lcu

lato

r to

fin

d th

e ti

me,

to t

he

nea

rest

ten

th o

f a

seco

nd,

that

it

wou

ldta

ke f

or t

he

wat

er b

allo

ons

to d

rop

22 f

eet.

(Do

not

act

ual

ly c

alcu

late

th

eti

me.

)S

amp

le a

nsw

er:

Mu

ltip

ly 2

2 by

2 (

giv

ing

44)

an

d d

ivid

eby

32.

Use

th

e ca

lcu

lato

r to

fin

d t

he

squ

are

roo

t o

f th

e re

sult

.R

ou

nd

th

is s

qu

are

roo

t to

th

e n

eare

st t

enth

.

Rea

din

g t

he

Less

on

1.C

ompl

ete

the

cond

itio

ns t

hat

mus

t be

met

for

a r

adic

al e

xpre

ssio

n to

be

in s

impl

ifie

d fo

rm.

•T

he

nis

as

as p

ossi

ble.

•T

he

con

tain

s n

o (o

ther

th

an 1

) th

at a

re n

th

pow

ers

of a

(n)

or p

olyn

omia

l.

•T

he

radi

can

d co

nta

ins

no

.

•N

o ap

pear

in

th

e .

2.a.

Wh

at a

re c

onju

gate

s of

rad

ical

exp

ress

ion

s u

sed

for?

to r

atio

nal

ize

bin

om

ial

den

om

inat

ors

b.

How

wou

ld y

ou u

se a

con

juga

te t

o si

mpl

ify

the

radi

cal

expr

essi

on

?

Mu

ltip

ly n

um

erat

or

and

den

om

inat

or

by 3

��

2�.c.

In o

rder

to

sim

plif

y th

e ra

dica

l ex

pres

sion

in

par

t b,

two

mu

ltip

lica

tion

s ar

e

nece

ssar

y.T

he m

ulti

plic

atio

n in

the

num

erat

or w

ould

be

done

by

the

met

hod

,an

d th

e m

ult

ipli

cati

on i

n t

he

den

omin

ator

wou

ld b

e do

ne

by f

indi

ng

the

of

.

Hel

pin

g Y

ou

Rem

emb

er

3.O

ne w

ay t

o re

mem

ber

som

ethi

ng is

to

expl

ain

it t

o an

othe

r pe

rson

.Whe

n ra

tion

aliz

ing

the

deno

min

ator

in

the

expr

essi

on

,man

y st

uden

ts t

hink

the

y sh

ould

mul

tipl

y nu

mer

ator

and

den

omin

ator

by

.How

wou

ld y

ou e

xpla

in t

o a

clas

smat

e w

hy

this

is

inco

rrec

t

and

wh

at h

e sh

ould

do

inst

ead.

Sam

ple

an

swer

:B

ecau

se y

ou

are

wo

rkin

g w

ith

cub

e ro

ots

,no

t sq

uar

e ro

ots

,yo

u n

eed

to

mak

e th

e ra

dic

and

in t

he

den

om

inat

or

a p

erfe

ct c

ub

e,n

ot

a p

erfe

ct s

qu

are.

Mu

ltip

ly n

um

erat

or

and

den

om

inat

or

by

to m

ake

the

den

om

inat

or

3 �8�,

wh

ich

eq

ual

s 2.

3 �4�

� 3 �4 �

3 �2�

� 3 �2�

1� 3 �

2�

squ

ares

two

dif

fere

nce

FO

IL

1 �

�2�

� 3 �

�2�

den

om

inat

or

rad

ical

s

frac

tio

ns

inte

ger

fact

ors

rad

ican

d

smal

lin

dex

©G

lenc

oe/M

cGra

w-H

ill27

4G

lenc

oe A

lgeb

ra 2

Sp

ecia

l Pro

du

cts

wit

h R

adic

als

Not

ice

that

(�3�)

(�3�)

�3,

or (�

3�)2

�3.

In g

ener

al,(

�x�)

2�

xw

hen

x �

0.

Als

o,n

otic

e th

at (�

9�)(�

4�)�

�36�

.

In g

ener

al,(

�x�)

(�y�)

��

xy�w

hen

xan

d y

are

not

neg

ativ

e.

You

can

use

th

ese

idea

s to

fin

d th

e sp

ecia

l pr

odu

cts

belo

w.

(�a�

��

b�)(�

a��

�b�)

�(�

a�)2

�(�

b�)2

�a

�b

(�a�

��

b�)2

�(�

a�)2

�2�

ab��

(�b�)

2�

a�

2�ab�

�b

(�a�

��

b�)2

�(�

a�)2

�2�

ab��

(�b�)

2�

a�

2�ab�

�b

Fin

d t

he

pro

du

ct:(

�2�

��

5�)( �

2��

�5�)

.

(�2�

��

5�)(�

2��

�5�)

�(�

2�)2

�(�

5�)2

�2

�5

��

3

Eva

luat

e ( �

2��

�8�)

2 .

(�2�

��

8�)2

�(�

2�)2

�2�

2��8�

�(�

8�)2

�2

�2�

16��

8 �

2 �

2(4)

�8

�2

�8

�8

�18

Mu

ltip

ly.

1.(�

3��

�7�)

(�3�

��

7�)�

42.

(�10�

��

2�)(�

10��

�2�)

8

3.(�

2x��

�6�)

(�2x�

��

6�)2x

�6

4.(�

3��

27)2

12

5.(�

1000

��

�10�

)212

106.

(�y�

��

5�)(�

y��

�5�)

y�

5

7.(�

50��

�x�)

250

�10

�2x�

�x

8.(�

x��

20)2

x�

4�5x�

�20

You

can

ext

end

thes

e id

eas

to p

atte

rns

for

sum

s an

d di

ffer

ence

s of

cu

bes.

Stu

dy t

he

patt

ern

bel

ow.

(3 �8�

� 3 �

x�)(3 �

82 ��

3 �8x�

�3 �

x2 �)�

3 �83 �

� 3 �

x3 ��

8 �

x

Mu

ltip

ly.

9.(3 �

2��

3 �5�)

(3 �22 �

�3 �

10��

3 �52 �

)�

3

10.(

3 �y�

�3 �

w�)(

3 �y2 �

�3 �

yw��

3 �w

2�

)y

�w

11.

(3 �7�

�3 �

20�)(

3 �72 �

�3 �

140

��

3 �20

2�

)27

12.(

3 �11�

�3 �

8�)(3 �

112

��

3 �88�

�3 �

82 �)

3

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-6

5-6

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Rat

ion

al E

xpo

nen

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-7

5-7

©G

lenc

oe/M

cGra

w-H

ill27

5G

lenc

oe A

lgeb

ra 2

Lesson 5-7

Rat

ion

al E

xpo

nen

ts a

nd

Rad

ical

s

Def

init

ion

of

b� n1 �

For

any

rea

l num

ber

ban

d an

y po

sitiv

e in

tege

r n,

b�1 n��

n �b�,

exc

ept

whe

n b

0

and

nis

eve

n.

Def

init

ion

of

b�m n�

For

any

non

zero

rea

l num

ber

b, a

nd a

ny in

tege

rs m

and

n, w

ith n

�1,

b�m n��

n �bm �

�(n �

b�)m

, ex

cept

whe

n b

0

and

nis

eve

n.

Wri

te 2

8�1 2�in

rad

ical

for

m.

Not

ice

that

28

�0.

28�1 2�

��

28��

�22

�7

��

�22 �

��

7��

2�7�

Eva

luat

e �

��1 3� .

Not

ice

that

�8

0,

�12

5

0,an

d 3

is o

dd.

��1 3�

� � �2 � 5�

2� �

53 ��

8�

� 3 ��

125

��

8� �

125

�8

� �12

5Ex

ampl

e1Ex

ampl

e1Ex

ampl

e2Ex

ampl

e2

Exer

cises

Exer

cises

Wri

te e

ach

exp

ress

ion

in

rad

ical

for

m.

1.11

�1 7�

2.15

�1 3�

3.30

0�3 2�

7 �11�

3 �15�

�30

03�

Wri

te e

ach

rad

ical

usi

ng

rati

onal

exp

onen

ts.

4.�

47�5.

3 �3a

5 b2

�6.

4 �16

2p5

�

47�1 2�

3�1 3� a�5 3� b�2 3�

3

2�1 4�

p�5 4�

Eva

luat

e ea

ch e

xpre

ssio

n.

