chapter 8 resource masters - math class · ©glencoe/mcgraw-hill iv glencoe algebra 1 teacher’s...
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Chapter 8Resource Masters
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks in both English and Spanish.
Study Guide and Intervention Workbook 0-07-827753-1Study Guide and Intervention Workbook (Spanish) 0-07-827754-XSkills Practice Workbook 0-07-827747-7Skills Practice Workbook (Spanish) 0-07-827749-3Practice Workbook 0-07-827748-5Practice Workbook (Spanish) 0-07-827750-7
ANSWERS FOR WORKBOOKS The answers for Chapter 8 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, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 1. 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: 0-07-827732-9 Algebra 1Chapter 8 Resource Masters
2 3 4 5 6 7 8 9 10 024 11 10 09 08 07 06 05 04 03
Glencoe/McGraw-Hill
© Glencoe/McGraw-Hill iii Glencoe Algebra 1
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 8-1Study Guide and Intervention . . . . . . . . 455–456Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 457Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 458Reading to Learn Mathematics . . . . . . . . . . 459Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 460
Lesson 8-2Study Guide and Intervention . . . . . . . . 461–462Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 463Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 464Reading to Learn Mathematics . . . . . . . . . . 465Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 466
Lesson 8-3Study Guide and Intervention . . . . . . . . 467–468Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 469Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 470Reading to Learn Mathematics . . . . . . . . . . 471Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 472
Lesson 8-4Study Guide and Intervention . . . . . . . . 473–474Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 475Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 476Reading to Learn Mathematics . . . . . . . . . . 477Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 478
Lesson 8-5Study Guide and Intervention . . . . . . . 479–480Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 481Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Reading to Learn Mathematics . . . . . . . . . . 483Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 484
Lesson 8-6Study Guide and Intervention . . . . . . . . 485–486Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 487Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 488Reading to Learn Mathematics . . . . . . . . . . 489Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 490
Lesson 8-7Study Guide and Intervention . . . . . . . . 491–492Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 493Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 494Reading to Learn Mathematics . . . . . . . . . . 495Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 496
Lesson 8-8Study Guide and Intervention . . . . . . . . 497–498Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 499Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 500Reading to Learn Mathematics . . . . . . . . . . 501Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 502
Chapter 8 AssessmentChapter 8 Test, Form 1 . . . . . . . . . . . . 503–504Chapter 8 Test, Form 2A . . . . . . . . . . . 505–506Chapter 8 Test, Form 2B . . . . . . . . . . . 507–508Chapter 8 Test, Form 2C . . . . . . . . . . . 509–510Chapter 8 Test, Form 2D . . . . . . . . . . . 511–512Chapter 8 Test, Form 3 . . . . . . . . . . . . 513–514Chapter 8 Open-Ended Assessment . . . . . . 515Chapter 8 Vocabulary Test/Review . . . . . . . 516Chapter 8 Quizzes 1 & 2 . . . . . . . . . . . . . . . 517Chapter 8 Quizzes 3 & 4 . . . . . . . . . . . . . . . 518Chapter 8 Mid-Chapter Test . . . . . . . . . . . . 519Chapter 8 Cumulative Review . . . . . . . . . . . 520Chapter 8 Standardized Test Practice . . 521–522
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A35
© Glencoe/McGraw-Hill iv Glencoe Algebra 1
Teacher’s Guide to Using theChapter 8 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 8 Resource Masters includes the core materials neededfor Chapter 8. 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 1 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 8-1.Encourage them to add these pages to theirAlgebra Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.
Study Guide and InterventionEach lesson in Algebra 1 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 1
Assessment OptionsThe assessment masters in the Chapter 8Resources 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 1. 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 470–471. 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 _____
88
© Glencoe/McGraw-Hill vii Glencoe Algebra 1
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 8.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
binomial
by·NOH·mee·uhl
constant
degree of a monomial
degree of a polynomial
FOIL method
monomial
mah·NOH·mee·uhl
negative exponent
polynomial
PAH·luh·NOH·mee·uhl
(continued on the next page)
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
⎧ ⎪ ⎨ ⎪ ⎩
© Glencoe/McGraw-Hill viii Glencoe Algebra 1
Vocabulary Term Found on Page Definition/Description/Example
Power of a Power
Power of a Product
Product of Powers
Power of a Quotient
Quotient of Powers
scientific notation
trinomial
try·NOH·mee·uhl
zero exponent
⎧ ⎪ ⎨ ⎪ ⎩
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
88
Study Guide and InterventionMultiplying Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
6-18-1
© Glencoe/McGraw-Hill 455 Glencoe Algebra 1
Less
on 8
-1
Multiply Monomials A monomial is a number, a variable, or a product of a numberand one or more variables. An expression of the form xn is called a power and representsthe product you obtain when x is used as a factor n times. To multiply two powers that havethe same base, add the exponents.
Product of Powers For any number a and all integers m and n, am ! an " am # n.
Simplify (3x6)(5x2).
(3x6)(5x2) " (3)(5)(x6 ! x2) Associative Property
" (3 ! 5)(x6 # 2) Product of Powers
" 15x8 Simplify.
The product is 15x8.
Simplify (!4a3b)(3a2b5).
($4a3b)(3a2b5) " ($4)(3)(a3 ! a2)(b ! b5)" $12(a3 # 2)(b1 # 5)" $12a5b6
The product is $12a5b6.
Example 1Example 1 Example 2Example 2
ExercisesExercises
Simplify.
1. y( y5) 2. n2 ! n7 3. ($7x2)(x4)
4. x(x2)(x4) 5. m ! m5 6. ($x3)($x4)
7. (2a2)(8a) 8. (rs)(rs3)(s2) 9. (x2y)(4xy3)
10. (2a3b)(6b3) 11. ($4x3)($5x7) 12. ($3j2k4)(2jk6)
13. (5a2bc3)! abc4" 14. ($5xy)(4x2)( y4) 15. (10x3yz2)($2xy5z)1%5
1%3
© Glencoe/McGraw-Hill 456 Glencoe Algebra 1
Powers of Monomials An expression of the form (xm)n is called a power of a powerand represents the product you obtain when xm is used as a factor n times. To find thepower of a power, multiply exponents.
Power of a Power For any number a and all integers m and n, (am)n " amn.
Power of a Product For any number a and all integers m and n, (ab)m " ambm.
Simplify (!2ab2)3(a2)4.
($2ab2)3(a2)4 " ($2ab2)3(a8) Power of a Power
" ($2)3(a3)(b2)3(a8) Power of a Product
" ($2)3(a3)(a8)(b2)3 Commutative Property
" ($2)3(a11)(b2)3 Product of Powers
" $8a11b6 Power of a Power
The product is $8a11b6.
Simplify.
1. (y5)2 2. (n7)4 3. (x2)5(x3)
4. $3(ab4)3 5. ($3ab4)3 6. (4x2b)3
7. (4a2)2(b3) 8. (4x)2(b3) 9. (x2y4)5
10. (2a3b2)(b3)2 11. ($4xy)3($2x2)3 12. ($3j2k3)2(2j2k)3
13. (25a2b)3! abc"214. (2xy)2($3x2)(4y4) 15. (2x3y2z2)3(x2z)4
16. ($2n6y5)($6n3y2)(ny)3 17. ($3a3n4)($3a3n)4 18. $3(2x)4(4x5y)2
1%5
Study Guide and Intervention (continued)
Multiplying Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
ExampleExample
ExercisesExercises
Skills PracticeMultiplying Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
© Glencoe/McGraw-Hill 457 Glencoe Algebra 1
Less
on 8
-1
Determine whether each expression is a monomial. Write yes or no. Explain.
1. 11
2. a $ b
3.
4. y
5. j3k
6. 2a # 3b
Simplify.
7. a2(a3)(a6) 8. x(x2)(x7)
9. (y2z)(yz2) 10. (!2k2)(!3k)
11. (e2f4)(e2f 2) 12. (cd2)(c3d2)
13. (2x2)(3x5) 14. (5a7)(4a2)
15. (4xy3)(3x3y5) 16. (7a5b2)(a2b3)
17. ($5m3)(3m8) 18. ($2c4d)($4cd)
19. (102)3 20. (p3)12
21. ($6p)2 22. ($3y)3
23. (3pq2)2 24. (2b3c4)2
GEOMETRY Express the area of each figure as a monomial.
25. 26. 27.
4p
9p3cd
cdx2
x5
p2%q2
© Glencoe/McGraw-Hill 458 Glencoe Algebra 1
Determine whether each expression is a monomial. Write yes or no. Explain.
1.
2.
Simplify.
3. ($5x2y)(3x4) 4. (2ab2c2)(4a3b2c2)
5. (3cd4)($2c2) 6. (4g3h)($2g5)
7. ($15xy4)!$ xy3" 8. ($xy)3(xz)
9. ($18m2n)2!$ mn2" 10. (0.2a2b3)2
11. ! p"212. ! cd3"2
13. (0.4k3)3 14. [(42)2]2
GEOMETRY Express the area of each figure as a monomial.
15. 16. 17.
GEOMETRY Express the volume of each solid as a monomial.
18. 19. 20.
21. COUNTING A panel of four light switches can be set in 24 ways. A panel of five lightswitches can set in twice this many ways. In how many ways can five light switches be set?
22. HOBBIES Tawa wants to increase her rock collection by a power of three this year andthen increase it again by a power of two next year. If she has 2 rocks now, how manyrocks will she have after the second year?
7g2
3g
m3nmn3
n
3h23h2
3h2
6ac3
4a2c
5x3
6a2b4
3ab2
1%4
2%3
1%6
1%3
b3c2%2
21a2%7b
Practice Multiplying Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
Reading to Learn MathematicsMultiplying Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
© Glencoe/McGraw-Hill 459 Glencoe Algebra 1
Less
on 8
-1
Pre-Activity Why does doubling speed quadruple braking distance?
Read the introduction to Lesson 8-1 at the top of page 410 in your textbook.
Find two examples in the table to verify the statement that when speed isdoubled, the braking distance is quadrupled. Write your examples in thetable.
Speed Braking Distance Speed Doubled Braking Distance (miles per hour) (feet) (miles per hour) Quadrupled (feet)
Reading the Lesson
1. Describe the expression 3xy using the terms monomial, constant, variable, and product.
2. Complete the chart by choosing the property that can be used to simplify eachexpression. Then simplify the expression.
Expression Property Expression Simplified
Product of Powers
35 ! 32 Power of a Power
Power of a Product
Product of Powers
(a3)4 Power of a Power
Power of a Product
Product of Powers
($4xy)5 Power of a Power
Power of a Product
Helping You Remember
3. Write an example of each of the three properties of powers discussed in this lesson.Then, using the examples, explain how the property is used to simplify them.
© Glencoe/McGraw-Hill 460 Glencoe Algebra 1
An Wang An Wang (1920–1990) was an Asian-American who became one of thepioneers of the computer industry in the United States. He grew up inShanghai, China, but came to the United States to further his studiesin science. In 1948, he invented a magnetic pulse controlling devicethat vastly increased the storage capacity of computers. He laterfounded his own company, Wang Laboratories, and became a leader inthe development of desktop calculators and word processing systems.In 1988, Wang was elected to the National Inventors Hall of Fame.
Digital computers store information as numbers. Because theelectronic circuits of a computer can exist in only one of two states,open or closed, the numbers that are stored can consist of only twodigits, 0 or 1. Numbers written using only these two digits are calledbinary numbers. To find the decimal value of a binary number, youuse the digits to write a polynomial in 2. For instance, this is how tofind the decimal value of the number 10011012. (The subscript 2 indicates that this is a binary number.)
10011012 " 1 & 2%6% # 0 & 2%
5% # 0 & 2%
4% # 1 & 2%
3% # 1 & 2%
2% # 0 & 2%
1% # 1 & 2%
0%
" 1 & 6%4% # 0 & 3%2% # 0 & 1%6% # 1 & 8% # 1 & 4% # 0 & 2% # 1 & 1%" 64 # 0 # 0 # 8 # 4 # 0 # 1
" 77
Find the decimal value of each binary number.
1. 11112 2. 100002 3. 110000112 4. 101110012
Write each decimal number as a binary number.
5. 8 6. 11 7. 29 8. 117
9. The chart at the right shows a set of decimal code numbers that is used widely in storing letters of the alphabet in a computer's memory.Find the code numbers for the letters of your name. Then write the code for your name using binary numbers.
The American Standard Guide for Information Interchange (ASCII)
A 65 N 78 a 97 n 110B 66 O 79 b 98 o 111C 67 P 80 c 99 p 112D 68 Q 81 d 100 q 113E 69 R 82 e 101 r 114F 70 S 83 f 102 s 115G 71 T 84 g 103 t 116H 72 U 85 h 104 u 117I 73 V 86 i 105 v 118J 74 W 87 j 106 w 119K 75 X 88 k 107 x 120L 76 Y 89 l 108 y 121M 77 Z 90 m 109 z 122
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
Study Guide and InterventionDividing Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
6-28-2
© Glencoe/McGraw-Hill 461 Glencoe Algebra 1
Less
on 8
-2
Quotients of Monomials To divide two powers with the same base, subtract theexponents.
Quotient of Powers For all integers m and n and any nonzero number a, " am $ n.
Power of a Quotient For any integer m and any real numbers a and b, b ' 0, ! "m" .am
%bm
a%b
am%an
Simplify . Assume
neither a nor b is equal to zero.
" ! "! " Group powers with the same base.
" (a4 $ 1)(b7 $ 2) Quotient of Powers
" a3b5 Simplify.
The quotient is a3b5 .
b7%b2
a4%a
a4b7%ab2
a4b7"ab2 Simplify ! "3
.
Assume that b is not equal to zero.
! "3" Power of a Quotient
" Power of a Product
" Power of a Power
" Quotient of Powers
The quotient is .8a9b9%27
8a9b9%27
8a9b15%27b6
23(a3)3(b5)3%%
(3)3(b2)3
(2a3b5)3%
(3b2)32a3b5%
3b2
2a3b5"
3b2Example 1Example 1 Example 2Example 2
ExercisesExercises
Simplify. Assume that no denominator is equal to zero.
1. 2. 3.
4. 5. 6.
7. 8. ! "39. ! "3
10. ! "411. ! "4
12. r7s7t2%s3r3t2
3r6s3%2r5s
2v5w3%v4w3
4p4q4%3p2q2
2a2b%a
xy6%y4x
$2y7%14y5
x5y3%x5y2
a2%a
p5n4%p2n
m6%m4
55%52
© Glencoe/McGraw-Hill 462 Glencoe Algebra 1
Negative Exponents Any nonzero number raised to the zero power is 1; for example,($0.5)0 " 1. Any nonzero number raised to a negative power is equal to the reciprocal of the number raised to the opposite power; for example, 6$3 " . These definitions can be used to simplify expressions that have negative exponents.
Zero Exponent For any nonzero number a, a0 " 1.
Negative Exponent Property For any nonzero number a and any integer n, a$n " and " an.
The simplified form of an expression containing negative exponents must contain onlypositive exponents.
Simplify . Assume that the denominator is not equal to zero.
" ! "! "! "! " Group powers with the same base.
" (a$3 $ 2)(b6 $ 6)(c5) Quotient of Powers and Negative Exponent Properties
" a$5b0c5 Simplify.
" ! "(1)c5 Negative Exponent and Zero Exponent Properties
" Simplify.
The solution is .
Simplify. Assume that no denominator is equal to zero.