7.�

27�2 3�

8.9.

(0.0

004)

�1 2�

90.

02

10.8

�2 3�

�4�3 2�

11.

12.

321 � 2

1 � 4

16�

�1 2�

� (0.2

5)�1 2�

144�

�1 2�

� 27�

�1 3�

1 � 105��1 2�

� 2�5�

©G

lenc

oe/M

cGra

w-H

ill27

6G

lenc

oe A

lgeb

ra 2

Sim

plif

y Ex

pre

ssio

ns

All

th

e pr

oper

ties

of

pow

ers

from

Les

son

5-1

app

ly t

o ra

tion

alex

pon

ents

.Wh

en y

ou s

impl

ify

expr

essi

ons

wit

h r

atio

nal

exp

onen

ts,l

eave

th

e ex

pon

ent

inra

tion

al f

orm

,an

d w

rite

th

e ex

pres

sion

wit

h a

ll p

osit

ive

expo

nen

ts.A

ny

expo

nen

ts i

n t

he

den

omin

ator

mu

st b

e po

siti

ve i

nte

gers

Wh

en y

ou s

impl

ify

radi

cal

expr

essi

ons,

you

may

use

rat

ion

al e

xpon

ents

to

sim

plif

y,bu

t yo

ur

answ

er s

hou

ld b

e in

rad

ical

for

m.U

se t

he

smal

lest

in

dex

poss

ible

.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Rat

ion

al E

xpo

nen

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-7

5-7

Sim

pli

fy y

�2 3�

y�3 8� .

y�2 3��

y�3 8��

y�2 3��

�3 8��

y�2 25 4�

Sim

pli

fy

4 �14

4x6

�.

4 �14

4x6

��

(144

x6)�1 4�

�(2

4�

32�

x6)�1 4�

�(2

4 )�1 4�

�(3

2 )�1 4�

�(x

6 )�1 4�

�2

�3�1 2�

�x�3 2�

�2x

�(3

x)�1 2�

�2x

�3x�

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sim

pli

fy e

ach

exp

ress

ion

.

1.x�4 5�

�x�6 5�

2.� y�2 3� ��3 4�

3.p�4 5�

�p� 17 0�

x2

y�1 2�

p�3 2�

4.� m

��6 5� ��2 5�

5.x�

�3 8��

x�4 3�6.

� s��1 6� ��

�4 3�

x�2 23 4�

s�2 9�

7.8.

� a�2 3� ��6 5�

�� a

�2 5� �39.

p�2 3�

a2

10.

6 �12

8�

11.

4 �49�

12.

5 �28

8�

2�2�

�7�

25 �

9�

13.�

32��

3�16�

14.

3 �25�

��

125

�15

.6 �

16�

48�

2�25

6 �5�

3 �4�

16.

17.�

�3 �48�

18.

6 �48�

�a�

6 �b

5 ��

bx

�3�

�6 �

35 ��

� 6

a3 �

b4 ��

ab3

�x

�3 �

3��

�12�

x�5 6�

� xx��1 2�

� x��1 3�

p � p�1 3�1� m

3


An

swer

s


Skil

ls P

ract

ice

Rat

ion

al E

xpo

nen

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-7

5-7

©G

lenc

oe/M

cGra

w-H

ill27

7G

lenc

oe A

lgeb

ra 2

Lesson 5-7

Wri

te e

ach

exp

ress

ion

in

rad

ical

for

m.

1.3�1 6�

6 �3�

2.8�1 5�

5 �8�

3.12

�2 3�3 �

122

�o

r (3 �

12�)2

4.(s

3 )�3 5�

s5 �

s4 �

Wri

te e

ach

rad

ical

usi

ng

rati

onal

exp

onen

ts.

5.�

51�51

�1 2�6.

3 �37�

37�1 3�

7.4 �

153

�15

�3 4�8.

3 �6x

y2�

6�1 3� x�1 3� y

�2 3�

Eva

luat

e ea

ch e

xpre

ssio

n.

9.32

�1 5�2

10.8

1�1 4�3

11.2

7��1 3�

12.4

��1 2�

13.1

6�3 2�64

14.(

�24

3)�4 5�

81

15.2

7�1 3��

27�5 3�

729

16. �

��3 2�

Sim

pli

fy e

ach

exp

ress

ion

.

17.c

�1 52 ��

c�3 5�c

318

.m�2 9�

�m

�1 96 �m

2

19.� q

�1 2� �3q

�3 2�20

.p�

�1 5�

21.x

�� 16 1�

22.

x� 15 2�

23.

24.

25.

12 �64�

�2�

26.

8 �49

a8b

�2 �

| a|4 �

7b�

n�2 3�

� nn

�1 3�

� n�1 6�

�n

�1 2�

y�1 4�

� yy�

�1 2�

� y�1 4�

x�2 3�

� x�1 4�

x� 15 1�

� x

p�4 5�

� p8 � 274 � 9

1 � 21 � 3

©G

lenc

oe/M

cGra

w-H

ill27

8G

lenc

oe A

lgeb

ra 2

Wri

te e

ach

exp

ress

ion

in

rad

ical

for

m.

1.5�1 3�

2.6�2 5�

3.m

�4 7�4.

(n3 )�2 5�

3 �5�

5 �62 �

or

(5 �6�)

27 �

m4

�o

r (7 �

m�)4

n5 �

n�

Wri

te e

ach

rad

ical

usi

ng

rati

onal

exp

onen

ts.

5.�

79�6.

4 �15

3�

7.3 �

27m

6 n�

4 �8.

5�2a

10b

�

79�1 2�

153�1 4�

3m2 n

�4 3�5

2�1 2�

| a5

| b�1 2�

Eva

luat

e ea

ch e

xpre

ssio

n.

9.81

�1 4�3

10.1

024�

�1 5�11

.8�

�5 3�

12.�

256�

�3 4��

13.(

�64

)��2 3�

14.2

7�1 3��

27�4 3�

243

15. �

��2 3�16

.17

.� 25�1 2� ��

64�

�1 3� � �

Sim

pli

fy e

ach

exp

ress

ion

.

18.g

�4 7��

g�3 7�g

19.s

�3 4��

s�1 43 �s

420

.� u�

�1 3� ��4 5�

u� 14 5�

21.y

��1 2�

22.b

��3 5�

23.

q�1 5�

24.

25.

26.10 �

85 �2�

2�27

.�12�

�5 �

123

�28

.4 �

6��

34 �

6�29

.

1210 �

12�3�

6�

30.E

LEC

TRIC

ITY

Th

e am

oun

t of

cu

rren

t in

am

pere

s I

that

an

app

lian

ce u

ses

can

be

calc

ula

ted

usi

ng

the

form

ula

I�

��1 2� ,w

her

e P

is t

he

pow

er i

n w

atts

an

d R

is t

he

resi

stan

ce i

n o

hm

s.H

ow m

uch

cu

rren

t do

es a

n a

ppli

ance

use

if

P�

500

wat

ts a

nd

R�

10 o

hm

s? R

oun

d yo

ur

answ

er t

o th

e n

eare

st t

enth

.7.

1 am

ps

31.B

USI

NES

SA

com

pan

y th

at p

rodu

ces

DV

Ds

use

s th

e fo

rmu

la C

�88

n�1 3�

�33

0 to

calc

ula

te t

he

cost

Cin

dol

lars

of

prod

uci

ng

nD

VD

s pe

r da

y.W

hat

is

the

com

pan

y’s

cost

to p

rodu

ce 1

50 D

VD

s pe

r da

y? R

oun

d yo

ur

answ

er t

o th

e n

eare

st d

olla

r.$7

98

P � R

a�3b�

�3b

a� �

3b�

2z�

2z�1 2�

��

z�

12z

�1 2�

� z�1 2��

1

t�1 11 2�

� 5t�2 3�

� 5t�1 2�

�t�

�3 4�

q�3 5�

� q�2 5�

b�2 5�

� b

y�1 2�

� y

5 � 416 � 49

64�2 3�

� 343�2 3�

25 � 3612

5� 21

6

1 � 161 � 64

1 � 321 � 4

Pra

ctic

e (

Ave

rag

e)

Rat

ion

al E

xpo

nen

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-7

5-7



Readin

g t

o L

earn

Math

em

ati

csR

atio

nal

Exp

on

ents

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-7

5-7

©G

lenc

oe/M

cGra

w-H

ill27

9G

lenc

oe A

lgeb

ra 2

Lesson 5-7

Pre-

Act

ivit

yH

ow d

o ra

tion

al e

xpon

ents

ap

ply

to

astr

onom

y?

Rea

d th

e in

trod

uct

ion

to

Les

son

5-7

at

the

top

of p

age

257

in y

our

text

book

.

The

for

mul

a in

the

int

rodu

ctio

n co

ntai

ns t

he e

xpon

ent

.Wha

t do

you

thi

nk

it m

igh

t m

ean

to

rais

e a

nu

mbe

r to

th

e po

wer

?

Sam

ple

an

swer

:Tak

e th

e fi

fth

ro

ot

of

the

nu

mb

er a

nd

sq

uar

e it

.

Rea

din

g t

he

Less

on

1.C

ompl

ete

the

foll

owin

g de

fin

itio

ns

of r

atio

nal

exp

onen

ts.

•F

or a

ny

real

nu

mbe

r b

and

for

any

posi

tive

in

tege

r n

,b� n1 �

�ex

cept

wh

en b

and

nis

.

•F

or a

ny

non

zero

rea

l n

um

ber

b,an

d an

y in

tege

rs m

and

n,w

ith

n,

b�m n��

�

,exc

ept

wh

en b

and

nis

.

2.C

ompl

ete

the

con

diti

ons

that

mu

st b

e m

et i

n o

rder

for

an

exp

ress

ion

wit

h r

atio

nal

expo

nen

ts t

o be

sim

plif

ied.

•It

has

no

expo

nen

ts.

•It

has

no

expo

nen

ts i

n t

he

.

•It

is

not

a

frac

tion

.

•T

he

of a

ny

rem

ain

ing

is t

he

nu

mbe

r po

ssib

le.

3.M

arga

rita

an

d P

ierr

e w

ere

wor

kin

g to

geth

er o

n t

hei

r al

gebr

a h

omew

ork.

On

e ex

erci

se

aske

d th

em t

o ev

alu

ate

the

expr

essi

on 2

7�4 3� .Mar

gari

ta t

hou

ght

that

th

ey s

hou

ld r

aise

27

to

the

fou

rth

pow

er f

irst

an

d th

en t

ake

the

cube

roo

t of

th

e re

sult

.Pie

rre

thou

ght

that

they

sh

ould

tak

e th

e cu

be r

oot

of 2

7 fi

rst

and

then

rai

se t

he

resu

lt t

o th

e fo

urt

h p

ower

.W

hos

e m

eth

od i

s co

rrec

t?B

oth

met

ho

ds

are

corr

ect.

Hel

pin

g Y

ou

Rem

emb

er

4.S

ome

stu

den

ts h

ave

trou

ble

rem

embe

rin

g w

hic

h p

art

of t

he

frac

tion

in

a r

atio

nal

expo

nen

t gi

ves

the

pow

er a

nd

wh

ich

par

t gi

ves

the

root

.How

can

you

r kn

owle

dge

ofin

tege

r ex

pon

ents

hel

p yo

u t

o ke

ep t

his

str

aigh

t?S

amp

le a

nsw

er:

An

inte

ger

ex

po

nen

t ca

n b

e w

ritt

en a

s a

rati

on

al e

xpo

nen

t.F

or

exam

ple

,23

�2�3 1� .

You

kn

ow

th

at t

his

mea

ns

that

2 is

rai

sed

to

th

e th

ird

po

wer

,so

th

en

um

erat

or

mu

st g

ive

the

po

wer

,an

d,t

her

efo

re,t

he

den

om

inat

or

mu

stg

ive

the

roo

t.

leas

tra

dic

alin

dexco

mp

lex

den

om

inat

or

frac

tio

nal

neg

ativ

e

even

�0

(n �b�)

mn �

bm �

�1

even

�0

n �b �

2 � 5

2 � 5

©G

lenc

oe/M

cGra

w-H

ill28

0G

lenc

oe A

lgeb

ra 2

Les

ser-

Kn

ow

n G

eom

etri

c F

orm

ula

sM

any

geom

etri

c fo

rmu

las

invo

lve

radi

cal

expr

essi

ons.

Mak

e a

dra

win

g to

ill

ust

rate

eac

h o

f th

e fo

rmu

las

give

n o

n t

his

pag

e.T

hen

eva

luat

e th

e fo

rmu

la f

or t

he

give

n v

alu

e of

th

e va

riab

le.R

oun

d

answ

ers

to t

he

nea

rest

hu

nd

red

th.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-7

5-7

1.T

he

area

of

an i

sosc

eles

tri

angl

e.T

wo

side

s h

ave

len

gth

a;t

he

oth

er s

ide

has

len

gth

c.F

ind

Aw

hen

a�

6 an

d c

�7.

A�

� 4c � �4a

2�

�c2 �

A�

17.0

6

3.T

he

area

of

a re

gula

r pe

nta

gon

wit

h a

side

of

len

gth

a.F

ind

Aw

hen

a�

4.

A�

�a 42 ��

25 �

�10

�5�

�A

�27

.53

5.T

he

volu

me

of a

reg

ula

r te

trah

edro

nw

ith

an

edg

e of

len

gth

a.F

ind

Vw

hen

a

�2.

V�

� 1a 23 ��

2�

V�

0.94

7.H

eron

’s F

orm

ula

for

th

e ar

ea o

f a

tria

ngl

e u

ses

the

sem

i-pe

rim

eter

s,

wh

ere

s�

�a�

2b�

c�

.Th

e si

des

of t

he

tria

ngl

e h

ave

len

gth

s a,

b,an

d c.

Fin

d A

wh

en a

�3,

b�

4,an

d c

�5.

A�

�s(

s�

�a)

(s�

�b)

(s�

�c)�

A�

6

2.T

he

area

of

an e

quil

ater

al t

rian

gle

wit

ha

side

of

len

gth

a.F

ind

Aw

hen

a�

8.

A�

�a 42 ��

3�

A�

27.7

1

4.T

he

area

of

a re

gula

r h

exag

on w

ith

asi

de o

f le

ngt

h a

.Fin

d A

wh

en a

�9.

A�

�3 2a2 ��

3�

A�

210.

44

6.T

he

area

of

the

curv

ed s

urf

ace

of a

rig

ht

con

e w

ith

an

alt

itu

de o

f h

and

radi

us

ofba

se r

.Fin

d S

wh

en r

�3

and

h�

6.

S�

�r�

r2�

h�

2 �S

�63

.22

8.T

he r

adiu

s of

a c

ircl

e in

scri

bed

in a

giv

entr

ian

gle

also

use

s th

e se

mi-

peri

met

er.

Fin

d r

wh

en a

�6,

b�

7,an

d c

�9.

r�

r�

1.91

rb ca

�s(

s�

�a)

(s�

�b)

(s�

�c)�

��

�s

r

h

aa

aa

aa

a

a

a

a

b

c

aa

a

aa

a

aa

a

a

a

a

c

a


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Rad

ical

Eq

uat

ion

s an

d In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-8

5-8

©G

lenc

oe/M

cGra

w-H

ill28

1G

lenc

oe A

lgeb

ra 2

Lesson 5-8

Solv

e R

adic

al E

qu

atio

ns

Th

e fo

llow

ing

step

s ar

e u

sed

in s

olvi

ng

equ

atio

ns

that

hav

eva

riab

les

in t

he

radi

can

d.S

ome

alge

brai

c pr

oced

ure

s m

ay b

e n

eede

d be

fore

you

use

th

ese

step

s.

Ste

p 1

Isol

ate

the

radi

cal o

n on

e si

de o

f th

e eq

uatio

n.S

tep

2To

elim

inat

e th

e ra

dica

l, ra

ise

each

sid

e of

the

equ

atio

n to

a p

ower

equ

al t

o th

e in

dex

of t

he r

adic

al.

Ste

p 3

Sol

ve t

he r

esul

ting

equa

tion.

Ste

p 4

Che

ck y

our

solu

tion

in th

e or

igin

al e

quat

ion

to m

ake

sure

that

you

hav

e no

t obt

aine

d an

y ex

tran

eous

roo

ts.

Sol

ve 2

�4x

��

8��

4 �

8.

2�4x

�8

��

4 �

8O

rigin

al e

quat

ion

2�4x

�8

��

12A

dd 4

to

each

sid

e.

�4x

�8

��

6Is

olat

e th

e ra

dica

l.

4x�

8 �

36S

quar

e ea

ch s

ide.