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. ! "0 12.($2mn2)$3%%
4m$6n44m2n2%8m$1!
s$3t$5%(s2t3)$1
(3st)2u$4%%s$1t2u7
(6a$1b)2%%
(b2)4
x4y0%x$2
(a2b3)2%(ab)$2
($x$1y)0%%4w$1y2
b$4%b$5
p$8%p3
m%m$4
22%2$3
c5%4a5
c5%4a5
1%a5
1%4
1%4
1%4
1%c$5
b6%b6
a$3%a2
4%16
4a$3b6%%16a2b6c$5
4a$3b6""16a2b6c$5
1%a$n
1%an
1%63
Study Guide and Intervention (continued)
Dividing Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
ExampleExample
ExercisesExercises
Skills PracticeDividing Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
© Glencoe/McGraw-Hill 463 Glencoe Algebra 1
Less
on 8
-2
Simplify. Assume that no denominator is equal to zero.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. ! "214. 4$4
15. 8$2 16. ! "$2
17. ! "$118.
19. k0(k4)(k$6) 20. k$1(!$6)(m3)
21. 22. ! "0
23. 24.
25. 26. 48x6y7z5%%$6xy5z6
$15w0u$1%%
5u3
15x6y$9%5xy$11
f$5g4%h$2
16p5q2%2p3q3
f$7%f4
h3%h$6
9%11
5%3
4p7%7s2
32x3y2z5%%$8xyz2
$21w5u2%%
7w4u5
m7n2%m3n2
a3b5%ab2
w4u3%w4u
12n5%36n
9d7%3d6
m%m3
r3s2%r3s4
x4%x2
912%98
65%64
© Glencoe/McGraw-Hill 464 Glencoe Algebra 1
Simplify. Assume that no denominator is equal to zero.
1. 2. 3.
4. 5. 6.
7. ! "38. ! "2
9.
10. x3( y$5)(x$8) 11. p(q$2)(r$3) 12. 12$2
13. ! "$214. ! "$4
15.
16. 17. 18. ! "0
19. 20. 21.
22. 23. 24.
25. ! "$526. ! "$1
27. ! "$2
28. BIOLOGY A lab technician draws a sample of blood. A cubic millimeter of the bloodcontains 223 white blood cells and 225 red blood cells. What is the ratio of white bloodcells to red blood cells?
29. COUNTING The number of three-letter “words” that can be formed with the Englishalphabet is 263. The number of five-letter “words” that can be formed is 265. How manytimes more five-letter “words” can be formed than three-letter “words”?
2x3y2z%3x4yz$2
7c$3d3%c5de$4
q$1r3%qr$2
(2a$2b)$3%%
5a2b4( j$1k3)$4%
j3k3
m$2n$5%%(m4n3)$1
r4%(3r)3
$12t$1u5v$4%%
2t$3uv56f$2g3h5%%54f$2g$5h3
x$3y5%4$3
8c3d2f4%%4c$1d2f$3
$15w0u$1%%
5u3
22r3s2%11r2s$3
4%3
3%7
$4c2%24c5
6w5%7p6s3
4f 3g%3h6
8y7z6%4y6z5
5c2d3%$4c2d
m5np%m4p
xy2%xy
a4b6%ab3
88%84
Practice Dividing Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
Reading to Learn MathematicsDividing Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
© Glencoe/McGraw-Hill 465 Glencoe Algebra 1
Less
on 8
-2
Pre-Activity How can you compare pH levels?
Read the introduction to Lesson 8-2 at the top of page 417 in your textbook.
• In the formula c " ! "pH, identify the base and the exponent.
• How do you think c will change as the exponent increases?
Reading the Lesson
1. Explain what the statement " am $ n means.
2. To find c in the formula c " ! "pH, you can find the power of the numerator, the power of
the denominator, and divide. This is an example of what property?
3. Use the Quotient of Powers Property to explain why 30 " 1.
4. Consider the expression 4$3.
a. Explain why the expression 4$3 is not simplified.
b. Define the term reciprocal.
c. 4$3 is the reciprocal of what power of 4?
d. What is the simplified form of 4$3?
Helping You Remember
5. Describe how you would help a friend who needs to simplify the expression .4x2%2x5
1%10
am%an
1%10
© Glencoe/McGraw-Hill 466 Glencoe Algebra 1
Patterns with PowersUse your calculator, if necessary, to complete each pattern.
a. 210 ! b. 510 ! c. 410 !
29 ! 59 ! 49 !
28 ! 58 ! 48 !
27 ! 57 ! 47 !
26 ! 56 ! 46 !
25 ! 55 ! 45 !
24 ! 54 ! 44 !
23 ! 53 ! 43 !
22 ! 52 ! 42 !
21 ! 51 ! 41 !
Study the patterns for a, b, and c above. Then answer the questions.
1. Describe the pattern of the exponents from the top of each column to the bottom.
2. Describe the pattern of the powers from the top of the column to the bottom.
3. What would you expect the following powers to be?20 50 40
4. Refer to Exercise 3. Write a rule. Test it on patterns that you obtain using 22, 25, and 24as bases.
Study the pattern below. Then answer the questions.
03 ! 0 02 ! 0 01 ! 0 00 ! 0"1 does not exist. 0"2 does not exist. 0"3 does not exist.
5. Why do 0"1, 0"2, and 0"3 not exist?
6. Based upon the pattern, can you determine whether 00 exists?
7. The symbol 00 is called an indeterminate, which means that it has no unique value.Thus it does not exist as a unique real number. Why do you think that 00 cannot equal 1?
?
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
Study Guide and InterventionScientific Notation
NAME ______________________________________________ DATE ____________ PERIOD _____
6-38-3
© Glencoe/McGraw-Hill 467 Glencoe Algebra 1
Less
on 8
-3
Scientific Notation Keeping track of place value in very large or very small numberswritten in standard form may be difficult. It is more efficient to write such numbers inscientific notation. A number is expressed in scientific notation when it is written as aproduct of two factors, one factor that is greater than or equal to 1 and less than 10 and onefactor that is a power of ten.
Scientific NotationA number is in scientific notation when it is in the form a # 10n, where 1 $ a % 10 and n is an integer.
Express 3.52 ! 104 instandard notation.
3.52 # 104 ! 3.52 # 10,000! 35,200
The decimal point moved 4 places to theright.
Express 37,600,000 inscientific notation.
37,600,000 ! 3.76 # 107
The decimal point moved 7 places so that itis between the 3 and the 7. Since 37,600,000 & 1, the exponent is positive.
Express 6.21 ! 10"5 instandard notation.
6.21 # 10"5 ! 6.21 #
! 6.21 # 0.00001! 0.0000621
The decimal point moved 5 places to the left.
Express 0.0000549 inscientific notation.
0.0000549 ! 5.49 # 10"5
The decimal point moved 5 places so that itis between the 5 and the 4. Since 0.0000549 % 1, the exponent is negative.
1'105
Example 1Example 1 Example 2Example 2
Example 3Example 3 Example 4Example 4
ExercisesExercises
Express each number in standard notation.
1. 3.65 # 105 2. 7.02 # 10"4 3. 8.003 # 108
4. 7.451 # 106 5. 5.91 # 100 6. 7.99 # 10"1
7. 8.9354 # 1010 8. 8.1 # 10"9 9. 4 # 1015
Express each number in scientific notation.
10. 0.0000456 11. 0.00001 12. 590,000,000
13. 0.00000000012 14. 0.000080436 15. 0.03621
16. 433 # 104 17. 0.0042 # 10"3 18. 50,000,000,000
© Glencoe/McGraw-Hill 468 Glencoe Algebra 1
Products and Quotients with Scientific Notation You can use properties ofpowers to compute with numbers written in scientific notation.
Evaluate (6.7 ! 103)(2 ! 10"5). Express the result in scientific andstandard notation.
(6.7 # 103)(2 # 10"5) ! (6.7 # 2)(103 # 10"5) Associative Property
! 13.4 # 10"2 Product of Powers
! (1.34 # 101) # 10"2 13.4 ! 1.34 # 101
! 1.34 # (101 # 10"2) Associative Property
! 1.34 # 10"1 or 0.134 Product of Powers
The solution is 1.34 # 10"1 or 0.134.
Evaluate . Express the result in scientific and
standard notation.
! ! "! " Associative Property
! 0.368 # 103 Quotient of Powers
! (3.68 # 10"1) # 103 0.368 ! 3.68 # 10"1
! 3.68 # (10"1 # 103) Associative Property
! 3.68 # 102 or 368 Product of Powers
The solution is 3.68 # 102 or 368.
Evaluate. Express each result in scientific and standard notation.
1. 2. 3. (3.2 # 10"2)(2.0 # 102)
4. 5. (7.7 # 105)(2.1 # 102) 6.
7. (3.3 # 105)(1.5 # 10"4) 8. 9.
10. FUEL CONSUMPTION North America burned 4.5 # 1016 BTU of petroleum in 1998.At this rate, how many BTU’s will be burned in 9 years? Source: The New York Times 2001 Almanac
11. OIL PRODUCTION If the United States produced 6.25 # 109 barrels of crude oil in1998, and Canada produced 1.98 # 109 barrels, what is the quotient of their productionrates? Write a statement using this quotient. Source: The New York Times 2001 Almanac
4 # 10"4''2.5 # 102
3.3 # 10"12''1.1 # 10"14
9.72 # 108''7.2 # 1010
1.2672 # 10"8''
2.4 # 10"12
3 # 10"12''2 # 10"15
1.4 # 104''2 # 102
108'105
1.5088'4.1
1.5088 # 108''4.1 # 105
1.5088 ! 108##
4.1 ! 105
Study Guide and Intervention (continued)
Scientific Notation
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeScientific Notation
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
© Glencoe/McGraw-Hill 469 Glencoe Algebra 1
Less
on 8
-3
Express each number in standard notation.
1. 4 # 103 2. 2 # 108 3. 3.2 # 105
4. 3 # 10"6 5. 9 # 10"2 6. 4.7 # 10"7
ASTRONOMY Express the number in each statement in standard notation.
7. The diameter of Jupiter is 1.42984 # 105 kilometers.
8. The surface density of the main ring around Jupiter is 5 # 10"6 grams per centimetersquared.
9. The minimum distance from Mars to Earth is 5.45 # 107 kilometers.
Express each number in scientific notation.
10. 41,000,000 11. 65,100 12. 283,000,000
13. 264,701 14. 0.019 15. 0.000007
16. 0.000010035 17. 264.9 18. 150 # 102
Evaluate. Express each result in scientific and standard notation.
19. (3.1 # 107)(2 # 10"5) 20. (5 # 10"2)(1.4 # 10"4)
21. (3 # 103)(4.2 # 10"1) 22. (3 # 10"2)(5.2 # 109)
23. (2.4 # 102)(4 # 10"10) 24. (1.5 # 10"4)(7 # 10"5)
25. 26.7.2 # 10"5''4 # 10"3
5.1 # 106''1.5 # 102
© Glencoe/McGraw-Hill 470 Glencoe Algebra 1
Express each number in standard notation.
1. 7.3 # 107 2. 2.9 # 103 3. 9.821 # 1012
4. 3.54 # 10"1 5. 7.3642 # 104 6. 4.268 # 10"6
PHYSICS Express the number in each statement in standard notation.
7. An electron has a negative charge of 1.6 # 10"19 Coulomb.
8. In the middle layer of the sun’s atmosphere, called the chromosphere, the temperatureaverages 2.78 # 104 degrees Celsius.
Express each number in scientific notation.
9. 915,600,000,000 10. 6387 11. 845,320 12. 0.00000000814
13. 0.00009621 14. 0.003157 15. 30,620 16. 0.0000000000112
17. 56 # 107 18. 4740 # 105 19. 0.076 # 10"3 20. 0.0057 # 103
Evaluate. Express each result in scientific and standard notation.
21. (5 # 10"2)(2.3 # 1012) 22. (2.5 # 10"3)(6 # 1015)
23. (3.9 # 103)(4.2 # 10"11) 24. (4.6 # 10"4)(3.1 # 10"1)
25. 26. 27.
28. 29. 30.
31. BIOLOGY A cubic millimeter of human blood contains about 5 # 106 red blood cells. Anadult human body may contain about 5 # 106 cubic millimeters of blood. About how manyred blood cells does such a human body contain?
32. POPULATION The population of Arizona is about 4.778 # 106 people. The land area isabout 1.14 # 105 square miles. What is the population density per square mile?
2.015 # 10"3''
3.1 # 1021.68 # 104''8.4 # 10"4
1.82 # 105''9.1 # 107
1.17 # 102''5 # 10"1
6.72 # 103''4.2 # 108
3.12 # 103''1.56 # 10"3
Practice Scientific Notation
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
Reading to Learn MathematicsScientific Notation
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
© Glencoe/McGraw-Hill 471 Glencoe Algebra 1
Less
on 8
-3
Pre-Activity Why is scientific notation important in astronomy?
Read the introduction to Lesson 8-3 at the top of page 425 in your textbook.
In the table, each mass is written as the product of a number and a powerof 10. Look at the first factor in each product. How are these factors alike?
Reading the Lesson
1. Is the number 0.0543 # 104 in scientific notation? Explain.
2. Complete each sentence to change from scientific notation to standard notation.
a. To express 3.64 # 106 in standard notation, move the decimal point
places to the .
b. To express 7.825 # 10"3 in standard notation, move the decimal point
places to the .
3. Complete each sentence to change from standard notation to scientific notation.
a. To express 0.0007865 in scientific notation, move the decimal point places
to the right and write .
b. To express 54,000,000,000 in scientific notation, move the decimal point
places to the left and write .
4. Write positive or negative to complete each sentence.
a. powers of 10 are used to express very large numbers in
scientific notation.
b. powers of 10 are used to express very small numbers in
scientific notation.
Helping You Remember
5. Describe the method you would use to estimate how many times greater the mass ofSaturn is than the mass of Pluto.
© Glencoe/McGraw-Hill 472 Glencoe Algebra 1
Converting Metric UnitsScientific notation is convenient to use for unit conversions in the metric system.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
How many kilometersare there in 4,300,000 meters?
Divide the measure by the number ofmeters (1000) in one kilometer. Expressboth numbers in scientific notation.
! 4.3 # 103
The answer is 4.3 3 103 km.
4.3 # 106''1 # 103
Convert 3700 grams intomilligrams.
Multiply by the number of milligrams(1000) in 1 gram.
(3.7 # 103)(1 # 103) ! 3.7 # 106
There are 3.7 # 106 mg in 3700 g.
Example 1Example 1 Example 2Example 2
Complete the following. Express each answer in scientific notation.
1. 250,000 m ! km 2. 375 km ! m
3. 247 m ! cm 4. 5000 m ! mm
5. 0.0004 km ! m 6. 0.01 mm ! m
7. 6000 m ! mm 8. 340 cm ! km
9. 52,000 mg ! g 10. 420 kL ! L
Solve.
11. The planet Mars has a diameter of 6.76 # 103 km. What is thediameter of Mars in meters? Express the answer in both scientificand decimal notation.
12. The distance from earth to the sun is 149,590,000 km. Lighttravels 3.0 # 108 meters per second. How long does it take lightfrom the sun to reach the earth in minutes? Round to the nearest hundredth.
13. A light-year is the distance that light travels in one year. (SeeExercise 12.) How far is a light year in kilometers? Express youranswer in scientific notation. Round to the nearest hundredth.
Study Guide and InterventionPolynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
6-48-4
© Glencoe/McGraw-Hill 473 Glencoe Algebra 1
Less
on 8
-4
Degree of a Polynomial A polynomial is a monomial or a sum of monomials. Abinomial is the sum of two monomials, and a trinomial is the sum of three monomials.Polynomials with more than three terms have no special name. The degree of a monomialis the sum of the exponents of all its variables. The degree of the polynomial is the sameas the degree of the monomial term with the highest degree.
State whether each expression is a polynomial. If the expression isa polynomial, identify it as a monomial, binomial, or trinomial. Then give thedegree of the polynomial.