4x�

28S

ubtr

act

8 fr

om e

ach

side

.

x�

7D

ivid

e ea

ch s

ide

by 4

.

Ch

eck

2�4(

7) �

�8�

�4

�8

2�36�

�4

�8

2(6)

�4

�8

8 �

8T

he

solu

tion

x�

7 ch

ecks

.

Sol

ve �

3x�

�1�

��

5x��

1.

�3x

�1

��

�5x�

�1

Orig

inal

equ

atio

n

3x�

1�

5x�

2�5x�

�1

Squ

are

each

sid

e.

2�5x�

�2x

Sim

plify

.

�5x�

�x

Isol

ate

the

radi

cal.

5x�

x2S

quar

e ea

ch s

ide.

x2�

5x�

0S

ubtr

act

5xfr

om e

ach

side

.

x(x

�5)

�0

Fac

tor.

x�

0 or

x�

5C

hec

k�

3(0)

��

1��

1,bu

t �

5(0)

��

1 �

�1,

so 0

is

not

a s

olu

tion

.�

3(5)

��

1��

4,an

d �

5(5)

��

1 �

4,so

th

eso

luti

on i

s x

�5.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sol

ve e

ach

eq

uat

ion

.

1.3

�2x

�3�

�5

2.2�

3x�

4�

�1

�15

3.8

��

x�

1�

�2

15n

o s

olu

tio

n

4.�

5 �

x�

�4

�6

5.12

��

2x�

1�

�4

6.�

12 �

�x�

�0

�95

no

so

luti

on

12

7.�

21��

�5x

�4

��

08.

10 �

�2x�

�5

9.�

x2�

7�

x��

�7x

�9

�

512

.5n

o s

olu

tio

n

10.4

3 �2x

�1

�1�

�2

�10

11.2

�x

�11

��

�x

�2

��

�3x

�6

�12

.�9x

�1

�1�

�x

�1

814

3,4

�3�

�3

©G

lenc

oe/M

cGra

w-H

ill28

2G

lenc

oe A

lgeb

ra 2

Solv

e R

adic

al In

equ

alit

ies

A r

adic

al i

neq

ual

ity

is a

n i

neq

ual

ity

that

has

a v

aria

ble

in a

rad

ican

d.U

se t

he

foll

owin

g st

eps

to s

olve

rad

ical

in

equ

alit

ies.

Ste

p 1

If th

e in

dex

of t

he r

oot

is e

ven,

iden

tify

the

valu

es o

f th

e va

riabl

e fo

r w

hich

the

rad

ican

d is

non

nega

tive.

Ste

p 2

Sol

ve t

he in

equa

lity

alge

brai

cally

.S

tep

3Te

st v

alue

s to

che

ck y

our

solu

tion.

Sol

ve 5

��

20x

��

4�

�3.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Rad

ical

Eq

uat

ion

s an

d In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-8

5-8

Exam

ple

Exam

ple

Sin

ce t

he

radi

can

d of

a s

quar

e ro

otm

ust

be

grea

ter

than

or

equ

al t

oze

ro,f

irst

sol

ve20

x�

4 �

0.20

x�

4 �

020

x�

�4

x�

�1 � 5

Now

sol

ve 5

��

20x

��

4��

�3.

5 �

�20

x�

�4�

��

3O

rigin

al in

equa

lity

�20

x�

�4�

�8

Isol

ate

the

radi

cal.

20x

�4

�64

Elim

inat

e th

e ra

dica

l by

squa

ring

each

sid

e.

20x

�60

Sub

trac

t 4

from

eac

h si

de.

x�

3D

ivid

e ea

ch s

ide

by 2

0.

It a

ppea

rs t

hat

��

x�

3 is

th

e so

luti

on.T

est

som

e va

lues

.

x�

�1

x�

0x

�4

�20

(�1

�)

�4

�is

not

a r

eal

5 �

�20

(0)

��

4�

�3,

so

the

5 �

�20

(4)

��

4�

��

4.2,

so

num

ber,

so t

he in

equa

lity

isin

equa

lity

is s

atis

fied.

th

e in

equa

lity

is n

otno

t sa

tisfie

d.sa

tisfie

d

Th

eref

ore

the

solu

tion

��

x�

3 ch

ecks

.

Sol

ve e

ach

in

equ

alit

y.

1.�

c�

2�

�4

�7

2.3�

2x�

1�

�6

15

3.�

10x

��

9��

2 �

5

c

11�

x�

5x

�4

4.5

3 �x

�2

��

8

25.

8 �

�3x

�4

��

36.

�2x

�8

��

4 �

2

x�

6�

�x

�7

x�

14

7.9

��

6x�

3�

�6

8.�

49.

2�5x

�6

��

1

5

��

x�

1x

8

�x

�3

10.�

2x�

1�

2��

4 �

1211

.�2d

��

1��

�d�

�5

12.4

�b

�3

��

�b

�2

��

10

x

260

�d

�4

b

6

6 � 51 � 2

20�

��

3x�

1�4 � 3

1 � 2

1 � 5

1 � 5

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Rad

ical

Eq

uat

ion

s an

d In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-8

5-8

©G

lenc

oe/M

cGra

w-H

ill28

3G

lenc

oe A

lgeb

ra 2

Lesson 5-8

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

1.�

x��

525

2.�

x��

3 �

716

3.5�

j��1

4.v�1 2�

�1

�0

no

so

luti

on

5.18

�3y

�1 2��

25n

o s

olu

tio

n6.

3 �2w�

�4

32

7.�

b�

5�

�4

218.

�3n

��

1��

58

9.3 �

3r�

6�

�3

1110

.2 �

�3p

��

7��

63

11.�

k�

4�

�1

�5

4012

.(2

d�

3)�1 3�

�2

13.(

t�

3)�1 3�

�2

1114

.4 �

(1 �

7u)�1 3�

�0

�9

15.�

3z�

2�

��

z�

4�

no

so

luti

on

16.�

g�

1�

��

2g�

�7�

8

17.�

x�

1�

�4�

x�

1�

no

so

luti

on

18.5

��

s�

3�

�6

3 �

s�

4

19.�

2 �

�3x

�3

�

7�

1 �

x�

2620

.��

2a�

�4�

��

6�

2 �

a�

16

21.2

�4r

�3

��

10r

�7

22.4

��

3x�

1�

�3

��

x�

0

23.�

y�

4�

�3

�3

y

3224

.�3�

11r

��

3��

�15

��

r�

23 � 11

1 � 3

5 � 2

1 � 25

©G

lenc

oe/M

cGra

w-H

ill28

4G

lenc

oe A

lgeb

ra 2

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

1.�

x��

864

2.4

��

x��

31

3.�

2p��

3 �

104.

4�3h�

�2

�0

5.c�1 2�

�6

�9

96.

18 �

7h�1 2�

�12

no

so

luti

on

7.3 �

d�

2�

�7

341

8.5 �

w�

7�

�1

8

9.6

�3 �

q�

4�

�9

3110

.4 �

y�

9�

�4

�0

no

so

luti

on

11.�

2m�

�6�

�16

�0

131

12.

3 �4m

��

1��

2 �

2

13. �

8n�

�5�

�1

�2

14.�

1 �

4�

t��8

��

6�

15.�

2t�

5�

�3

�3

16.(

7v�

2)�1 4�

�12

�7

no

so

luti

on

17.(

3g�

1)�1 2�

�6

�4

3318

.(6u

�5)

�1 3��

2 �

�3

�20

19.�

2d�

�5�

��

d�

1�

420

.�4r

�6

��

�r�

2

21.�

6x�

4�

��

2x�

1�

0�22

.�2x

�5

��

�2x

�1

�n

o s

olu

tio

n

23.3

�a�

�12

a

1624

.�z

�5

��

4 �

13�

5 �

z�

76

25.8

��

2q��

5n

o s

olu

tio

n26

.�2a

��

3�

5�

a�

14

27.9

��

c�

4�

�6

c

528

.3 �

x�

1�

�

2x

��

7

29.S

TATI

STIC

SS

tati

stic

ian

s u

se t

he

form

ula

��

�v�

to c

alcu

late

a s

tan

dard

dev

iati

on �

,w

her

e v

is t

he

vari

ance

of

a da

ta s

et.F

ind

the

vari

ance

wh

en t

he

stan

dard

dev

iati

on

is 1

5.22

5

30.G

RA

VIT

ATI

ON

Hel

ena

drop

s a

ball

fro

m 2

5 fe

et a

bove

a l

ake.