Expression Polynomial?Monomial, Binomial, Degree of the
or Trinomial? Polynomial
3x " 7xyzYes. 3x " 7xyz ! 3x ( ("7xyz),
binomial 3which is the sum of two monomials
"25 Yes. "25 is a real number. monomial 0
7n3 ( 3n"4No. 3n"4 ! , which is not
none of these —a monomial
Yes. The expression simplifies to9x3 ( 4x ( x ( 4 ( 2x 9x3 ( 7x ( 4, which is the sum trinomial 3
of three monomials
State whether each expression is a polynomial. If the expression is a polynomial,identify it as a monomial, binomial, or trinomial.
1. 36 2. ( 5
3. 7x " x ( 5 4. 8g2h " 7gh ( 2
5. ( 5y " 8 6. 6x ( x2
Find the degree of each polynomial.
7. 4x2y3z 8. "2abc 9. 15m
10. s ( 5t 11. 22 12. 18x2 ( 4yz " 10y
13. x4 " 6x2 " 2x3 " 10 14. 2x3y2 " 4xy3 15. "2r8s4 ( 7r2s " 4r7s6
16. 9x2 ( yz8 17. 8b ( bc5 18. 4x4y " 8zx2 ( 2x5
19. 4x2 " 1 20. 9abc ( bc " d5 21. h3m ( 6h4m2 " 7
1'4y2
3'q2
3'n4
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 474 Glencoe Algebra 1
Write Polynomials in Order The terms of a polynomial are usually arranged so that the powers of one variable are in ascending (increasing) order or descending(decreasing) order.
Study Guide and Intervention (continued)
Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
Arrange the terms ofeach polynomial so that the powers ofx are in ascending order.
a. x4 " x2 $ 5x3
"x2 ( 5x3 ( x4
b. 8x3y " y2 $ 6x2y $ xy2
"y2 ( xy2 ( 6x2y ( 8x3y
Arrange the terms ofeach polynomial so that the powers ofx are in descending order.
a. x4 $ 4x5 " x2
4x5 ( x4 " x2
b. "6xy $ y3 " x2y2 $ x4y2
x4y2 " x2y2 " 6xy ( y3
Example 1Example 1 Example 2Example 2
ExercisesExercises
Arrange the terms of each polynomial so that the powers of x are in ascending order.
1. 5x ( x2 ( 6 2. 6x ( 9 " 4x2 3. 4xy ( 2y ( 6x2
4. 6y2x " 6x2y ( 2 5. x4 ( x3( x2 6. 2x3 " x ( 3x7
7. "5cx ( 10c2x3( 15cx2 8. "4nx " 5n3x3( 5 9. 4xy ( 2y ( 5x2
Arrange the terms of each polynomial so that the powers of x are in descending order.
10. 2x ( x2 " 5 11. 20x " 10x2 ( 5x3 12. x2 ( 4yx" 10x5
13. 9bx ( 3bx2 " 6x3 14. x3 ( x5 " x2 15. ax2 ( 8a2x5 " 4
16. 3x3y " 4xy2 " x4y2 ( y5 17. x4 ( 4x3 " 7x5 ( 1
18. "3x6 " x5 ( 2x8 19. "15cx2 ( 8c2x5 ( cx
20. 24x2y " 12x3y2 ( 6x4 21. "15x3 ( 10x4y2 ( 7xy2
Skills PracticePolynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
© Glencoe/McGraw-Hill 475 Glencoe Algebra 1
Less
on 8
-4
State whether each expression is a polynomial. If the expression is a polynomial,identify it as a monomial, a binomial, or a trinomial.
1. 5mn ( n2 2. 4by ( 2b " by 3. "32
4. 5. 5x2 " 3x"4 6. 2c2 ( 8c ( 9 " 3
GEOMETRY Write a polynomial to represent the area of each shaded region.
7. 8.
Find the degree of each polynomial.
9. 12 10. 3r4 11. b ( 6
12. 4a3 " 2a 13. 5abc " 2b2 ( 1 14. 8x5y4 " 2x8
Arrange the terms of each polynomial so that the powers of x are in ascendingorder.
15. 3x ( 1 ( 2x2 16. 5x " 6 ( 3x2
17. 9x2 ( 2 ( x3 ( x 18. "3 ( 3x3 " x2 ( 4x
19. 7r5x ( 21r4 " r2x2 " 15x3 20. 3a2x4 ( 14a2 " 10x3 ( ax2
Arrange the terms of each polynomial so that the powers of x are in descendingorder.
21. x2 ( 3x3 ( 27 " x 22. 25 " x3 ( x
23. x " 3x2 ( 4 ( 5x3 24. x2 ( 64 " x ( 7x3
25. 2cx ( 32 " c3x2 ( 6x3 26. 13 " x3y3 ( x2y2 ( x
r
a
x
by
3x'7
© Glencoe/McGraw-Hill 476 Glencoe Algebra 1
State whether each expression is a polynomial. If the expression is a polynomial,identify it as a monomial, a binomial, or a trinomial.
1. 7a2b ( 3b2 " a2b 2. y3 ( y2 " 9 3. 6g2h3k
GEOMETRY Write a polynomial to represent the area of each shaded region.
4. 5.
Find the degree of each polynomial.
6. x ( 3x4 " 21x2 ( x3 7. 3g2h3 ( g3h
8. "2x2y ( 3xy3 ( x2 9. 5n3m " 2m3 ( n2m4 ( n2
10. a3b2c ( 2a5c ( b3c2 11. 10s2t2 ( 4st2 " 5s3t2
Arrange the terms of each polynomial so that the powers of x are in ascendingorder.
12. 8x2 " 15 ( 5x5 13. 10bx " 7b2 ( x4 ( 4b2x3
14. "3x3y ( 8y2 ( xy4 15. 7ax " 12 ( 3ax3 (a2x2
Arrange the terms of each polynomial so that the powers of x are in descendingorder.
16. 13x2 " 5 ( 6x3 " x 17. 4x ( 2x5 " 6x3 ( 2
18. g2x " 3gx3 ( 7g3 ( 4x2 19. "11x2y3 ( 6y " 2xy (2x4
20. 7a2x2 ( 17 " a3x3 ( 2ax 21. 12rx3 ( 9r6 ( r2x ( 8x6
22. MONEY Write a polynomial to represent the value of t ten-dollar bills, f fifty-dollarbills, and h one-hundred-dollar bills.
23. GRAVITY The height above the ground of a ball thrown up with a velocity of 96 feet persecond from a height of 6 feet is 6 ( 96t " 16t2 feet, where t is the time in seconds.According to this model, how high is the ball after 7 seconds? Explain.
da
b
b
1'5
Practice Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
Reading to Learn MathematicsPolynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
© Glencoe/McGraw-Hill 477 Glencoe Algebra 1
Less
on 8
-4
Pre-Activity How are polynomials useful in modeling data?
Read the introduction to Lesson 8-4 at the top of page 432 in your textbook.
• How many terms does t4 ! 9t3 " 26t ! 18t " 76 have?
• What could you call a polynomial with just one term?
Reading the Lesson
1. What is the meaning of the prefixes mono-, bi-, and tri-?
2. Write examples of words that begin with the prefixes mono-, bi-, and tri-.
3. Complete the table.
monomial binomial trinomialpolynomial with more
than three terms
Example 3r 2t 2x2 " 3x 5x2 " 3x " 2 7s2 " s4 " 2s3 ! s " 5
Number of Terms
4. What is the degree of the monomial 3xy2z?
5. What is the degree of the polynomial 4x4 " 2x3y3 " y2 " 14? Explain how you foundyour answer.
Helping You Remember
6. Use a dictionary to find the meaning of the terms ascending and descending. Write theirmeanings and then describe a situation in your everyday life that relates to them.
© Glencoe/McGraw-Hill 478 Glencoe Algebra 1
Polynomial FunctionsSuppose a linear equation such as 23x " y # 4 is solved for y. Then an equivalent equation,y # 3x " 4, is found. Expressed in this way, y is a function of x, or f(x) # 3x " 4. Notice that the right side of the equation is a binomial of degree 1.
Higher-degree polynomials in x may also form functions. An example is f(x) # x3 " 1, which is a polynomial function of degree 3. You can graph this function using a table of ordered pairs, as shown at the right.
For each of the following polynomial functions, make a tableof values for x and y ! f(x). Then draw the graph on the grid.
1. f(x) # 1 ! x2 2. f(x) # x2 ! 5
3. f(x) # x2 " 4x ! 1 4. f(x) # x3
x
y
Ox
y
O
x
y
Ox
y
O
x
y
O
x y
!1$12$ !2$
38$
!1 0
0 1
1 2
1$12$ 4 $
38$
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
Study Guide and InterventionAdding and Subtracting Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
6-58-5
© Glencoe/McGraw-Hill 479 Glencoe Algebra 1
Less
on 8
-5
Add Polynomials To add polynomials, you can group like terms horizontally or writethem in column form, aligning like terms vertically. Like terms are monomial terms thatare either identical or differ only in their coefficients, such as 3p and !5p or 2x2y and 8x2y.
Find (2x2 " x # 8) "(3x # 4x2 " 2).
Horizontal MethodGroup like terms.
(2x2 " x ! 8) " (3x ! 4x2 " 2)# [(2x2 " (!4x2)] " (x " 3x ) " [(!8) " 2)]# !2x2 " 4x ! 6.
The sum is !2x2 " 4x ! 6.
Find (3x2 " 5xy) "(xy " 2x2).
Vertical MethodAlign like terms in columns and add.
3x2 " 5xy(") 2x2 " xy Put the terms in descending order.
5x2 " 6xy
The sum is 5x2 " 6xy.
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find each sum.
1. (4a ! 5) " (3a " 6) 2. (6x " 9) " (4x2 ! 7)
3. (6xy " 2y " 6x) " (4xy ! x) 4. (x2 " y2) " (!x2 " y2)
5. (3p2 ! 2p " 3) " (p2 ! 7p " 7) 6. (2x2 " 5xy " 4y2) " (!xy ! 6x2 " 2y2)
7. (5p " 2q) " (2p2 ! 8q " 1) 8. (4x2 ! x " 4) " (5x " 2x2 " 2)
9. (6x2 " 3x) " (x2 ! 4x ! 3) 10. (x2 " 2xy " y2) " (x2 ! xy ! 2y2)
11. (2a ! 4b ! c) " (!2a ! b ! 4c) 12. (6xy2 " 4xy) " (2xy ! 10xy2 " y2)
13. (2p ! 5q) " (3p " 6q) " (p ! q) 14. (2x2 ! 6) " (5x2 " 2) " (!x2 ! 7)
15. (3z2 " 5z) " (z2 " 2z) " (z ! 4) 16. (8x2 " 4x " 3y2 " y) " (6x2 ! x " 4y)
© Glencoe/McGraw-Hill 480 Glencoe Algebra 1
Subtract Polynomials You can subtract a polynomial by adding its additive inverse.To find the additive inverse of a polynomial, replace each term with its additive inverse or opposite.
Find (3x2 " 2x # 6) # (2x " x2 " 3).
Study Guide and Intervention (continued)
Adding and Subtracting Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
ExampleExampleHorizontal Method
Use additive inverses to rewrite as addition.Then group like terms.
(3x2 " 2x ! 6) ! (2x " x2 " 3)# (3x2 " 2x ! 6) " [(!2x)" (!x2) " (!3)]# [3x2 " (!x2)] " [2x " (!2x)] " [!6 " (!3)]# 2x2 " (!9)# 2x2 ! 9
The difference is 2x2 ! 9.
Vertical Method
Align like terms in columns andsubtract by adding the additive inverse.
3x2 " 2x ! 6(!) x2 " 2x " 3
3x2 " 2x ! 6(") !x2 ! 2x ! 3
2x2 ! 9The difference is 2x2 ! 9.
ExercisesExercises
Find each difference.
1. (3a ! 5) ! (5a " 1) 2. (9x " 2) ! (!3x2 ! 5)
3. (9xy " y ! 2x) ! (6xy ! 2x) 4. (x2 " y2) ! (!x2 " y2)
5. (6p2 " 4p " 5) ! (2p2 ! 5p " 1) 6. (6x2 " 5xy ! 2y2) ! (!xy ! 2x2 ! 4y2)
7. (8p ! 5q) ! (!6p2 " 6q ! 3) 8. (8x2 ! 4x ! 3) ! (!2x ! x2 " 5)
9. (3x2 ! 2x) ! (3x2 " 5x ! 1) 10. (4x2 " 6xy " 2y2) ! (!x2 " 2xy ! 5y2)
11. (2h ! 6j ! 2k) ! (!7h ! 5j ! 4k) 12. (9xy2 " 5xy) ! (!2xy ! 8xy2)
13. (2a ! 8b) ! (!3a " 5b) 14. (2x2 ! 8) ! (!2x2 ! 6)
15. (6z2 " 4z " 2) ! (4z2 " z) 16. (6x2 ! 5x " 1) ! (!7x2 ! 2x " 4)
Skills PracticeAdding and Subtracting Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
© Glencoe/McGraw-Hill 481 Glencoe Algebra 1
Less
on 8
-5
Find each sum or difference.
1. (2x ! 3y) ! (4x ! 9y) 2. (6s ! 5t) ! (4t ! 8s)
3. (5a ! 9b) " (2a ! 4b) 4. (11m " 7n) " (2m ! 6n)
5. (m2 " m) ! (2m ! m2) 6. (x2 " 3x) " (2x2 ! 5x)
7. (d2 " d ! 5) " (2d ! 5) 8. (2e2 " 5e) ! (7e " 3e2)
9. (5f ! g " 2) ! ("2f ! 3) 10. (6k2 ! 2k ! 9) ! (4k2 " 5k)
11. (x3 " x ! 1) " (3x " 1) 12. (b2 ! ab " 2) " (2b2 ! 2ab)
13. (7z2 ! 4 " z) " ("5 ! 3z2) 14. (5 ! 4n ! 2m) ! ("6m " 8)
15. (4t2 ! 2) ! ("4 ! 2t) 16. (3g3 ! 7g) " (4g ! 8g3)
17. (2a2 ! 8a ! 4) " (a2 " 3) 18. (3x2 " 7x ! 5) " ("x2 ! 4x)
19. (7z2 ! z ! 1) " ("4z ! 3z2 " 3) 20. (2c2 ! 7c ! 4) ! (c2 ! 1 " 9c)
21. (n2 ! 3n ! 2) " (2n2 " 6n " 2) 22. (a2 ! ab " 3b2) ! (b2 ! 4a2 " ab)
23. (!2 " 5! " 6) ! (2!2 ! 5 ! !) 24. (2m2 ! 5m ! 1) " (4m2 " 3m " 3)
25. (x2 " 6x ! 2) " ("5x2 ! 7x " 4) 26. (5b2 " 9b " 5) ! (b2 " 6 ! 2b)
27. (2x2 " 6x " 2) ! (x2 ! 4x) ! (3x2 ! x ! 5)
© Glencoe/McGraw-Hill 482 Glencoe Algebra 1
Find each sum or difference.
1. (4y ! 5) ! ("7y " 1) 2. ("x2 ! 3x) " (5x ! 2x2)
3. (4k2 ! 8k ! 2) " (2k ! 3) 4. (2m2 ! 6m) ! (m2 " 5m ! 7)
5. (2w2 " 3w ! 1) ! (4w " 7) 6. (g3 ! 2g2) " (6g " 4g2 ! 2g3)
7. (5a2 ! 6a ! 2) " (7a2 " 7a ! 5) 8. ("4p2 " p ! 9) ! (p2 ! 3p " 1)
9. (x3 " 3x ! 1) " (x3 ! 7 " 12x) 10. (6c2 " c ! 1) " ("4 ! 2c2 ! 8c)
11. ("b3 ! 8bc2 ! 5) " (7bc2 " 2 ! b3) 12. (5n2 " 3n ! 2) ! ("n ! 2n2 " 4)
13. (4y2 ! 2y " 8) " (7y2 ! 4 " y) 14. (w2 " 4w " 1) ! ("5 ! 5w2 " 3w)
15. (4u2 " 2u " 3) ! (3u2 " u ! 4) 16. (5b2 " 8 ! 2b) " (b ! 9b2 ! 5)
17. (4d2 ! 2d ! 2) ! (5d2 " 2 " d) 18. (8x2 ! x " 6) " ("x2 ! 2x " 3)
19. (3h2 ! 7h " 1) " (4h ! 8h2 ! 1) 20. (4m2 " 3m ! 10) ! (m2 ! m " 2)
21. (x2 ! y2 " 6) " (5x2 " y2 " 5) 22. (7t2 ! 2 " t) ! (t2 " 7 " 2t)
23. (k3 " 2k2 ! 4k ! 6) " ("4k ! k2 " 3) 24. (9j2 ! j ! jk) ! ("3j2 " jk " 4j)
25. (2x ! 6y " 3z) ! (4x ! 6z " 8y) ! (x " 3y ! z)
26. (6f 2 " 7f " 3) " (5f 2 " 1 ! 2f ) " (2f 2 " 3 ! f )
27. BUSINESS The polynomial s3 " 70s2 ! 1500s " 10,800 models the profit a companymakes on selling an item at a price s. A second item sold at the same price brings in aprofit of s3 " 30s2 ! 450s " 5000. Write a polynomial that expresses the total profitfrom the sale of both items.
28. GEOMETRY The measures of two sides of a triangle are given.If P is the perimeter, and P # 10x ! 5y, find the measure of the third side.
3x ! 4y
5x " y
Practice Adding and Subtracting Polynomials
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8-58-5
Reading to Learn MathematicsAdding and Subtracting Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
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© Glencoe/McGraw-Hill 483 Glencoe Algebra 1
Less
on 8
-5
Pre-Activity How can adding polynomials help you model sales?
Read the introduction to Lesson 8-5 at the top of page 439 in your textbook.
What operation would you use to find how much more the traditional toysales R were than the video games sales V ?
Reading the Lesson
1. Use the example ("3x3 ! 4x2 ! 5x ! 1) ! ("5x3 " 2x2 ! 2x " 7).
a. Show what is meant by grouping like terms horizontally.
b. Show what is meant by aligning like terms vertically.
c. Choose one method, then add the polynomials.
2. How is subtracting a polynomial like subtracting a rational number?
3. An algebra student got the following exercise wrong on his homework. What was his error?
(3x5 " 3x4 ! 2x3 " 4x2 ! 5) " (2x5 " x3 ! 2x2 " 4)# [3x5 ! ("2x5)] ! ("3x4) ! [2x3 ! ("x3)] ! ["4x2 ! ("2x2)] ! (5 ! 4)# x5 " 3x4 ! x3 " 6x2 ! 9
Helping You Remember
4. How is adding and subtracting polynomials vertically like adding and subtractingdecimals vertically?
© Glencoe/McGraw-Hill 484 Glencoe Algebra 1
Circular Areas and VolumesArea of Circle Volume of Cylinder Volume of Cone
A # !r2 V # !r2h V # !r2h
Write an algebraic expression for each shaded area. (Recall that the diameter of a circle is twice its radius.)
1. 2. 3.
Write an algebraic expression for the total volume of each figure.
4. 5.
Each figure has a cylindrical hole with a radius of 2 inches and a height of 5 inches. Find each volume.
6. 7.
3x
4x
5x
7x
3x—2
x ! a
x
x ! b
x
2x5x
3x2x2x x
y x x yxx
r
h
r
h
r
1$3
Enrichment
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8-58-5
Study Guide and InterventionMultiplying a Polynomial by a Monomial
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
© Glencoe/McGraw-Hill 485 Glencoe Algebra 1
Less
on 8
-6
Product of Monomial and Polynomial The Distributive Property can be used tomultiply a polynomial by a monomial. You can multiply horizontally or vertically. Sometimesmultiplying results in like terms. The products can be simplified by combining like terms.
Find "3x2(4x2 ! 6x " 8).
Horizontal Method"3x2(4x2 ! 6x " 8)
# "3x2(4x2) ! ("3x2)(6x) " ("3x2)(8)# "12x4 ! ("18x3) " ("24x2)# "12x4 " 18x3 ! 24x2
Vertical Method4x2 ! 6x " 8
(%) "3x2
"12x4 " 18x3 ! 24x2
The product is "12x4 " 18x3 ! 24x2.
Simplify "2(4x2 ! 5x) "x(x2 ! 6x).
"2(4x2 ! 5x) " x( x2 ! 6x)# "2(4x2) ! ("2)(5x) ! ("x)(x2) ! ("x)(6x)# "8x2 ! ("10x) ! ("x3) ! ("6x2)# ("x3) ! ["8x2 ! ("6x2)] ! ("10x)# "x3 " 14x2 " 10x
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find each product.
1. x(5x ! x2) 2. x(4x2 ! 3x ! 2) 3. "2xy(2y ! 4x2)
4. "2g( g2 " 2g ! 2) 5. 3x(x4 ! x3! x2) 6. "4x(2x3 " 2x ! 3)
7. "4cx(10 ! 3x) 8. 3y("4x " 6x3" 2y) 9. 2x2y2(3xy ! 2y ! 5x)
Simplify.
10. x(3x " 4) " 5x 11. "x(2x2 " 4x) " 6x2
12. 6a(2a " b) ! 2a("4a ! 5b) 13. 4r(2r2 " 3r ! 5) ! 6r(4r2 ! 2r ! 8)
14. 4n(3n2 ! n " 4) " n(3 " n) 15. 2b(b2 ! 4b ! 8) " 3b(3b2 ! 9b " 18)
16. "2z(4z2 " 3z ! 1) " z(3z2 ! 2z " 1) 17. 2(4x2 " 2x) " 3("6x2 ! 4) ! 2x(x " 1)
© Glencoe/McGraw-Hill 486 Glencoe Algebra 1
Solve Equations with Polynomial Expressions Many equations containpolynomials that must be added, subtracted, or multiplied before the equation can be solved.
Solve 4(n " 2) ! 5n # 6(3 " n) ! 19.
4(n " 2) ! 5n # 6(3 " n) ! 19 Original equation
4n " 8 ! 5n # 18 " 6n ! 19 Distributive Property
9n " 8 # 37 " 6n Combine like terms.
15n " 8 # 37 Add 6n to both sides.
15n # 45 Add 8 to both sides.
n # 3 Divide each side by 15.
The solution is 3.
Solve each equation.
1. 2(a " 3) # 3("2a ! 6) 2. 3(x ! 5) " 6 # 18
3. 3x(x " 5) " 3x2 # "30 4. 6(x2 ! 2x) # 2(3x2 ! 12)
5. 4(2p ! 1) " 12p # 2(8p ! 12) 6. 2(6x ! 4) ! 2 # 4(x " 4)
7. "2(4y " 3) " 8y ! 6 # 4(y " 2) 8. c(c ! 2) " c(c " 6) # 10c " 12
9. 3(x2 " 2x) # 3x2 ! 5x " 11 10. 2(4x ! 3) ! 2 # "4(x ! 1)
11. 3(2h " 6) " (2h ! 1) # 9 12. 3(y ! 5) " (4y " 8) # "2y ! 10
13. 3(2a " 6) " ("3a " 1) # 4a " 2 14. 5(2x2 " 1) " (10x2 " 6) # "(x ! 2)
15. 3(x ! 2) ! 2(x ! 1) # "5(x " 3) 16. 4(3p2 ! 2p) " 12p2 # 2(8p ! 6)
Study Guide and Intervention (continued)
Multiplying a Polynomial by a Monomial
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8-68-6
ExampleExample
ExercisesExercises
Skills PracticeMultiplying a Polynomial by a Monomial
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© Glencoe/McGraw-Hill 487 Glencoe Algebra 1
Less
on 8
-6
Find each product.
1. a(4a ! 3) 2. "c(11c ! 4)
3. x(2x " 5) 4. 2y(y " 4)
5. "3n(n2 ! 2n) 6. 4h(3h " 5)
7. 3x(5x2 " x ! 4) 8. 7c(5 " 2c2 ! c3)
9. "4b(1 " 9b " 2b2) 10. 6y("5 " y ! 4y2)
11. 2m2(2m2 ! 3m " 5) 12. "3n2("2n2 ! 3n ! 4)
Simplify.
13. w(3w ! 2) ! 5w 14. f(5f " 3) " 2f
15. "p(2p " 8) " 5p 16. y2("4y ! 5) " 6y2
17. 2x(3x2 ! 4) " 3x3 18. 4a(5a2 " 4) ! 9a
19. 4b("5b " 3) " 2(b2 " 7b " 4) 20. 3m(3m ! 6) " 3(m2 ! 4m ! 1)
Solve each equation.
21. 3(a ! 2) ! 5 # 2a ! 4 22. 2(4x ! 2) " 8 # 4(x ! 3)
23. 5( y ! 1) ! 2 # 4( y ! 2) " 6 24. 4(b ! 6) # 2(b ! 5) ! 2
25. 6(m " 2) ! 14 # 3(m ! 2) " 10 26. 3(c ! 5) " 2 # 2(c ! 6) ! 2
© Glencoe/McGraw-Hill 488 Glencoe Algebra 1
Find each product.
1. 2h(!7h2 ! 4h) 2. 6pq(3p2 " 4q) 3. !2u2n(4u ! 2n)
4. 5jk(3jk " 2k) 5. !3rs(!2s2 " 3r) 6. 4mg2(2mg " 4g)
7. ! m(8m2 " m ! 7) 8. ! n2(!9n2 " 3n " 6)
Simplify.
9. !2!(3! ! 4) " 7! 10. 5w(!7w " 3) " 2w(!2w2 " 19w " 2)
11. 6t(2t ! 3) ! 5(2t2 " 9t ! 3) 12. !2(3m3 " 5m " 6) " 3m(2m2 " 3m " 1)
13. !3g(7g ! 2) " 3(g2 " 2g " 1) ! 3g(!5g " 3)
14. 4z2(z ! 7) ! 5z(z2 ! 2z ! 2) " 3z(4z ! 2)
Solve each equation.
15. 5(2s ! 1) " 3 # 3(3s " 2) 16. 3(3u " 2) " 5 # 2(2u ! 2)
17. 4(8n " 3) ! 5 # 2(6n " 8) " 1 18. 8(3b " 1) # 4(b " 3) ! 9
19. h(h ! 3) ! 2h # h(h ! 2) ! 12 20. w(w " 6) " 4w # !7 " w(w " 9)
21. t(t " 4) ! 1 # t(t " 2) " 2 22. u(u ! 5) " 8u # u(u " 2) ! 4
23. NUMBER THEORY Let x be an integer. What is the product of twice the integer addedto three times the next consecutive integer?
INVESTMENTS For Exercises 24–26, use the following information.Kent invested $5,000 in a retirement plan. He allocated x dollars of the money to a bondaccount that earns 4% interest per year and the rest to a traditional account that earns 5%interest per year.
24. Write an expression that represents the amount of money invested in the traditionalaccount.
25. Write a polynomial model in simplest form for the total amount of money T Kent hasinvested after one year. (Hint: Each account has A " IA dollars, where A is the originalamount in the account and I is its interest rate.)
26. If Kent put $500 in the bond account, how much money does he have in his retirementplan after one year?
2$3
1$4
Practice Multiplying a Polynomial by a Monomial
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
Reading to Learn MathematicsMultiplying a Polynomial by a Monomial
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8-68-6
© Glencoe/McGraw-Hill 489 Glencoe Algebra 1
Less
on 8
-6
Pre-Activity How is finding the product of a monomial and a polynomial relatedto finding the area of a rectangle?
Read the introduction to Lesson 8-6 at the top of page 444 in your textbook.
You may recall that the formula for the area of a rectangle is A # !w. In
this rectangle, ! # and w # . How would you
substitute these values in the area formula?
Reading the Lesson
1. Refer to Lesson 8-6.
a. How is the Distributive Property used to multiply a polynomial by a monomial?
b. Use the Distributive Property to complete the following.
2y2(3y2 " 2y ! 7) # 2y2( ) " 2y2( ) ! 2y2( )
# " !
!3x3(x3 ! 2x2 " 3) # ! "
# " !
2. What is the difference between simplifying an expression and solving an equation?
Helping You Remember
3. Use the equation 2x(x ! 5) " 3x(x " 3) # 5x(x " 7) ! 9 to show how you would explainthe process of solving equations with polynomial expressions to another algebra student.
© Glencoe/McGraw-Hill 490 Glencoe Algebra 1
Figurate NumbersThe numbers below are called pentagonal numbers. They are the numbers of dots or disksthat can be arranged as pentagons.
1. Find the product n(3n ! 1).
2. Evaluate the product in Exercise 1 for values of n from 1 through 4.
3. What do you notice?
4. Find the next six pentagonal numbers.
5. Find the product n(n " 1).
6. Evaluate the product in Exercise 5 for values of n from 1 through 5. On another sheet ofpaper, make drawings to show why these numbers are called the triangular numbers.
7. Find the product n(2n ! 1).
8. Evaluate the product in Exercise 7 for values of n from 1 through 5. Draw thesehexagonal numbers.
9. Find the first 5 square numbers. Also, write the general expression for any squarenumber.
The numbers you have explored above are called the plane figurate numbers because theycan be arranged to make geometric figures. You can also create solid figurate numbers.
10. If you pile 10 oranges into a pyramid with a triangle as a base, you get one of thetetrahedral numbers. How many layers are there in the pyramid? How many orangesare there in the bottom layers?
11. Evaluate the expression n3 " n2 " n for values of n from 1 through 5 to find the
first five tetrahedral numbers.
1$3
1$2
1$6
1$2
1$2
●●
221251
Enrichment
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Study Guide and InterventionMultiplying Polynomials
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8-78-7
© Glencoe/McGraw-Hill 491 Glencoe Algebra 1
Less
on 8
-7
Multiply Binomials To multiply two binomials, you can apply the Distributive Propertytwice. A useful way to keep track of terms in the product is to use the FOIL method asillustrated in Example 2.
Find (x ! 3)(x " 4).
Horizontal Method(x ! 3)(x " 4)
# x(x " 4) ! 3(x " 4)# (x)(x) ! x("4) ! 3(x)! 3("4)# x2 " 4x ! 3x " 12# x2 " x " 12
Vertical Method
x ! 3($) x " 4
"4x " 12x2 ! 3xx2 " x " 12
The product is x2 " x " 12.
Find (x " 2)(x ! 5) usingthe FOIL method.
(x " 2)(x ! 5)First Outer Inner Last
# (x)(x) ! (x)(5) ! ("2)(x) ! ("2)(5)# x2 ! 5x ! ("2x) " 10# x2 ! 3x " 10
The product is x2 ! 3x " 10.
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find each product.
1. (x ! 2)(x ! 3) 2. (x " 4)(x ! 1) 3. (x " 6)(x " 2)
4. (p " 4)(p ! 2) 5. (y ! 5)(y ! 2) 6. (2x " 1)(x ! 5)
7. (3n " 4)(3n " 4) 8. (8m " 2)(8m ! 2) 9. (k ! 4)(5k " 1)
10. (3x ! 1)(4x ! 3) 11. (x " 8)("3x ! 1) 12. (5t ! 4)(2t " 6)
13. (5m " 3n)(4m " 2n) 14. (a " 3b)(2a " 5b) 15. (8x " 5)(8x ! 5)
16. (2n " 4)(2n ! 5) 17. (4m " 3)(5m " 5) 18. (7g " 4)(7g ! 4)
© Glencoe/McGraw-Hill 492 Glencoe Algebra 1
Multiply Polynomials The Distributive Property can be used to multiply any two polynomials.