Th

e fo

rmu

la

t�

�25

��

h�de

scri

bes

the

tim

e t

in s

econ

ds t

hat

th

e ba

ll i

s h

feet

abo

ve t

he

wat

er.

How

man

y fe

et a

bove

th

e w

ater

wil

l th

e ba

ll b

e af

ter

1 se

con

d?9

ft

1 � 4

3 � 2

7 � 2

41 � 2

3 � 47 � 4

63 � 4

1 � 1249 � 2

Pra

ctic

e (

Ave

rag

e)

Rad

ical

Eq

uat

ion

s an

d In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-8

5-8


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csR

adic

al E

qu

atio

ns

and

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-8

5-8

©G

lenc

oe/M

cGra

w-H

ill28

5G

lenc

oe A

lgeb

ra 2

Lesson 5-8

Pre-

Act

ivit

yH

ow d

o ra

dic

al e

qu

atio

ns

app

ly t

o m

anu

fact

uri

ng?

Rea

d th

e in

trod

uct

ion

to

Les

son

5-8

at

the

top

of p

age

263

in y

our

text

book

.

Exp

lain

how

you

wou

ld u

se t

he

form

ula

in

you

r te

xtbo

ok t

o fi

nd

the

cost

of

prod

ucin

g 12

5,00

0 co

mpu

ter

chip

s.(D

escr

ibe

the

step

s of

the

cal

cula

tion

in t

heor

der

in w

hich

you

wou

ld p

erfo

rm t

hem

,but

do

not

actu

ally

do

the

calc

ulat

ion.

)

Sam

ple

an

swer

:R

aise

125

,000

to

th

e p

ow

er b

y ta

kin

g t

he

cub

e ro

ot

of

125,

000

and

sq

uar

ing

th

e re

sult

(o

r ra

ise

125,

000

to t

he

po

wer

by

ente

rin

g 1

25,0

00 ^

(2/

3) o

n a

cal

cula

tor)

.

Mu

ltip

ly t

he

nu

mb

er y

ou

get

by

10 a

nd

th

en a

dd

150

0.

Rea

din

g t

he

Less

on

1.a.

Wh

at i

s an

ext

ran

eou

s so

luti

onof

a r

adic

al e

quat

ion

?S

amp

le a

nsw

er:

a n

um

ber

that

sat

isfi

es a

n e

qu

atio

n o

bta

ined

by

rais

ing

bo

th s

ides

of

the

ori

gin

aleq

uat

ion

to

a h

igh

er p

ow

er b

ut

do

es n

ot

sati

sfy

the

ori

gin

al e

qu

atio

n

b.

Des

crib

e tw

o w

ays

you

can

ch

eck

the

prop

osed

sol

uti

ons

of a

rad

ical

equ

atio

n i

n o

rder

to d

eter

min

e w

het

her

an

y of

th

em a

re e

xtra

neo

us

solu

tion

s.S

amp

le a

nsw

er:

On

ew

ay is

to

ch

eck

each

pro

po

sed

so

luti

on

by

sub

stit

uti

ng

it in

to t

he

ori

gin

al e

qu

atio

n.A

no

ther

way

is t

o u

se a

gra

ph

ing

cal

cula

tor

to g

rap

hb

oth

sid

es o

f th

e o

rig

inal

eq

uat

ion

.See

wh

ere

the

gra

ph

s in

ters

ect.

Th

is c

an h

elp

yo

u id

enti

fy s

olu

tio

ns

that

may

be

extr

aneo

us.

2.C

ompl

ete

the

step

s th

at s

hou

ld b

e fo

llow

ed i

n o

rder

to

solv

e a

radi

cal

ineq

ual

ity.

Ste

p 1

If t

he

of t

he

root

is

,ide

nti

fy t

he

valu

es o

f

the

vari

able

for

wh

ich

th

e is

.

Ste

p 2

Sol

ve t

he

alge

brai

call

y.

Ste

p 3

Tes

t to

ch

eck

you

r so

luti

on.

Hel

pin

g Y

ou

Rem

emb

er

3.O

ne

way

to

rem

embe

r so

met

hin

g is

to

expl

ain

it

to a

not

her

per

son

.Su

ppos

e th

at y

our

frie

nd

Leo

ra t

hin

ks t

hat

sh

e do

es n

ot n

eed

to c

hec

k h

er s

olu

tion

s to

rad

ical

equ

atio

ns

bysu

bsti

tuti

on b

ecau

se s

he

know

s sh

e is

ver

y ca

refu

l an

d se

ldom

mak

es m

ista

kes

in h

erw

ork.

How

can

you

exp

lain

to

her

th

at s

he

shou

ld n

ever

thel

ess

chec

k ev

ery

prop

osed

solu

tion

in

th

e or

igin

al e

quat

ion

?S

amp

le a

nsw

er:

Sq

uar

ing

bo

th s

ides

of

aneq

uat

ion

can

pro

du

ce a

n e

qu

atio

n t

hat

is n

ot

equ

ival

ent

to t

he

ori

gin

alo

ne.

Fo

r ex

amp

le,t

he

on

ly s

olu

tio

n o

f x

�5

is 5

,bu

t th

e sq

uar

edeq

uat

ion

x2

�25

has

tw

o s

olu

tio

ns,

5 an

d �

5.

valu

es

ineq

ual

ity

no

nn

egat

ive

rad

ican

d

even

ind

ex

2 � 3

2 � 3

©G

lenc

oe/M

cGra

w-H

ill28

6G

lenc

oe A

lgeb

ra 2

Tru

th T

able

sIn

mat

hem

atic

s,th

e ba

sic

oper

atio

ns

are

addi

tion

,su

btra

ctio

n,m

ult

ipli

cati

on,

divi

sion

,fin

din

g a

root

,an

d ra

isin

g to

a p

ower

.In

log

ic,t

he

basi

c op

erat

ion

s ar

e th

e fo

llow

ing:

not

(),

and

(�

),or

(�

),an

dim

plie

s(→

).

If P

and

Qar

e st

atem

ents

,th

en

Pm

ean

s n

ot P

;Q

mea

ns

not

Q;P

� Q

mea

ns

Pan

d Q

;P �

Qm

ean

s P

or Q

;an

d P

→ Q

mea

ns

Pim

plie

s Q

.Th

e op

erat

ion

s ar

e de

fin

ed b

y tr

uth

tab

les.

On

th

e le

ft b

elow

is

the

tru

th t

able

for

th

e st

atem

ent

P

.Not

ice

that

th

ere

are

two

poss

ible

con

diti

ons

for

P,t

rue

(T)

or f

alse

(F

).If

Pis

tru

e,

Pis

fal

se;i

f P

is f

alse

,P

is t

rue.

Als

o sh

own

are

th

e tr

uth

tab

les

for

P �

Q,P

�Q

,an

d P

→ Q

.

P

PP

QP

� Q

PQ

P �

QP

QP

→ Q

TF

TT

TT

TT

TT

TF

TT

FF

TF

TT

FF

FT

FF

TT

FT

TF

FF

FF

FF

FT

You

can

use

th

is i

nfo

rmat

ion

to

fin

d ou

t u

nde

r w

hat

con

diti

ons

a co

mpl

ex

stat

emen

t is

tru

e. Un

der

wh

at c

ond

itio

ns

is

P

Qtr

ue?

Cre

ate

the

tru

th t

able

for

th

e st

atem

ent.

Use

th

e in

form

atio

n f

rom

th

e tr

uth

ta

ble

abov

e fo

r P

Q

to c

ompl

ete

the

last

col

um

n.

PQ

P

P

�Q

TT

FT

Wh

en o

ne

stat

emen

t is

T

FF

Ftr

ue

and

on

e is

fal

se,

FT

TT

the

con

jun

ctio

n is

tru

e.F

FT

T

The

tru

th t

able

indi

cate

s th

at

P �

Qis

tru

e in

all

case

s ex

cept

whe

re P

is t

rue

and

Qis

fal

se.

Use

tru

th t

able

s to

det

erm

ine

the

con

dit

ion

s u

nd

er w

hic

h e

ach

sta

tem

ent

is t

rue.

1.

P �

Q

2.

P →

(P →

Q)

all e

xcep

t w

her

e b

oth

all

Pan

d Q

are

tru

e3.

(P �

Q)

�(

P �

Q

)4.