Find (3x ! 2)(2x2 " 4x ! 5).
(3x ! 2)(2x2 " 4x ! 5)# 3x(2x2 " 4x ! 5) ! 2(2x2 " 4x ! 5) Distributive Property
# 6x3 " 12x2 ! 15x ! 4x2 " 8x ! 10 Distributive Property
# 6x3 " 8x2 ! 7x ! 10 Combine like terms.
The product is 6x3 " 8x2 ! 7x ! 10.
Find each product.
1. (x ! 2)(x2 " 2x ! 1) 2. (x ! 3)(2x2 ! x " 3)
3. (2x " 1)(x2 " x ! 2) 4. (p " 3)(p2 " 4p ! 2)
5. (3k ! 2)(k2 ! k " 4) 6. (2t ! 1)(10t2 " 2t " 4)
7. (3n " 4)(n2 ! 5n " 4) 8. (8x " 2)(3x2 ! 2x " 1)
9. (2a ! 4)(2a2 " 8a ! 3) 10. (3x " 4)(2x2 ! 3x ! 3)
11. (n2 ! 2n " 1)(n2 ! n ! 2) 12. (t2 ! 4t " 1)(2t2 " t " 3)
13. (y2 " 5y ! 3)(2y2 ! 7y " 4) 14. (3b2 " 2b ! 1)(2b2 " 3b " 4)
Study Guide and Intervention (continued)
Multiplying Polynomials
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8-78-7
ExampleExample
ExercisesExercises
Skills PracticeMultiplying Polynomials
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© Glencoe/McGraw-Hill 493 Glencoe Algebra 1
Less
on 8
-7
Find each product.
1. (m ! 4)(m ! 1) 2. (x ! 2)(x ! 2)
3. (b ! 3)(b ! 4) 4. (t ! 4)(t " 3)
5. (r ! 1)(r " 2) 6. (z " 5)(z ! 1)
7. (3c ! 1)(c " 2) 8. (2x " 6)(x ! 3)
9. (d " 1)(5d " 4) 10. (2! ! 5)(! " 4)
11. (3n " 7)(n ! 3) 12. (q ! 5)(5q " 1)
13. (3b ! 3)(3b " 2) 14. (2m ! 2)(3m " 3)
15. (4c ! 1)(2c ! 1) 16. (5a " 2)(2a " 3)
17. (4h " 2)(4h " 1) 18. (x " y)(2x " y)
19. (e ! 4)(e2 ! 3e " 6) 20. (t ! 1)(t2 ! 2t ! 4)
21. (k ! 4)(k2 ! 3k " 6) 22. (m ! 3)(m2 ! 3m ! 5)
GEOMETRY Write an expression to represent the area of each figure.
23. 24.
2x ! 4
2x ! 5
4x " 1
© Glencoe/McGraw-Hill 494 Glencoe Algebra 1
Find each product.
1. (q ! 6)(q ! 5) 2. (x ! 7)(x ! 4) 3. (s ! 5)(s " 6)
4. (n " 4)(n " 6) 5. (a " 5)(a " 8) 6. (w " 6)(w " 9)
7. (4c ! 6)(c " 4) 8. (2x " 9)(2x ! 4) 9. (4d " 5)(2d " 3)
10. (4b ! 3)(3b " 4) 11. (4m ! 2)(4m " 3) 12. (5c " 5)(7c ! 9)
13. (6a " 3)(7a " 4) 14. (6h " 3)(4h " 2) 15. (2x " 2)(5x " 4)
16. (3a " b)(2a " b) 17. (4g ! 3h)(2g ! 3h) 18. (4x ! y)(4x ! y)
19. (m ! 5)(m2 ! 4m " 8) 20. (t ! 3)(t2 ! 4t ! 7)
21. (2h ! 3)(2h2 ! 3h ! 4) 22. (3d ! 3)(2d2 ! 5d " 2)
23. (3q ! 2)(9q2 " 12q ! 4) 24. (3r ! 2)(9r2 ! 6r ! 4)
25. (3c2 ! 2c " 1)(2c2 ! c ! 9) 26. (2!2 ! ! ! 3)(4!2 ! 2! " 2)
27. (2x2 " 2x " 3)(2x2 " 4x ! 3) 28. (3y2 ! 2y ! 2)(3y2 " 4y " 5)
GEOMETRY Write an expression to represent the area of each figure.
29. 30.
31. NUMBER THEORY Let x be an even integer. What is the product of the next twoconsecutive even integers?
32. GEOMETRY The volume of a rectangular pyramid is one third the product of the areaof its base and its height. Find an expression for the volume of a rectangular pyramidwhose base has an area of 3x2 ! 12x ! 9 square feet and whose height is x ! 3 feet.
3x ! 2
5x " 4
x ! 1
4x ! 2
2x " 2
Practice Multiplying Polynomials
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8-78-7
Reading to Learn MathematicsMultiplying Polynomials
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8-78-7
© Glencoe/McGraw-Hill 495 Glencoe Algebra 1
Less
on 8
-7
Pre-Activity How is multiplying binomials similar to multiplying two-digitnumbers?
Read the introduction to Lesson 8-7 at the top of page 452 in your textbook.
In your own words, explain how the distributive property is used twice tomultiply two-digit numbers.
Reading the Lesson
1. How is multiplying binomials similar to multiplying two-digit numbers?
2. Complete the table using the FOIL method.
Product of!
Product of!
Product of!
Product of First Terms Outer Terms Inner Terms Last Terms
(x ! 5)(x " 3) # ! ! !
# ! ! !
# ! "
(3y ! 6)(y " 2) # ! ! !
# ! ! !
# "
Helping You Remember
3. Think of a method for remembering all the product combinations used in the FOILmethod for multiplying two binomials. Describe your method using words or a diagram.
© Glencoe/McGraw-Hill 496 Glencoe Algebra 1
Pascal’s TriangleThis arrangement of numbers is called Pascal’s Triangle.It was first published in 1665, but was known hundreds of years earlier.
1. Each number in the triangle is found by adding two numbers.What two numbers were added to get the 6 in the 5th row?
2. Describe how to create the 6th row of Pascal’s Triangle.
3. Write the numbers for rows 6 through 10 of the triangle.
Row 6:
Row 7:
Row 8:
Row 9:
Row 10:
Multiply to find the expanded form of each product.
4. (a ! b)2
5. (a ! b)3
6. (a ! b)4
Now compare the coefficients of the three products in Exercises 4–6 with Pascal’s Triangle.
7. Describe the relationship between the expanded form of (a ! b)n and Pascal’s Triangle.
8. Use Pascal’s Triangle to write the expanded form of (a ! b)6.
11 1
1 2 11 3 3 1
1 4 6 4 1
Enrichment
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Study Guide and InterventionSpecial Products
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© Glencoe/McGraw-Hill 497 Glencoe Algebra 1
Less
on 8
-8
Squares of Sums and Differences Some pairs of binomials have products thatfollow specific patterns. One such pattern is called the square of a sum. Another is called thesquare of a difference.
Square of a sum (a ! b)2 # (a ! b)(a ! b) # a2 ! 2ab ! b2
Square of a difference (a " b)2 # (a " b)(a " b) # a2 " 2ab ! b2
Find (3a ! 4)(3a ! 4).
Use the square of a sum pattern, with a #3a and b # 4.
(3a ! 4)(3a ! 4) # (3a)2 ! 2(3a)(4) ! (4)2
# 9a2 ! 24a ! 16
The product is 9a2 ! 24a ! 16.
Find (2z " 9)(2z " 9).
Use the square of a difference pattern witha # 2z and b # 9.
(2z " 9)(2z " 9) # (2z)2 " 2(2z)(9) ! (9)(9)# 4z2 " 36z ! 81
The product is 4z2 " 36z ! 81.
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find each product.
1. (x " 6)2 2. (3p ! 4)2 3. (4x " 5)2
4. (2x " 1)2 5. (2h ! 3)2 6. (m ! 5)2
7. (c ! 3)2 8. (3 " p)2 9. (x " 5y)2
10. (8y ! 4)2 11. (8 ! x)2 12. (3a " 2b)2
13. (2x " 8)2 14. (x2 ! 1)2 15. (m2 " 2)2
16. (x3 " 1)2 17. (2h2 " k2)2 18. ! x ! 3"2
19. (x " 4y2)2 20. (2p ! 4q)2 21. ! x " 2"22%3
1%4
© Glencoe/McGraw-Hill 498 Glencoe Algebra 1
Product of a Sum and a Difference There is also a pattern for the product of asum and a difference of the same two terms, (a ! b)(a " b). The product is called thedifference of squares.
Product of a Sum and a Difference (a ! b)(a " b) # a2 " b2
Find (5x ! 3y)(5x " 3y).
(a ! b)(a " b) # a2 " b2 Product of a Sum and a Difference
(5x ! 3y)(5x " 3y) # (5x)2 " (3y)2 a # 5x and b # 3y
# 25x2 " 9y2 Simplify.
The product is 25x2 " 9y2.
Find each product.
1. (x " 4)(x ! 4) 2. (p ! 2)(p " 2) 3. (4x " 5)(4x ! 5)
4. (2x " 1)(2x ! 1) 5. (h ! 7)(h " 7) 6. (m " 5)(m ! 5)
7. (2c " 3)(2c ! 3) 8. (3 " 5q)(3 ! 5q) 9. (x " y)(x ! y)
10. ( y " 4x)( y ! 4x) 11. (8 ! 4x)(8 " 4x) 12. (3a " 2b)(3a ! 2b)
13. (3y " 8)(3y ! 8) 14. (x2 " 1)(x2 ! 1) 15. (m2 " 5)(m2 ! 5)
16. (x3 " 2)(x3 ! 2) 17. (h2 " k2)(h2 ! k2) 18. ! x ! 2"! x " 2"
19. (3x " 2y2)(3x ! 2y2) 20. (2p " 5s)(2p ! 5s) 21. ! x " 2y"! x ! 2y"4%3
4%3
1%4
1%4
Study Guide and Intervention (continued)
Special Products
NAME ______________________________________________ DATE ____________ PERIOD _____
8-88-8
ExampleExample
ExercisesExercises
Skills PracticeSpecial Products
NAME ______________________________________________ DATE ____________ PERIOD _____
8-88-8
© Glencoe/McGraw-Hill 499 Glencoe Algebra 1
Less
on 8
-8
Find each product.
1. (n ! 3)2 2. (x ! 4)(x ! 4)
3. ( y " 7)2 4. (t " 3)(t " 3)
5. (b ! 1)(b " 1) 6. (a " 5)(a ! 5)
7. (p " 4)2 8. (z ! 3)(z " 3)
9. (! ! 2)(! ! 2) 10. (r " 1)(r " 1)
11. (3g ! 2)(3g " 2) 12. (2m " 3)(2m ! 3)
13. (6 ! u)2 14. (r ! s)2
15. (3q ! 1)(3q " 1) 16. (c " e)2
17. (2k " 2)2 18. (w ! 3h)2
19. (3p " 4)(3p ! 4) 20. (t ! 2u)2
21. (x " 4y)2 22. (3b ! 7)(3b " 7)
23. (3y " 3g)(3y ! 3g) 24. (s2 ! r2)2
25. (2k ! m2)2 26. (3u2 " n)2
27. GEOMETRY The length of a rectangle is the sum of two whole numbers. The width ofthe rectangle is the difference of the same two whole numbers. Using these facts, write averbal expression for the area of the rectangle.
© Glencoe/McGraw-Hill 500 Glencoe Algebra 1
Find each product.
1. (n ! 9)2 2. (q ! 8)2 3. (! " 10)2
4. (r " 11)2 5. ( p ! 7)2 6. (b ! 6)(b " 6)
7. (z ! 13)(z " 13) 8. (4e ! 2)2 9. (5w " 4)2
10. (6h " 1)2 11. (3s ! 4)2 12. (7v " 2)2
13. (7k ! 3)(7k " 3) 14. (4d " 7)(4d ! 7) 15. (3g ! 9h)(3g " 9h)
16. (4q ! 5t)(4q " 5t) 17. (a ! 6u)2 18. (5r ! s)2
19. (6c " m)2 20. (k " 6y)2 21. (u " 7p)2
22. (4b " 7v)2 23. (6n ! 4p)2 24. (5q ! 6s)2
25. (6a " 7b)(6a ! 7b) 26. (8h ! 3d)(8h " 3d) 27. (9x ! 2y2)2
28. (3p3 ! 2m)2 29. (5a2 " 2b)2 30. (4m3 " 2t)2
31. (6e3 " c)2 32. (2b2 " g)(2b2 ! g) 33. (2v2 ! 3e2)(2v2 ! 3e2)
34. GEOMETRY Janelle wants to enlarge a square graph that she has made so that a sideof the new graph will be 1 inch more than twice the original side s. What trinomialrepresents the area of the enlarged graph?
GENETICS For Exercises 35 and 36, use the following information.In a guinea pig, pure black hair coloring B is dominant over pure white coloring b. Supposetwo hybrid Bb guinea pigs, with black hair coloring, are bred.
35. Find an expression for the genetic make-up of the guinea pig offspring.
36. What is the probability that two hybrid guinea pigs with black hair coloring will producea guinea pig with white hair coloring?
Practice Special Products
NAME ______________________________________________ DATE ____________ PERIOD _____
8-88-8
Reading to Learn MathematicsSpecial Products
NAME ______________________________________________ DATE ____________ PERIOD _____
8-88-8
© Glencoe/McGraw-Hill 501 Glencoe Algebra 1
Less
on 8
-8
Pre-Activity When is the product of two binomials also a binomial?
Read the introduction to Lesson 8-8 at the top of page 458 in your textook.
What is a meant by the term trinomial product?
Reading the Lesson
1. Refer to the Key Concepts boxes on pages 458, 459, and 460.
a. When multiplying two binomials, there are three special products. What are the threespecial products that may result when multiplying two binomials?
b. Explain what is meant by the name of each special product.
c. Use the examples in the Key Concepts boxes to complete the table.
Symbols Product Example Product
Square of a Sum
Square of a Difference
Product of a Sum and a Difference
2. What is another phrase that describes the product of the sum and difference of two terms?
Helping You Remember
3. Explain how FOIL can help you remember how many terms are in the special productsstudied in this lesson.
© Glencoe/McGraw-Hill 502 Glencoe Algebra 1
Sums and Differences of Cubes Recall the formulas for finding some special products:
Perfect-square trinomials: (a ! b)2 " a2 ! 2ab ! b2 or (a # b)2 " a2 # 2ab ! b2
Difference of two squares: (a ! b)(a # b) " a2 # b2
A pattern also exists for finding the cube of a sum (a ! b)3.
1. Find the product of (a ! b)(a ! b)(a ! b).
2. Use the pattern from Exercise 1 to evaluate (x ! 2)3.
3. Based on your answer to Exercise 1, predict the pattern for the cube of a difference (a # b)3.
4. Find the product of (a # b)(a # b)(a # b) and compare it to your answer for Exercise 3.
5. Use the pattern from Exercise 4 to evaluate (x ! 4)3.
Find each product.
6. (x ! 6)3 7. (x # 10)3
8. (3x # y)3 9. (2x # y)3
10. (4x ! 3y)3 11. (5x ! 2)3
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-88-8
© Glencoe/McGraw-Hill A2 Glencoe Algebra 1
Stu
dy
Gu
ide
and I
nte
rven
tion
Mul
tiply
ing
Mon
omia
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
6-1
8-1
©G
lenc
oe/M
cGra
w-H
ill45
5G
lenc
oe A
lgeb
ra 1
Lesson 8-1
Mul
tipl
y M
onom
ials
A m
onom
iali
s a
num
ber,
a va
riab
le,o
r a
prod
uct
of a
num
ber
and
one
or m
ore
vari
able
s.A
n ex
pres
sion
of
the
form
xn
is c
alle
d a
pow
eran
d re
pres
ents
the
prod
uct
you
obta
in w
hen
xis
use
d as
a f
acto
r n
tim
es.T
o m
ulti
ply
two
pow
ers
that
hav
eth
e sa
me
base
,add
the
exp
onen
ts.
Pro
duct
of
Pow
ers
For
any
num
ber
aan
d al
l int
eger
s m
and
n, a
m!
an"
am#
n.
Sim
pli
fy (
3x6 )
(5x2
).
(3x6
)(5x
2 ) "
(3)(
5)(x
6!
x2)
Ass
ocia
tive
Pro
pert
y
"(3
!5)
(x6
# 2
)P
rodu
ct o
f Pow
ers
"15
x8S
impl
ify.
The
pro
duct
is 1
5x8 .
Sim
pli
fy (
!4a
3 b)(
3a2 b
5 ).
($4a
3 b)(
3a2 b
5 )"
($4)
(3)(
a3!
a2)(
b!
b5)
"$
12(a
3 #
2)(
b1 #
5)
"$
12a5
b6
The
pro
duct
is $
12a5
b6.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Sim
pli
fy.
1.y(
y5)
2.n2
!n7
3.($
7x2 )
(x4 )
y6
n9
!7x
6
4.x(
x2)(
x4)
5.m
!m
56.
($x3
)($
x4)
x7m
6x7
7.(2
a2)(
8a)
8.(r
s)(r
s3)(
s2)
9.(x
2 y)(
4xy3
)
16a3
r2s6
4x3 y
4
10.
(2a3
b)(6
b3)
11.(
$4x
3 )($
5x7 )
12.(
$3j
2 k4 )
(2jk
6 )
4a3 b
420
x10
!6j
3 k10
13.(
5a2 b
c3) !
abc4"
14.(
$5x
y)(4
x2)(
y4)
15.(
10x3
yz2 )
($2x
y5z)
a3b
2 c7
!20
x3 y
5!
20x
4 y6 z
3
1 % 5
1 % 3
©G
lenc
oe/M
cGra
w-H
ill45
6G
lenc
oe A
lgeb
ra 1
Pow
ers
of M
onom
ials
An
expr
essi
on o
f th
e fo
rm (x
m)n
is c
alle
d a
pow
er o
f a
pow
eran
d re
pres
ents
the
pro
duct
you
obt
ain
whe
n xm
is u
sed
as a
fac
tor
nti
mes
.To
find
the
pow
er o
f a
pow
er,m
ulti
ply
expo
nent
s.
Pow
er o
f a
Pow
erF
or a
ny n
umbe
r a
and
all i
nteg
ers
man
d n,
(am
)n"
amn .
Pow
er o
f a
Pro
duct
For
any
num
ber
aan
d al
l int
eger
s m
and
n, (
ab)m
"am
bm.
Sim
pli
fy (
!2a
b2)3
(a2 )
4 .
($2a
b2)3
(a2 )
4"
($2a
b2)3
(a8 )
Pow
er o
f a P
ower
"($
2)3 (
a3)(
b2)3
(a8 )
Pow
er o
f a P
rodu
ct
"($
2)3 (
a3)(
a8)(
b2)3
Com
mut
ativ
e P
rope
rty
"($
2)3 (
a11 )
(b2 )
3P
rodu
ct o
f Pow
ers
"$
8a11
b6P
ower
of a
Pow
er
The
pro
duct
is $
8a11
b6.
Sim
pli
fy.
1.(y
5 )2
2.(n
7 )4
3.(x
2 )5 (
x3)
y10
n28
x13
4.$
3(ab
4 )3
5.($
3ab4
)36.
(4x2
b)3
!3a
3 b12
!27
a3 b
1264
x6 b
3
7.(4
a2)2
(b3 )
8.(4
x)2 (
b3)
9.(x
2 y4 )
5
16a4
b3
16x
2 b3
x10 y
20
10.(
2a3 b
2 )(b
3 )2
11.(
$4x
y)3 (
$2x
2 )3
12.(
$3j
2 k3 )
2 (2j
2 k)3
2a3 b
851
2x9 y
372
j10 k
9
13.(
25a2
b)3 !
abc "2
14.(
2xy)
2 ($
3x2 )
(4y4
)15
.(2x
3 y2 z
2 )3 (
x2z)
4
625a
8 b5 c
2!
48x
4 y6
8x17
y6 z
10
16.(
$2n
6 y5 )
($6n
3 y2 )
(ny)
317
.($
3a3 n
4 )($
3a3 n
)418
.$3(
2x)4
(4x5
y)2
12n1
2 y10
!24
3a15
n8
!76
8x14
y2
1 % 5Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Mul
tiply
ing
Mon
omia
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-1
8-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 8-1)
© Glencoe/McGraw-Hill A3 Glencoe Algebra 1
Ans
wer
s
Skil
ls P
ract
ice
Mul
tiply
ing
Mon
omia
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-1
8-1
©G
lenc
oe/M
cGra
w-H
ill45
7G
lenc
oe A
lgeb
ra 1
Lesson 8-1
Det
erm
ine
wh
eth
er e
ach
exp
ress
ion
is
a m
onom
ial.
Wri
te y
esor
no.
Exp
lain
.
1.11
Yes;
11 is
a r
eal n
umbe
r an
d an
exa
mpl
e of
a c
onst
ant.
2.a
$b
No;
This
is t
he d
iffer
ence
,not
the
pro
duct
,of
two
vari
able
s.
3.N
o;Th
is is
the
quo
tient
,not
the
pro
duct
,of
two
vari
able
s.
4.y
Yes;
Sin
gle
vari
able
s ar
e m
onom
ials
.
5.j3
kYe
s;Th
is is
the
pro
duct
of
two
vari
able
s.
6.2a
#3b
No;
This
is t
he s
um o
f tw
o m
onom
ials
.
Sim
pli
fy.
7.a2
(a3 )
(a6 )
a11
8.x(
x2)(
x7)
x10
9.(y
2 z)(
yz2 )
y3z3
10.(
!2 k2 )
(!3 k
)!5 k
3
11.(
e2f4
)(e2
f2)
e4f6
12.(
cd2 )
(c3 d
2 )c4
d4
13.(
2x2 )
(3x5
)6x
714
.(5a
7 )(4
a2)
20a9
15.(
4xy3
)(3x
3 y5 )
12x
4 y8
16.(
7a5 b
2 )(a
2 b3 )
7a7 b
5
17.(
$5m
3 )(3
m8 )
!15
m11
18.(
$2c
4 d)(
$4c
d)8c
5 d2
19.(
102 )
310
6or
1,0
00,0
0020
.(p3
)12
p36
21.(
$6p
)236
p2
22.(
$3y
)3!
27y
3
23.(
3pq2
)29p
2 q4
24.(
2b3 c
4 )2
4b6 c
8
GEO
MET
RYE
xpre
ss t
he
area
of
each
fig
ure
as
a m
onom
ial.
25.
26.
27.
x7c2
d2
18p
44p 9p3
cd
cdx2
x5
p2% q2
©G
lenc
oe/M
cGra
w-H
ill45
8G
lenc
oe A
lgeb
ra 1
Det
erm
ine
wh
eth
er e
ach
exp
ress
ion
is
a m
onom
ial.
Wri
te y
esor
no.
Exp
lain
.
1.N
o;th
is in
volv
es t
he q
uotie
nt,n
ot t
he p
rodu
ct,o
f va
riab
les.
2.Ye
s;th
is is
the
pro
duct
of
a nu
mbe
r,,a
nd t
wo
vari
able
s.
Sim
pli
fy.
3.($
5x2 y
)(3x
4 )!
15x
6 y4.
(2ab
2 c2 )
(4a3
b2c2
)8a
4 b4 c
4
5.(3
cd4 )
($2c
2 )!
6c3 d
46.
(4g3
h)($
2g5 )
!8g
8 h
7.($
15xy
4 )!$
xy3 "
5x2 y
78.
($xy
)3(x
z)!
x4 y
3 z
9.($
18m
2 n)2!$
mn2
"!54
m5 n
410
.(0.
2a2 b
3 )2
0.04
a4b
6
11. !
p "2p
212
. !cd
3 "2c2
d6
13.(
0.4k
3 )3
0.06
4k9
14.[
(42 )
2 ]2
48or
65,
536
GEO
MET
RYE
xpre
ss t
he
area
of
each
fig
ure
as
a m
onom
ial.
15.
16.
17.
18a
3 b6
(25x
6 )"
12a3
c4
GEO
MET
RYE
xpre
ss t
he
volu
me
of e
ach
sol
id a
s a
mon
omia
l.
18.
19.
20.
27h
6m
4 n5
(63g
4 )"
21.C
OU
NTI
NG
A p
anel
of
four
ligh
t sw
itch
es c
an b
e se
t in
24
way
s.A
pan
el o
f fi
ve li
ght
swit
ches
can
set
in t
wic
e th
is m
any
way
s.In
how
man
y w
ays
can
five
ligh
t sw
itch
es
be s
et?
25or
32
22.H
OB
BIE
STa
wa
wan
ts t
o in
crea
se h
er r
ock
colle
ctio
n by
a p
ower
of
thre
e th
is y
ear
and
then
incr
ease
it a
gain
by
a po
wer
of
two
next
yea
r.If
she
has
2 r
ocks
now
,how
man
yro
cks
will
she
hav
e af
ter
the
seco
nd y
ear?
26or
64
7g2
3g
m3 n
mn3
n
3h2
3h2
3h2
6ac3
4a2 c
5x3
6a2 b
4
3ab2
1 # 161 % 4
4 # 92 % 3
1 % 6
1 % 3
1 # 2b3 c
2%
2
21a2
%7b
Pra
ctic
e (A
vera
ge)
Mul
tiply
ing
Mon
omia
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-1
8-1
Answers (Lesson 8-1)
© Glencoe/McGraw-Hill A4 Glencoe Algebra 1
Rea
din
g t
o L
earn
Math
emati
csM
ultip
lyin
g M
onom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-1
8-1
©G
lenc
oe/M
cGra
w-H
ill45
9G
lenc
oe A
lgeb
ra 1
Lesson 8-1
Pre-
Act
ivit
yW
hy
doe
s d
oubl
ing
spee
d q
uad
rup
le b
rak
ing
dis
tan
ce?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
1 at
the
top
of
page
410
in y
our
text
book
.
Fin
d tw
o ex
ampl
es in
the
tab
le t
o ve
rify
the
sta
tem
ent
that
whe
n sp
eed
isdo
uble
d,th
e br
akin
g di
stan
ce is
qua
drup
led.
Wri
te y
our
exam
ples
in t
heta
ble.
Spe
edB
raki
ng D
ista
nce
Spe
ed D
oubl
ed
Bra
king
Dis
tanc
e (m
iles
per
hour
)(f
eet)
(mile
s pe
r ho
ur)
Qua
drup
led
(fee
t)
2020
4080
3045
6018
0
Read
ing
the
Less
on
1.D
escr
ibe
the
expr
essi
on 3
xyus
ing
the
term
s m
onom
ial,
cons
tant
,var
iabl
e,an
d pr
oduc
t.
The
mon
omia
l 3xy
is t
he p
rodu
ct o
f th
e co
nsta
nt 3
and
the
var
iabl
es x
and
y.
2.C
ompl
ete
the
char
t by
cho
osin
g th
e pr
oper
ty t
hat
can
be u
sed
to s
impl
ify
each
expr
essi
on.T
hen
sim
plif
y th
e ex
pres
sion
.
Exp
ress
ion
Pro
pert
yE
xpre
ssio
n S
impl
ified
Pro
duct
of P
ower
s
35!
32P
ower
of a
Pow
er37
or 2
187
Pow
er o
f a P
rodu
ct
Pro
duct
of P
ower
s
(a3 )
4P
ower
of a
Pow
era1
2
Pow
er o
f a P
rodu
ct
Pro
duct
of P
ower
s
($4x
y)5
Pow
er o
f a P
ower
$
1024
x5 y
5
Pow
er o
f a P
rodu
ct
Hel
ping
You
Rem
embe
r
3.W
rite
an
exam
ple
of e
ach
of t
he t
hree
pro
pert
ies
of p
ower
s di
scus
sed
in t
his
less
on.
The
n,us
ing
the
exam
ples
,exp
lain
how
the
pro
pert
y is
use
d to
sim
plif
y th
em.
Sam
ple
answ
er:F
or z
2$
z5 ,
sinc
e th
e ba
ses
are
the
sam
e,us
e th
eP
rodu
ct o
f P
ower
s P
rope
rty
and
add
the
expo
nent
s to
get
z7 .
For
(a4 )
3 ,us
e th
e P
ower
of
a P
ower
Pro
pert
y.M
ultip
ly t
he e
xpon
ents
to
get
a12 .
For
(3rs
)3,u
se t
he P
ower
of
a P
rodu
ct P
rope
rty.
Rai
se t
he c
onst
ant
and
each
var
iabl
e to
the
pow
er t
o ge
t 27
r3s
3 .
©G
lenc
oe/M
cGra
w-H
ill46
0G
lenc
oe A
lgeb
ra 1
An
Wan
g A
n W
ang
(192
0–19
90)
was
an
Asi
an-A
mer
ican
who
bec
ame
one
of t
hepi
onee
rs o
f th
e co
mpu
ter
indu
stry
in t
he U
nite
d St
ates
.He
grew
up
inSh
angh
ai,C
hina
,but
cam
e to
the
Uni
ted
Stat
es t
o fu
rthe
r hi
s st
udie
sin
sci
ence
.In
1948
,he
inve
nted
a m
agne
tic
puls
e co
ntro
lling
dev
ice
that
vas
tly
incr
ease
d th
e st
orag
e ca
paci
ty o
f co
mpu
ters
.He
late
rfo
unde
d hi
s ow
n co
mpa
ny,W
ang
Lab
orat
orie
s,an
d be
cam
e a
lead
er in
the
deve
lopm
ent
of d
eskt
op c
alcu
lato
rs a
nd w
ord
proc
essi
ng s
yste
ms.
In 1
988,
Wan
g w
as e
lect
ed t
o th
e N
atio
nal I
nven
tors
Hal
l of
Fam
e.
Dig
ital
com
pute
rs s
tore
info
rmat
ion
as n
umbe
rs.B
ecau
se t
heel
ectr
onic
cir
cuit
s of
a c
ompu
ter
can
exis
t in
onl
y on
e of
tw
o st
ates
,op
en o
r cl
osed
,the
num
bers
tha
t ar
e st
ored
can
con
sist
of
only
tw
odi
gits
,0 o
r 1.
Num
bers
wri
tten
usi
ng o
nly
thes
e tw
o di
gits
are
cal
led
bin
ary
nu
mbe
rs.T
o fi
nd t
he d
ecim
al v
alue
of a
bin
ary
num
ber,
you
use
the
digi
ts t
o w
rite
a p
olyn
omia
l in
2.F
or in
stan
ce,t
his
is h
ow t
ofi
nd t
he d
ecim
al v
alue
of
the
num
ber
1001
101 2
.(T
he s
ubsc
ript
2
indi
cate
s th
at t
his
is a
bin
ary
num
ber.)