(P →

Q)

�(Q

→ P

)al

lal

l

5.(P

→ Q

)�

(Q →

P)

6.(

P �

Q

)→

(P

�Q

)b

oth

Pan

d Q

are

tru

e;al

lb

oth

Pan

d Q

are

fals

e

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-8

5-8

Exam

ple

Exam

ple



To

solv

e a

quad

rati

c eq

uat

ion

th

at d

oes

not

hav

e re

al s

olu

tion

s,yo

u c

an u

se t

he

fact

th

ati2

��

1 to

fin

d co

mpl

ex s

olu

tion

s.

Sol

ve 2

x2�

24 �

0.

2x2

�24

�0

Orig

inal

equ

atio

n

2x2

��

24S

ubtr

act

24 f

rom

eac

h si

de.

x2�

�12

Div

ide

each

sid

e by

2.

x�

�

�12

�Ta

ke t

he s

quar

e ro

ot o

f ea

ch s

ide.

x�

2i

�3�

��

12�

��

4��

��

1�

��

3�

Sim

pli

fy.

1.(�

4 �

2i)

�(6

�3i

)2.

(5 �

i) �

(3 �

2i)

3.(6

�3i

) �

(4 �

2i)

2 �

i2

�i

10 �

5i

4.(�

11 �

4i)

�(1

�5i

)5.

(8 �

4i)

�(8

�4i

)6.

(5 �

2i)

�(�

6 �

3i)

�12

�9i

1611

�5i

7.(1

2 �

5i)

�(4

�3i

)8.

(9 �

2i)

�(�

2 �

5i)

9.(1

5 �

12i)

�(1

1 �

13i)

8 �

8i7

�7i

26 �

25i

10.i

411

.i6

12.i

15

1�

1�

i

Sol

ve e

ach

eq

uat

ion

.

13.5

x2�

45 �

014

.4x2

�24

�0

15.�

9x2

�9

�3i

�i�

6��

i

Stu

dy G

uid

e a

nd I

nte

rven

tion

Co

mp

lex

Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-9

5-9

©G

lenc

oe/M

cGra

w-H

ill28

7G

lenc

oe A

lgeb

ra 2

Lesson 5-9

Ad

d a

nd

Su

btr

act

Co

mp

lex

Nu

mb

ers

Aco

mpl

ex n

umbe

r is

any

num

ber

that

can

be

writ

ten

in t

he f

orm

a�

bi,

Co

mp

lex

Nu

mb

erw

here

aan

d b

are

real

num

bers

and

iis

the

imag

inar

y un

it (i

2�

�1)

.a

is c

alle

d th

e re

al p

art,

and

bis

cal

led

the

imag

inar

y pa

rt.

Ad

dit

ion

an

d

Com

bine

like

ter

ms.

Su

btr

acti

on

of

(a�

bi)

�(c

�di

) �

(a�

c) �

(b�

d)i

Co

mp

lex

Nu

mb

ers

(a�

bi)

�(c

�di

) �

(a�

c) �

(b�

d)i

Sim

pli

fy (

6 �

i)

�(4

�5i

).

(6 �

i) �

(4 �

5i)

�(6

�4)

�(1

�5)

i�

10 �

4i

Sim

pli

fy (

8 �

3i)

�(6

�2i

).

(8 �

3i)

�(6

�2i

)�

(8 �

6) �

[3 �

(�2)

]i�

2 �

5i

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exam

ple

3Ex

ampl

e 3

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill28

8G

lenc

oe A

lgeb

ra 2

Mu

ltip

ly a

nd

Div

ide

Co

mp

lex

Nu

mb

ers

Mu

ltip

licat

ion

of

Co

mp

lex

Nu

mb

ers

Use

the

def

initi

on o

f i2

and

the

FO

ILm

etho

d:(a

�bi

)(c

�di

) �

(ac

�bd

) �

(ad

�bc

)i

To

divi

de b

y a

com

plex

nu

mbe

r,fi

rst

mu

ltip

ly t

he

divi

den

d an

d di

viso

r by

th

e co

mp

lex

con

juga

teof

th

e di

viso

r.

Co

mp

lex

Co

nju

gat

ea

�bi

and

a�

biar

e co

mpl

ex c

onju

gate

s. T

he p

rodu

ct o

f co

mpl

ex c

onju

gate

s is

al

way

s a

real

num

ber.

Sim

pli

fy (

2 �

5i)

(�

4 �

2i).

(2 �

5i)

�(�

4 �

2i)

�2(

�4)

�2(

2i)

�(�

5i)(

�4)

�(�

5i)(

2i)

FO

IL

��

8 �

4i�

20i

�10

i2M

ultip

ly.

��

8 �

24i

�10

(�1)

Sim

plify

.

�2

�24

iS

tand

ard

form

Sim

pli

fy

.

��

Use

the

com

plex

con

juga

te o

f th

e di

viso

r.

�M

ultip

ly.

�i2

��

1

��

iS

tand

ard

form

Sim

pli

fy.

1.(2

�i)

(3 �

i)7

�i

2.(5

�2i

)(4

�i)

18 �

13i

3.(4

�2i

)(1

�2i

)�

10i

4.(4

�6i

)(2

�3i

)26

5.(2

�i)

(5 �

i)11

�3i

6.(5

�3i

)(�

1 �

i)�

8 �

2i

7.(1

�i)

(2 �

2i)(

3 �

3i)

8.(4

�i)

(3 �

2i)(

2 �

i)9.

(5 �

2i)(

1 �

i)(3

�i)

12 �

12i

31 �

12i

16 �

18i

10.

�i

11.

��

i12

.�

�2i

13.

1 �

i14

.�

�2i

15.

��

i

16.

�17

.�

1 �

2i�

2�18

.�

2 i�

6 ��

3�

3 ��

3�

6��

i�3�

��

�2�

�i

4 �

i�2�

��

i�2�

3 i�

5 ��

72 � 7

3 �

i�5�

��

3 �

i�5�

31 � 418 � 41

3 �

4i� 4

�5i

1 � 2�

5 �

3i�2

�2i

4 �

2i� 3

�i

5 � 36

�5i

�3i

7 � 213 � 2

7 �

13i

�2i

1 � 23 � 2

5� 3

�i

11 � 133 � 133

�11

i�

13

6 �

9i�

2i�

3i2

��

�4

�9i

2

2 �

3i� 2

�3i

3 �

i� 2

�3i

3 �

i� 2

�3i

3 �

i� 2

�3i

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Co

mp

lex

Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-9

5-9

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Co

mp

lex

Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-9

5-9

©G

lenc

oe/M

cGra

w-H

ill28

9G

lenc

oe A

lgeb

ra 2

Lesson 5-9

Sim

pli

fy.

1.�

�36

�6i

2.�

�19

6�

14i

3.�

�81

x6�

9| x

3| i

4.�

�23

��

��

46�

�23

�2�

5.(3

i)(�

2i)(

5i)

30i

6.i1

1�

i

7.i6

5i

8.(7

�8i

) �

(�12

�4i

)�

5 �

12i

9.(�

3 �

5i)

�(1

8 �

7i)

15 �

2i10

.(10

�4i

) �

(7 �

3i)

3 �

7i

11.(

2 �

i)(2

�3i

)1

�8i

12.(

2 �

i)(3

�5i

)11

�7i

13.(

7 �

6i)(

2 �

3i)

�4

�33

i14

.(3

�4i

)(3

�4i

)25

15.

16.

Sol

ve e

ach

eq

uat

ion

.

17.3

x2�

3 �

0�

i18

.5x2

�12

5 �

0�

5i

19.4

x2�

20 �

0�

i�5�

20.�

x2�

16 �

0�

4i

21.x

2�

18 �

0�

3i�

2�22

.8x2

�96

�0

�2i

�3�

Fin

d t

he

valu

es o

f m

and

nth

at m

ake

each

eq

uat

ion

tru

e.

23.2

0 �

12i

�5m

�4n

i4,

�3

24.m

�16

i�

3 �

2ni

3,8

25.(

4 �

m)

�2n

i�

9 �

14i

5,7

26.(

3 �

n)

�(7

m�

14)i

�1

�7i

3,2

3 �

6i�

103i

� 4 �

2i�

6 �

8i�

38

�6i

�3i

©G

lenc

oe/M

cGra

w-H

ill29

0G

lenc

oe A

lgeb

ra 2

Sim

pli

fy.

1.�

�49

�7i

2.6�

�12

�12

i�3�

3.�

�12

1�

s8 �11

s4 i

4.�

�36

a�

3 b4

�5.

��

8�

��

�32

�6.

��

15�

��

�25

�6| a

| b2 i

�a�

�16

�5�

15�

7.(�

3i)(

4i)(

�5i

)8.

(7i)

2 (6i

)9.

i42

�60

i�

294i

�1

10.i

5511

.i89

12.(

5 �

2i)

�(�

13 �

8i)

�i

i�

8 �

10i

13.(

7 �

6i)

�(9

�11

i)14

.(�

12 �

48i)

�(1

5 �

21i)

15.(

10 �

15i)

�(4

8 �

30i)

16 �

5i3

�69

i�

38 �

45i

16.(

28 �

4i)

�(1

0 �

30i)

17.(

6 �

4i)(

6 �

4i)

18.(

8 �

11i)

(8 �

11i)

18 �

26i

52�

57 �

176i

19.(

4 �

3i)(

2 �

5i)

20.(

7 �

2i)(

9 �

6i)

21.

23 �

14i

75 �

24i

22.

23.

24.

�1

�i

Sol

ve e

ach

eq

uat

ion

.

25.5

n2

�35

�0

�i�

7�26

.2m

2�

10 �

0�

i�5�

27.4

m2

�76

�0

�i�

19�28

.�2m

2�

6 �

0�

i�3�

29.�

5m2

�65

�0

�i�

13�30

.x2

�12

�0

�4i

Fin

d t

he

valu

es o

f m

and

nth

at m

ake

each

eq

uat

ion

tru

e.

31.1

5 �

28i

�3m

�4n

i5,

�7

32.(

6 �

m)

�3n

i�

�12

�27

i18

,9

33.(

3m�

4) �

(3 �

n)i

�16

�3i

4,6

34.(

7 �

n)

�(4

m�

10)i

�3

�6i

1,�

4

35.E

LEC

TRIC

ITY

Th

e im

peda

nce

in

on

e pa

rt o

f a

seri

es c

ircu

it i

s 1

�3j

ohm

s an

d th

eim

peda

nce

in

an

oth

er p

art

of t

he

circ

uit

is

7 �

5joh

ms.

Add

th

ese

com

plex

nu

mbe

rs t

ofi

nd

the

tota

l im

peda

nce

in

th

e ci

rcu

it.

8 �

2j

oh

ms

36.E

LEC

TRIC

ITY

Usi

ng

the

form

ula

E�

IZ,f

ind

the

volt

age

Ein

a c

ircu

it w

hen

th

ecu

rren

t I

is 3

�j

amps

an

d th

e im

peda

nce

Zis

3 �

2joh

ms.

11 �

3j

volt

s

3 � 4

2 �

4i� 1

�3i

7 �

i�

53

�i

� 2 �

i14

�16

i�

�11

32

� 7 �

8i

�5

�6i

�2

6 �

5i�

�2i

Pra

ctic

e (

Ave

rag

e)

Co

mp

lex

Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-9

5-9



Readin

g t

o L

earn

Math

em

ati

csC

om

ple

x N

um

ber

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-9

5-9

©G

lenc

oe/M

cGra

w-H

ill29

1G

lenc

oe A

lgeb

ra 2

Lesson 5-9

Pre-

Act

ivit

yH

ow d

o co

mp

lex

nu

mb

ers

app

ly t

o p

olyn

omia

l eq

uat

ion

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

5-9

at

the

top

of p

age

270

in y

our

text

book

.

Sup

pose

the

num

ber

iis

def

ined

suc

h th

at i

2�

�1.

Com

plet

e ea

ch e

quat

ion.

2i2

�(2

i)2

�i4

�

Rea

din

g t

he

Less

on

1.C

ompl

ete

each

sta

tem

ent.

a.T

he

form

a�

biis

cal

led

the

of a

com

plex

nu

mbe

r.

b.

In t

he c

ompl

ex n

umbe

r 4

�5i

,the

rea

l par

t is

an

d th

e im

agin

ary

part

is

.

Th

is i

s an

exa

mpl

e of

a c

ompl

ex n

um

ber

that

is

also

a(n

) n

um

ber.

c.In

th

e co

mpl

ex n

um

ber

3,th

e re

al p

art

is

and

the

imag

inar

y pa

rt i

s .

Th

is i

s ex

ampl

e of

com

plex

nu

mbe

r th

at i

s al

so a

(n)

nu

mbe

r.

d.

In t

he

com

plex

nu

mbe

r 7i

,th

e re

al p

art

is

and

the

imag

inar

y pa

rt i

s .

Th

is i

s an

exa

mpl

e of

a c

ompl

ex n

um

ber

that

is

also

a(n

) n

um

ber.

2.G

ive

the

com

plex

con

juga

te o

f ea

ch n

um

ber.

a.3

�7i

b.

2 �

i

3.W

hy

are

com

plex

con

juga

tes

use

d in

div

idin

g co

mpl

ex n

um

bers

?T

he

pro

du

ct o

fco

mp

lex

con

jug

ates

is a

lway

s a

real

nu

mb

er.

4.E

xpla

in h

ow y

ou w

ould

use

com

plex

con

juga

tes

to f

ind

(3 �

7i)

(2

�i)

.W

rite

th

ed

ivis

ion

in f

ract

ion

fo

rm.T

hen

mu

ltip

ly n

um

erat

or

and

den

om

inat

or

by

2 �

i.

Hel

pin

g Y

ou

Rem

emb

er

5.H

ow c

an y

ou u

se w

hat

you

kn

ow a

bou

t si

mpl

ifyi

ng

an e

xpre

ssio

n s

uch

as

to

hel

p yo

u r

emem

ber

how

to

sim

plif

y fr

acti

ons

wit

h i

mag

inar

y n

um

bers

in

th

ede

nom

inat

or?

Sam

ple

an

swer

:In

bo

th c

ases

,yo

u c

an m

ult

iply

th

en

um

erat

or

and

den

om

inat

or

by t

he

con

jug

ate

of

the

den

om

inat

or.

1 �

�3�

� 2 �

�5�

2 �

i

3 �

7i

pu

re im

agin

ary

70

real

03

imag

inar

y

54

stan

dar

d f

orm

1�

4�

2

©G

lenc

oe/M

cGra

w-H

ill29

2G

lenc

oe A

lgeb

ra 2

Co

nju

gat

es a

nd

Ab

solu

te V

alu

eW

hen

stu

dyin

g co

mpl

ex n

um

bers

,it

is o

ften

con

ven

ien

t to

rep

rese

nt

a co

mpl

ex

nu

mbe

r by

a s

ingl

e va

riab

le.F

or e

xam

ple,

we

mig

ht

let

z�

x�

yi.W

e de

not

e th

e co

nju

gate

of

zby

z �.T

hu

s,z �

�x

�yi

.

We

can

def

ine

the

abso

lute

val

ue

of a

com

plex

nu

mbe

r as

fol

low

s.

z �

x�

yi �

�x2

�y

�2 �

Th

ere

are

man

y im

port

ant

rela

tion

ship

s in

volv

ing

con

juga

tes

and

abso

lute

va

lues

of

com

plex

nu

mbe

rs.

Sh

ow �z

�2�

zz �fo

r an

y co

mp

lex

nu

mb

er z

.

Let

z�

x�

yi.T

hen

,z

�(x

�yi

)(x

�yi

)�

x2�

y2

��

( x2

�y2

�)2 �

� z

2

Sh

ow

is t

he

mu

ltip

lica

tive

in

vers

e fo

r an

y n

onze

ro

com

ple

x n

um

ber

z.

We

know

z 2

�zz �.

If z

� 0

,th

en w

e h

ave

z ��

1.

Th

us,

is t

he

mu

ltip

lica

tive

in

vers

e of

z.

For

eac

h o

f th

e fo

llow

ing

com

ple

x n

um

ber

s,fi

nd

th

e ab

solu

te v

alu

e an

dm

ult

ipli

cati

ve i

nve

rse.

1.2i

2;�� 2i �

2.�

4 �

3i5;

��4 2� 5

3i�

3.12

�5i

13;�12

1� 695i

�

4.5

�12

i13

;�5

� 16912

i�

5.1

�i

�2�;

�1� 2

i�

6.�

3��

i2;

7.�

i8.