1001
101 2
"1
&2 %
6 %#
0 &
2 %5 %
#0
&2 %
4 %#
1 &
2 %3 %
#1
&2 %
2 %#
0 &
2 %1 %
#1
&2 %
0 %"
1 &
6 %4 %
#0
&3 %2 %
#0
&1 %6 %
#1
&8 %
#1
&4 %
#0
&2 %
#1
&1 %
"64
#0
#0
#8
#4
#0
#1
"77
Fin
d t
he
dec
imal
val
ue
of e
ach
bin
ary
nu
mbe
r.
1.11
112
152.
1000
0 216
3.11
0000
112
195
4.10
1110
012
185
Wri
te e
ach
dec
imal
nu
mbe
r as
a b
inar
y n
um
ber.
5.8
1000
6.11
1011
7.29
1110
18.
117
1110
101
9.T
he c
hart
at
the
righ
t sh
ows
a se
t of
dec
imal
co
de n
umbe
rs t
hat
is u
sed
wid
ely
in s
tori
ng
lett
ers
of t
he a
lpha
bet
in a
com
pute
r's
mem
ory.
Fin
d th
e co
de n
umbe
rs fo
r th
e le
tter
s of
you
r na
me.
The
n w
rite
the
cod
e fo
r yo
ur n
ame
usin
g bi
nary
num
bers
. Ans
wer
s w
ill v
ary.
The
Am
eric
an S
tand
ard
Gui
de fo
r In
form
atio
n In
terc
hang
e (A
SC
II)
A65
N78
a97
n11
0B
66O
79b
98o
111
C67
P80
c99
p11
2D
68Q
81d
100
q11
3E
69R
82e
101
r11
4F
70S
83f
102
s11
5G
71T
84g
103
t11
6H
72U
85h
104
u11
7I
73V
86i
105
v11
8J
74W
87j
106
w11
9K
75X
88k
107
x12
0L
76Y
89l
108
y12
1M
77Z
90m
109
z12
2
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-1
8-1
Answers (Lesson 8-1)
© Glencoe/McGraw-Hill A5 Glencoe Algebra 1
Ans
wer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Div
idin
g M
onom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
6-2
8-2
©G
lenc
oe/M
cGra
w-H
ill46
1G
lenc
oe A
lgeb
ra 1
Lesson 8-2
Quo
tien
ts o
f M
onom
ials
To d
ivid
e tw
o po
wer
s w
ith
the
sam
e ba
se,s
ubtr
act
the
expo
nent
s.
Quo
tient
of
Pow
ers
For
all
inte
gers
man
d n
and
any
nonz
ero
num
ber
a,
"am
$n .
Pow
er o
f a
Quo
tient
For
any
inte
ger
man
d an
y re
al n
umbe
rs a
and
b, b
'0,
!"m
".
am% bm
a % b
am% an
Sim
pli
fy
.Ass
um
e
nei
ther
an
or b
is e
qual
to
zero
.
"!
"!"
Gro
up p
ower
s w
ith th
e sa
me
base
.
"(a
4 $
1 )(b
7 $
2 )Q
uotie
nt o
f Pow
ers
"a3
b5S
impl
ify.
The
quo
tien
t is
a3 b
5.
b7% b2
a4% a
a4 b7
% ab2
a4 b
7# a
b2S
imp
lify
!"3 .
Ass
um
e th
at b
is n
ot e
qual
to
zero
.
!"3
"P
ower
of a
Quo
tient
"P
ower
of a
Pro
duct
"P
ower
of a
Pow
er
"Q
uotie
nt o
f Pow
ers
The
quo
tien
t is
.
8a9 b
9%
27
8a9 b
9%
27
8a9 b
15%27
b6
23 (a3 )
3 (b5 )
3%
%(3
)3 (b2 )
3
(2a3 b
5 )3
%(3
b2 )3
2a3 b
5%
3b2
2a3 b
5#
3b2
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Sim
pli
fy.A
ssu
me
that
no
den
omin
ator
is
equ
al t
o ze
ro.
1.53
or 1
252.
m2
3.p
3 n3
4.a
5.y
6.!
y2
7.y
28.
!"3
8a3 b
39.
!"3
p6 q
6
10. !
"416
v4
11. !
"4r4
s812
.r4
s4
r7 s7 t
2% s3 r
3 t2
81 # 163r
6 s3
% 2r5 s
2v5 w
3% v4 w
3
64 # 274p
4 q4
% 3p2 q
22a
2 b%
axy
6% y4 x
1 # 7$
2y7
% 14y5
x5 y3
% x5 y2
a2% a
p5 n4
% p2 nm
6% m
455% 52
©G
lenc
oe/M
cGra
w-H
ill46
2G
lenc
oe A
lgeb
ra 1
Neg
ativ
e Ex
pone
nts
Any
non
zero
num
ber
rais
ed t
o th
e ze
ro p
ower
is 1
;for
exa
mpl
e,($
0.5)
0"
1.A
ny n
onze
ro n
umbe
r ra
ised
to
a ne
gati
ve p
ower
is e
qual
to
the
reci
proc
al o
f th
e nu
mbe
r ra
ised
to
the
oppo
site
pow
er;f
or e
xam
ple,
6$3
".T
hese
def
init
ions
can
be
used
to
sim
plif
y ex
pres
sion
s th
at h
ave
nega
tive
exp
onen
ts.
Zero
Exp
onen
tF
or a
ny n
onze
ro n
umbe
r a,
a0
"1.
Neg
ativ
e E
xpon
ent
Pro
pert
yF
or a
ny n
onze
ro n
umbe
r a
and
any
inte
ger
n, a
$n
"an
d "
an.
The
sim
plif
ied
form
of
an e
xpre
ssio
n co
ntai
ning
neg
ativ
e ex
pone
nts
mus
t co
ntai
n on
lypo
siti
ve e
xpon
ents
.
Sim
pli
fy
.Ass
um
e th
at t
he
den
omin
ator
is
not
equ
al t
o ze
ro.
"!
"!"!
"!"
Gro
up p
ower
s w
ith th
e sa
me
base
.
"(a
$3
$2 )
(b6
$6 )
(c5 )
Quo
tient
of P
ower
s an
d N
egat
ive
Exp
onen
t Pro
pert
ies
"a$
5 b0 c
5S
impl
ify.
"!
"(1)c
5N
egat
ive
Exp
onen
t and
Zer
o E
xpon
ent P
rope
rtie
s
"S
impl
ify.
The
sol
utio
n is
.
Sim
pli
fy.A
ssu
me
that
no
den
omin
ator
is
equ
al t
o ze
ro.
1.25
or 3
22.
m5
3.
4.b
5.6.
a6b
8
7.x
68.
9.
10.
11. !
"01
12.
!m
3# 32
n10($
2mn2 )
$3
%%
4m$
6 n4
4m2 n
2% 8m
$1 !
1# st
2s$
3 t$
5% (s
2 t3 )
$1
9s3
# u11(3
st)2 u
$4
%%
s$1 t
2 u7
36# a2 b
6(6
a$1 b
)2%
%(b
2 )4
x4 y0
% x$2
(a2 b
3 )2
% (ab)
$2
w # 4y2
($x$
1 y)0
%%
4w$
1 y2
b$4
% b$5
1# p11
p$8
% p3m
% m$
422% 2$
3
c5% 4a
5
c5% 4a
51 % a51 % 41 % 41 % 4
1% c$
5b6% b6
a$3
% a24 % 16
4a$
3 b6
%%
16a2 b
6 c$
5
4a$
3 b6
##
16a
2 b6 c
$5
1% a$
n1 % an
1 % 63
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Div
idin
g M
onom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-2
8-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 8-2)
© Glencoe/McGraw-Hill A6 Glencoe Algebra 1
Skil
ls P
ract
ice
Div
idin
g M
onom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-2
8-2
©G
lenc
oe/M
cGra
w-H
ill46
3G
lenc
oe A
lgeb
ra 1
Lesson 8-2
Sim
pli
fy.A
ssu
me
that
no
den
omin
ator
is
equ
al t
o ze
ro.
1.61
or 6
2.94
or 6
561
3.x
24.
5.6.
3d
7.8.
u2
9.a2
b3
10.
m4
11.
!12
.!
4x2 y
z3
13. !
"214
.4$
4
15.8
$2
16. !
"$2
17. !
"$1
18.
h9
19.k
0 (k4
)(k$
6 )20
.k$
1 (!$
6 )(m
3 )
21.
22. !
"01
23.
24.
3x5 y
2
25.
!26
.!
8x5 y
2#
z48
x6 y7 z
5%
%$
6xy5 z
63 # u
4$
15w
0 u$
1%
%5u
3
15x6 y
$9
% 5xy$
11g
4 h2
#f5
f$5 g
4% h$
2
16p5 q
2% 2p
3 q3
1 # f11f$
7% f4
m3
# k!6
1 # k2
h3% h$
611 # 9
9 % 11
9 # 255 % 3
1 # 64
1# 25
616
p14# 49
s44p
7% 7s
2
32x3 y
2 z5
%%
$8x
yz2
3w # u3
$21
w5 u
2%
%7w
4 u5
m7 n
2% m
3 n2
a3 b5
% ab2
w4 u
3% w
4 un
4# 3
12n5
% 36n
9d7
% 3d6
1 # m2
m % m3
1 # s2r3 s
2% r3 s
4x4% x2
912% 98
65% 64
©G
lenc
oe/M
cGra
w-H
ill46
4G
lenc
oe A
lgeb
ra 1
Sim
pli
fy.A
ssu
me
that
no
den
omin
ator
is
equ
al t
o ze
ro.
1.84
or 4
096
2.a3
b3
3.y
4.m
n5.
!6.
2yz
7.!
"38.
!"2
9.!
10.x
3 (y$
5 )(x
$8 )
11.p
(q$
2 )(r
$3 )
12.1
2$2
13. !
"$2
14. !
"$4
15.
2rs5
16.
!17
.2c
4 f7
18. !
"01
19.
20.
!21
.
22.
23.
24.
25. !
"$5
26. !
"$1
27. !
"$2
28.B
IOLO
GY
A la
b te
chni
cian
dra
ws
a sa
mpl
e of
blo
od.A
cub
ic m
illim
eter
of
the
bloo
dco
ntai
ns 2
23w
hite
blo
od c
ells
and
225
red
bloo
d ce
lls.W
hat
is t
he r
atio
of
whi
te b
lood
cells
to
red
bloo
d ce
lls?
29. C
OU
NTI
NG
The
num
ber
of t
hree
-let
ter
“wor
ds”
that
can
be
form
ed w
ith
the
Eng
lish
alph
abet
is 2
63.T
he n
umbe
r of
five
-let
ter
“wor
ds”
that
can
be
form
ed is
265
.How
man
yti
mes
mor
e fi
ve-l
ette
r “w
ords
”ca
n be
for
med
tha
n th
ree-
lett
er “
wor
ds”?
676
1# 48
4
9x2
# 4y2 z
62x
3 y2 z
% 3x4 y
z$2
c8# 7d
2 e4
7c$
3 d3
% c5 de$
4q10# r25
q$1 r
3% qr
$2
a4# 40
b7
(2a$
2 b)$
3%
%5a
2 b4
j# k15
(j$
1 k3 )
$4
%j3 k
3m
2# n
2m
$2 n
$5
%%
(m4 n
3 )$
1
r # 27r4
% (3r)
36t
2 u4
#v9
$12
t$1 u
5 v$
4%
%2t
$3 u
v5g
8 h2
#9
6f$
2 g3 h
5%
%54
f$2 g
$5 h
3
x$3 y
5%4$
38c
3 d2 f
4%
%4c
$1 d
2 f$
33 # u4
$15
w0 u
$1
%%
5u3
22r3 s
2% 11
r2 s$
381 # 25
64 % 3
49 # 93 % 7
1# 14
4p
# q2 r
31
# x5 y5
1 # 6c3
$4c
2% 24
c536
w10
# 49p12
s66w
5% 7p
6 s3
64f9 g
3# 27
h184f
3 g% 3h
6
8y7 z
6% 4y
6 z5
5d2
#4
5c2 d
3% $
4c2 d
m5 n
p% m
4 p
xy2
% xya4 b
6% ab
388% 84
Pra
ctic
e (A
vera
ge)
Div
idin
g M
onom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-2
8-2
Answers (Lesson 8-2)
© Glencoe/McGraw-Hill A7 Glencoe Algebra 1
Ans
wer
s
Rea
din
g t
o L
earn
Math
emati
csD
ivid
ing
Mon
omia
ls
NA
ME
____
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____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-2
8-2
©G
lenc
oe/M
cGra
w-H
ill46
5G
lenc
oe A
lgeb
ra 1
Lesson 8-2
Pre-
Act
ivit
yH
ow c
an y
ou c
omp
are
pH
lev
els?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
2 at
the
top
of
page
417
in y
our
text
book
.
•In
the
for
mul
a c
"!
"pH,i
dent
ify
the
base
and
the
exp
onen
t.
base
%,e
xpon
ent
%pH
•H
ow d
o yo
u th
ink
cw
ill c
hang
e as
the
exp
onen
t in
crea
ses?
cw
ill d
ecre
ase.
Read
ing
the
Less
on
1.E
xpla
in w
hat
the
stat
emen
t "
am$
nm
eans
.
To d
ivid
e tw
o po
wer
s th
at h
ave
the
sam
e ba
se,s
ubtr
act
the
expo
nent
s.
2.To
find
cin
the
form
ula
c"
!"pH
,you
can
find
the
pow
er o
f the
num
erat
or,t
he p
ower
of
the
deno
min
ator
,and
div
ide.
Thi
s is
an
exam
ple
of w
hat
prop
erty
?P
ower
of
a Q
uotie
nt P
rope
rty
3.U
se t
he Q
uoti
ent
of P
ower
s P
rope
rty
to e
xpla
in w
hy 3
0"
1.S
ampl
e an
swer
:
%1.
The
Quo
tient
of
Pow
ers
Pro
pert
y sa
ys t
hat
whe
n yo
u di
vide
two
pow
ers
that
hav
e th
e sa
me
base
,you
sub
trac
t th
e ex
pone
nts.
So
%30
.
4.C
onsi
der
the
expr
essi
on 4
$3 .
a.E
xpla
in w
hy t
he e
xpre
ssio
n 4$
3is
not
sim
plif
ied.
An
expr
essi
on in
volv
ing
expo
nent
s is
not
con
side
red
sim
plifi
ed if
the
exp
ress
ion
cont
ains
nega
tive
expo
nent
s.b.
Def
ine
the
term
rec
ipro
cal.
The
reci
proc
al o
f a
num
ber
is 1
div
ided
by
the
num
ber.
c.4$
3is
the
rec
ipro
cal o
f w
hat
pow
er o
f 4?
43
d.W
hat
is t
he s
impl
ifie
d fo
rm o
f 4$
3 ?or
Hel
ping
You
Rem
embe
r
5.D
escr
ibe
how
you
wou
ld h
elp
a fr
iend
who
nee
ds t
o si
mpl
ify
the
expr
essi
on
.
Div
ide
the
cons
tant
s an
d gr
oup
pow
ers
with
the
sam
e ba
se t
o ge
t
!"!
".Use
the
Quo
tient
of
Pow
ers
Pro
pert
y to
get
(2)
(x2!
5 ) o
r (2
)(x!
3 ).
To s
impl
ify (
2)( x
!3 )
,use
the
Neg
ativ
e E
xpon
ent
Pro
pert
y to
get
(2) !
",or
.2 # x3
1 # x3x2# x5
4 # 2
4x2
% 2x5
1 # 641 # 43
34# 34
34# 34
1 % 10am % an
1 # 10
1 % 10
©G
lenc
oe/M
cGra
w-H
ill46
6G
lenc
oe A
lgeb
ra 1
Patt
erns
with
Pow
ers
Use
you
r ca
lcul
ator
,if
nece
ssar
y,to
com
plet
e ea
ch p
atte
rn.
a.21
0"
b.51
0"
c.41
0"
29"
59"
49"
28"
58"
48"
27"
57"
47"
26"
56"
46"
25"
55"
45"
24"
54"
44"
23"
53"
43"
22"
52"
42"
21"
51"
41"
Stu
dy
the
pat
tern
s fo
r a,
b,an
d c
abo
ve.T
hen
an
swer
th
e qu
esti
ons.