�i

9.�1 2�

�i

;1;

�i

1;�1 2�

�i

�3�

�3

�2�

�2

�2�

�2

�3�

�i�

3��

� 2�

6��

3

�3�

�2

�2�

�2

�2�

�2

�3�

�3

�3�

�3

�3�

�i

�4

z �� z

2

z �� z

2

z �� z

�2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-9

5-9

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8. B

C

A

C

D

B

D

A

�12

A

C

D

B

B

C

D

A

B

D

A

A

C

D

B

D

C

B

A

D

Chapter 5 Assessment Answer KeyForm 1 Form 2APage 293 Page 294 Page 295

An

swer

s

(continued on the next page)


9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

D

A

C

D

B

C

A

D

A

C

D

B

A

C

B

C

D

C

C

A

�9

A

C

D

B

A

C

D

B

C

D

A

A

Chapter 5 Assessment Answer KeyForm 2A (continued) Form 2BPage 296 Page 297 Page 298

(z � 3)(z � 3) �

(x � 2y)(x � 2y)


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:�65

�

�3177� � �

2127�j amps

6 � 12j ohms

about 2.67 � 104

people per mi2

y � 7

21

x�16

�or �6

x�

2m�35

�

47.693 in.

�xx

��

64

�

(2x � 3y)(z � 4)

x2 � x � 20 � �x

1�0

3�

5y2 � 12y � 21 � �2y

7�3

3�

2.8 � 106

15 � 16i

22 � 19i

14 � �6�

18�2� � 5�3�

2a2b�33b2�

7� x3 �y2

�27

�

6x2 � 7x � 20

14p2 � 3p � 12

3c2 � 14c � 12

�a9

2

bc6

6�

�75

t2r 4�

Chapter 5 Assessment Answer KeyForm 2CPage 299 Page 300

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:�45

�

�2137� � �

2177�j ohms

9 � 6j ohms

t � 1

8

x�110�

or �10x�

�5x�23

�

40.406 in.

�xx

��

45

�

(4x � y)(5x � 2)

x2 � 6x � 3 � �x

1�6

2�

4x2 � 3x � 3 � �2x

7� 1�

1.5 � 103

11 � 27i

4 � 8i

5 � 2�3�

7�2� � 3�5�

�4a2b2�3b�

2� x �y2

�35

�

10m2 � 7m � 6

6g3 � 3g2 � 7g � 8

7f 2 � 2f � 3

�b4a

c4

8�

�40

cd 2�

Chapter 5 Assessment Answer KeyForm 2DPage 301 Page 302

6.8 � 103

people per km2


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:�5147� � �

2127�j ohms

$4.85 � 103

��i �

3105��

��19

� � �4�

95��i

�2 x 2

565

�5 � 3�3�

2x2y �x�

5.760 m

m � 5

2x2 � x � 1

x2 � 3x � 9 � �x2

2�2x

3�x

2� 1

�

1.3 � 10�3

4 � 6i

5 � 14i

�x� � 3

4�15� � 9�5�

xy2�3x2y�

�3x2y

� 2x � 5 �

m2 � 4nm � 4n2

12p2 � �53

�pr � �156�r 2

��5xy2�

�16

1a6�

Chapter 5 Assessment Answer KeyForm 3Page 303 Page 304

An

swer

s

2(9w2 � n2) �(3w � n)(3w � n)

(x2 � 2y2)(x4 � 2x2y2 � 4y4)


Chapter 5 Assessment Answer KeyPage 305, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts ofoperations with polynomials; operations with radicalexpressions; and solving equations and inequalitiescontaining radicals.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Diagrams are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of operationswith polynomials; operations with radical expressions; andsolving equations and inequalities containing radicals.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Diagrams are mostly accurate and appropriate.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofoperations with polynomials; operations with radicalexpressions; and solving equations and inequalitiescontaining radicals.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Diagrams are mostly accurate.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work is shown to substantiate

the final computation.• Diagrams may be accurate but lack detail or explanation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the conceptsof solving systems of operations with polynomials;operations with radical expressions; and solving equationsand inequalities containing radicals.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Diagrams are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer may be given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations

Chapter 5 Assessment Answer KeyPage 305, Open-Ended Assessment

Sample Answers


1a. Student responses should indicate thatthe monthly profit for each companydepends on the number of sleds sold;one company may have a greater profitfor a given number of sleds, but theother company may have the greaterprofit for a different number of sleds.

1b. Student responses may vary but mustbe between 2 and 50. For a response ofx � 10 sleds, the A-Glide Company would earn a profit of �3(10)�� 19� � 7hundred dollars, or $700, whileSnowFun would earn a profit of 3 � �2(10)� � 7.47 hundred dollars, or$747.

1c. Students should indicate that Mark’sdecision to work for A-Glide means thatA-Glide has the greater monthly profitfor the number of sleds sold by eachcompany, so �3x � 1�9� � 3 � �2x�. The solution ofthis inequality is {x�x � 2 or x � 50}which means that A-Glide’s profits aregreater than SnowFun’s profits during amonth that one sled or more than 50sleds are sold.

2a. The length and width are 2x � 1 andx � 1 units.

2b. The perimeter can be found using theformula p � 2(l � w). Substituting 2x � 1 for length and x � 1 for width,p � 2(2x � 1 � x � 1)p � 2(3x � 2) � 6x � 4

2c. For a choice of 3, the length is 7 units,the width is 4 units, the perimeter is 22 units, and the area is 28 units2. Thevalue of x must be chosen so that thelength, width, perimeter, and area areall positive. The expressions 2x � 1,x � 1, 6x � 14, and 2x2 � 3x � 1 will all

be positive only if x � ��12�.

2d. 2x2 � 3x � 1 � (2x � 1)(x � 1)2e. Students should indicate that the

factors of the polynomial in part a arethe same as the dimensions of therectangle in part d.

2f. Student polynomials and tile modelswill vary. Sample answer:

3x2 � 4x � 1 � (3x � 1)(x � 1)Explanations should demonstrate anunderstanding that the length andwidth of the rectangle are the same asthe factors of the polynomial.

x

x 1

x 2

x

x 2

x

x 2

In addition to the scoring rubric found on page A34, the following sample answers may be used as guidance in evaluating open-ended assessment items.

An

swer

s


Chapter 5 Assessment Answer KeyVocabulary Test/Review Quiz (Lessons 5–1 through 5–3) Quiz (Lessons 5–6 and 5–7)

Page 306 Page 307 Page 308

1. binomial

2. complex number

3. Scientific notation

4. extraneous solution

5. radical inequalities

6. conjugates

7. terms

8. constant

9. power

10. rationalizing thedenominator

11. Sample answer: Asquare root of anumber b is anumber whosesquare is b.

12. Sample answer: Apure imaginarynumber is acomplex numberwhose real part is 0.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Quiz (Lessons 5–4 and 5–5)

Page 307

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Quiz (Lessons 5–8 and 5–9)

Page 308

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.�12

� � �12

�i

68 � 4i

�11 � 3i

�6�2�

4i �5�

�2i �5�x � 2

�15

� x 1

��12

�

no solution

6t2

�614�

2z�35

�

�8x5� or (�8

x�)5

�7 �

53�6��

11 � 11�5�

12 � 2�35�

14�3� � 39�2�

3m2 � n3 � �2m�

��21x0x��

�3.826

�3w3y2

�xx

��

34

�

3(2a � 3)(a � 2)

2(c � 7)(c � 7)

a2 � 3a � 1

m � 3 � �m

6� 4�

A

8x2 � 18x � 35

�3x � 3

9p � q

1.52 � 104

6.8 � 10�7

8x2y2

��2

y43n7

�


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.��

157� � �

1147�i

51/2z3/2

7 � x � y2

3x2 � x � 1 � �2x

1� 1�

1.5 � 1011

�3422

5� � �mn � � �169�

�2529�

s � 0t � 0

3s � 4t 500

(3, –2)

a2 � 7a � 10

(n � 3)2

5x3 � 10x2 � 2x � 3

(a � 5)(x � 2)(x � 2)

1.12 � 104

1.25 � 10�9 s

4.116

x2 � 5x � 1 � �x �

27

�

2x2 � 3x � 1

B

D

C

D

B

A

C

Chapter 5 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 309 Page 310

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 12.

13. 14.

15.

16.

17.

18. DCBA

DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 5 0

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

5 / 6 4

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

5 1 2

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

2 / 3

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Chapter 5 Assessment Answer KeyStandardized Test Practice

Page 311 Page 312

chapter 5 resource masters - ktl math...

Documents