1.D
escr
ibe
the
patt
ern
of t
he e
xpon
ents
fro
m t
he t
op o
f ea
ch c
olum
n to
the
bot
tom
.Th
e ex
pone
nts
decr
ease
by
one
from
eac
h ro
w t
o th
e on
e be
low
.
2.D
escr
ibe
the
patt
ern
of t
he p
ower
s fr
om t
he t
op o
f th
e co
lum
n to
the
bot
tom
.To
get
each
pow
er,d
ivid
e th
e po
wer
on
the
row
abo
ve b
y th
e ba
se (
2,5,
or 4
).
3.W
hat
wou
ld y
ou e
xpec
t th
e fo
llow
ing
pow
ers
to b
e?20
150
140
1
4.R
efer
to
Exe
rcis
e 3.
Wri
te a
rul
e.Te
st it
on
patt
erns
tha
t yo
u ob
tain
usi
ng 2
2,25
,and
24
as b
ases
.A
ny n
onze
ro n
umbe
r to
the
zer
o po
wer
equ
als
one.
Stu
dy
the
pat
tern
bel
ow.T
hen
an
swer
th
e qu
esti
ons.
03"
002
"0
01"
000
"0$
1do
es n
ot e
xist
.0$
2do
es n
ot e
xist
.0$
3do
es n
ot e
xist
.
5.W
hy d
o 0$
1 ,0$
2 ,an
d 0$
3no
t ex
ist?
Neg
ativ
e ex
pone
nts
are
not
defin
ed u
nles
s th
e ba
se is
non
zero
.
6.B
ased
upo
n th
e pa
tter
n,ca
n yo
u de
term
ine
whe
ther
00
exis
ts?
No,
sinc
e th
e pa
tter
n 0n
%0
brea
ks d
own
for
n&
1.
7.T
he s
ymbo
l 00
is c
alle
d an
in
dete
rmin
ate,
whi
ch m
eans
tha
t it
has
no
uniq
ue v
alue
.T
hus
it d
oes
not
exis
t as
a u
niqu
e re
al n
umbe
r.W
hy d
o yo
u th
ink
that
00
cann
ot e
qual
1?
Ans
wer
s w
ill v
ary.
One
ans
wer
is t
hat
if 00
%1,
then
1 %
%%
!"0 ,
whi
ch is
a f
alse
res
ult,
sinc
e di
visi
on b
y ze
ro is
not
allo
wed
.Thu
s,00
cann
ot e
qual
1.
1 # 010# 00
1 # 1
?
45
216
254
6412
58
256
625
1610
2431
2532
4096
15,6
2564
16,3
8478
,125
128
65,5
3639
0,62
525
626
2,14
41,
953,
125
512
1,04
8,57
69,
765,
625
1024
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-2
8-2
Answers (Lesson 8-2)
© Glencoe/McGraw-Hill A8 Glencoe Algebra 1
Stu
dy
Gu
ide
and I
nte
rven
tion
Sci
entif
ic N
otat
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
6-3
8-3
©G
lenc
oe/M
cGra
w-H
ill46
7G
lenc
oe A
lgeb
ra 1
Lesson 8-3
Scie
ntif
ic N
otat
ion
Kee
ping
tra
ck o
f pla
ce v
alue
in v
ery
larg
e or
ver
y sm
all n
umbe
rsw
ritt
en in
sta
ndar
d fo
rm m
ay b
e di
ffic
ult.
It is
mor
e ef
fici
ent
to w
rite
suc
h nu
mbe
rs in
scie
ntif
ic n
otat
ion.
A n
umbe
r is
exp
ress
ed in
sci
enti
fic
nota
tion
whe
n it
is w
ritt
en a
s a
prod
uct
of t
wo
fact
ors,
one
fact
or t
hat
is g
reat
er t
han
or e
qual
to
1 an
d le
ss t
han
10 a
nd o
nefa
ctor
tha
t is
a p
ower
of
ten.
Sci
entif
ic N
otat
ion
Anu
mbe
r is
in s
cien
tific
not
atio
n w
hen
it is
in th
e fo
rm a
&10
n , w
here
1 (
a)
10
and
nis
an
inte
ger.
Exp
ress
3.5
2 '
104
inst
and
ard
not
atio
n.
3.52
&10
4"
3.52
&10
,000
"35
,200
The
dec
imal
poi
nt m
oved
4 p
lace
s to
the
righ
t.
Exp
ress
37,
600,
000
insc
ien
tifi
c n
otat
ion
.
37,6
00,0
00 "
3.76
&10
7
The
dec
imal
poi
nt m
oved
7 p
lace
s so
tha
t it
is b
etw
een
the
3 an
d th
e 7.
Sinc
e 37
,600
,000
*1,
the
expo
nent
is p
osit
ive.
Exp
ress
6.2
1 '
10!
5in
stan
dar
d n
otat
ion
.
6.21
&10
$5
"6.
21 &
"6.
21 &
0.00
001
"0.
0000
621
The
dec
imal
poi
nt m
oved
5 p
lace
s to
the
left
.
Exp
ress
0.0
0005
49 i
nsc
ien
tifi
c n
otat
ion
.
0.00
0054
9 "
5.49
&10
$5
The
dec
imal
poi
nt m
oved
5 p
lace
s so
tha
t it
is b
etw
een
the
5 an
d th
e 4.
Sinc
e 0.
0000
549
)1,
the
expo
nent
is n
egat
ive.
1% 10
5
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exam
ple
3Ex
ampl
e 3
Exam
ple
4Ex
ampl
e 4
Exer
cises
Exer
cises
Exp
ress
eac
h n
um
ber
in s
tan
dar
d n
otat
ion
.
1.3.
65 &
105
2.7.
02 &
10$
43.
8.00
3 &
108
365,
000
0.00
0702
800,
300,
000
4.7.
451
&10
65.
5.91
&10
06.
7.99
&10
$1
7,45
1,00
05.
910.
799
7.8.
9354
&10
108.
8.1
&10
$9
9.4
&10
15
89,3
54,0
00,0
000.
0000
0000
814,
000,
000,
000,
000,
000
Exp
ress
eac
h n
um
ber
in s
cien
tifi
c n
otat
ion
.
10.0
.000
0456
11.0
.000
0112
.590
,000
,000
4.56
'10
!5
1 '
10!
55.
9 '
108
13.0
.000
0000
0012
14.0
.000
0804
3615
.0.0
3621
1.2
'10
!10
8.04
36 '
10!
53.
621
'10
!2
16.4
33 &
104
17.0
.004
2 &
10$
318
.50,
000,
000,
000
4.33
'10
64.
2 '
10!
65
'10
10
©G
lenc
oe/M
cGra
w-H
ill46
8G
lenc
oe A
lgeb
ra 1
Prod
ucts
and
Quo
tien
ts w
ith
Scie
ntif
ic N
otat
ion
You
can
use
prop
erti
es o
fpo
wer
s to
com
pute
wit
h nu
mbe
rs w
ritt
en in
sci
enti
fic
nota
tion
.
Eva
luat
e (6
.7 '
103 )
(2 '
10!
5 ).E
xpre
ss t
he
resu
lt i
n s
cien
tifi
c an
dst
and
ard
not
atio
n.
(6.7
&10
3 )(2
&10
$5 )
"(6
.7 &
2)(1
03&
10$
5 )A
ssoc
iativ
e P
rope
rty
"13
.4 &
10$
2P
rodu
ct o
f Pow
ers
"(1
.34
&10
1 ) &
10$
213
.4 "
1.34
&10
1
"1.
34 &
(101
&10
$2 )
Ass
ocia
tive
Pro
pert
y
"1.
34 &
10$
1or
0.1
34P
rodu
ct o
f Pow
ers
The
sol
utio
n is
1.3
4 &
10$
1or
0.1
34.
Eva
luat
e .E
xpre
ss t
he
resu
lt i
n s
cien
tifi
c an
d
stan
dar
d n
otat
ion
.
"!
"!"
Ass
ocia
tive
Pro
pert
y
"0.
368
&10
3Q
uotie
nt o
f Pow
ers
"(3
.68
&10
$1 )
&10
30.
368
"3.
68 &
10$
1
"3.
68 &
(10$
1&
103 )
Ass
ocia
tive
Pro
pert
y
"3.
68 &
102
or 3
68P
rodu
ct o
f Pow
ers
The
sol
utio
n is
3.6
8 &
102
or 3
68.
Eva
luat
e.E
xpre
ss e
ach
res
ult
in
sci
enti
fic
and
sta
nd
ard
not
atio
n.
1.2.
3.(3
.2 &
10$
2 )(2
.0 &
102 )
7 '
10;7
01.
5 '
103 ;
1500
6.4
'10
0 ;6.
4
4.5.
(7.7
&10
5 )(2
.1 &
102 )
6.
5.28
'10
3 ;52
801.
617
'10
8 ;16
1,70
0,00
01.
35 '
10!
2 ;0.
0135
7.(3
.3 &
105 )
(1.5
&10
$4 )
8.9.
4.95
'10
1 ;49
.53
'10
2 ;30
01.
6 '
10!
6 ;0.
0000
016
10.F
UEL
CO
NSU
MPT
ION
Nor
th A
mer
ica
burn
ed 4
.5 &
1016
BT
U o
f pe
trol
eum
in 1
998.
At
this
rat
e,ho
w m
any
BT
U’s
will
be
burn
ed in
9 y
ears
?So
urce
: The
New
York
Tim
es 2
001
Alm
anac
4.05
'10
17
11.O
IL P
RO
DU
CTI
ON
If t
he U
nite
d St
ates
pro
duce
d 6.
25 &
109
barr
els
of c
rude
oil
in19
98,a
nd C
anad
a pr
oduc
ed 1
.98
&10
9ba
rrel
s,w
hat
is t
he q
uoti
ent
of t
heir
pro
duct
ion
rate
s? W
rite
a s
tate
men
t us
ing
this
quo
tien
t.So
urce
: The
New
York
Tim
es 2
001
Alm
anac
Abo
ut 3
.16;
Sam
ple
answ
er:T
he U
nite
d S
tate
s pr
oduc
es m
ore
than
3
times
the
cru
de o
il of
Can
ada.
4 &
10$
4%
%2.
5 &
102
3.3
&10
$12
%%
1.1
&10
$14
9.72
&10
8%
%7.
2 &
1010
1.26
72 &
10$
8%
%2.
4 &
10$
12
3 &
10$
12%
%2
&10
$15
1.4
&10
4%
%2
&10
2
108
% 105
1.50
88%
4.1
1.50
88 &
108
%%
4.1
&10
5
1.50
88 '
108
##
4.1
'10
5
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Sci
entif
ic N
otat
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-3
8-3
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Answers (Lesson 8-3)
© Glencoe/McGraw-Hill A9 Glencoe Algebra 1
Ans
wer
s
Skil
ls P
ract
ice
Sci
entif
ic N
otat
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-3
8-3
©G
lenc
oe/M
cGra
w-H
ill46
9G
lenc
oe A
lgeb
ra 1
Lesson 8-3
Exp
ress
eac
h n
um
ber
in s
tan
dar
d n
otat
ion
.
1.4
&10
32.
2 &
108
3.3.
2 &
105
4000
200,
000,
000
320,
000
4.3
&10
$6
5.9
&10
$2
6.4.
7 &
10$
7
0.00
0003
0.09
0.00
0000
47
AST
RO
NO
MY
Exp
ress
th
e n
um
ber
in e
ach
sta
tem
ent
in s
tan
dar
d n
otat
ion
.
7.T
he d
iam
eter
of
Jupi
ter
is 1
.429
84 &
105
kilo
met
ers.
142,
984
8.T
he s
urfa
ce d
ensi
ty o
f th
e m
ain
ring
aro
und
Jupi
ter
is 5
&10
$6
gram
s pe
r ce
ntim
eter
squa
red.
0.00
0005
9.T
he m
inim
um d
ista
nce
from
Mar
s to
Ear
th is
5.4
5 &
107
kilo
met
ers.
54,5
00,0
00
Exp
ress
eac
h n
um
ber
in s
cien
tifi
c n
otat
ion
.
10.4
1,00
0,00
011
.65,
100
12.2
83,0
00,0
00
4.1
'10
76.
51 '
104
2.83
'10
8
13.2
64,7
0114
.0.0
1915
.0.0
0000
7
2.64
701
'10
51.
9 '
10!
27
'10
!6
16.0
.000
0100
3517
.264
.918
.150
&10
2
1.00
35 '
10!
52.
649
'10
21.
5 '
104
Eva
luat
e.E
xpre
ss e
ach
res
ult
in
sci
enti
fic
and
sta
nd
ard
not
atio
n.
19.(
3.1
&10
7 )(2
&10
$5 )
20.(
5 &
10$
2 )(1
.4 &
10$
4 )
6.2
'10
2 ;62
07.
0 '
10!
6 ;0.
0000
07
21.(
3 &
103 )
(4.2
&10
$1 )
22.(
3 &
10$
2 )(5
.2 &
109 )
1.26
'10
3 ;12
601.
56 '
108 ;
156,
000,
000
23.(
2.4
&10
2 )(4
&10
$10
)24
.(1.
5 &
10$
4 )(7
&10
$5 )
9.6
'10
!8 ;
0.00
0000
096
1.05
'10
!8 ;
0.00
0000
0105
25.
26.
3.4
'10
4 ;34
,000
1.8
'10
!2 ;
0.01
8
7.2
&10
$5
%%
4 &
10$
35.
1 &
106
%%
1.5
&10
2
©G
lenc
oe/M
cGra
w-H
ill47
0G
lenc
oe A
lgeb
ra 1
Exp
ress
eac
h n
um
ber
in s
tan
dar
d n
otat
ion
.
1.7.
3 &
107
2.2.
9 &
103
3.9.
821
&10
12
73,0
00,0
0029
009,
821,
000,
000,
000
4.3.
54 &
10$
15.
7.36
42 &
104
6.4.
268
&10
$6
0.35
473
,642
0.00
0004
268
PHY
SIC
SE
xpre
ss t
he
nu
mbe
r in
eac
h s
tate
men
t in
sta
nd
ard
not
atio
n.
7.A
n el
ectr
on h
as a
neg
ativ
e ch
arge
of 1
.6 &
10$
19C
oulo
mb.
0.00
0000
0000
0000
0000
16
8.In
the
mid
dle
laye
r of
the
sun
’s a
tmos
pher
e,ca
lled
the
chro
mos
pher
e,th
e te
mpe
ratu
reav
erag
es 2
.78
&10
4de
gree
s C
elsi
us.
27,8
00
Exp
ress
eac
h n
um
ber
in s
cien
tifi
c n
otat
ion
.
9.91
5,60
0,00
0,00
010
.638
711
.845
,320
12.0
.000
0000
0814
9.15
6 '
1011
6.38
7 '
103
8.45
32 '
105
8.14
'10
!9
13.0
.000
0962
114
.0.0
0315
715
.30,
620
16.0
.000
0000
0001
129.
621
'10
!5
3.15
7 '
10!
33.
062
'10
41.
12 '
10!
11
17.5
6 &
107
18.4
740
&10
519
.0.0
76 &
10$
320
.0.0
057
&10
3
5.6
'10
84.
74 '
108
7.6
'10
!5
5.7
or 5
.7 '
100
Eva
luat
e.E
xpre
ss e
ach
res
ult
in
sci
enti
fic
and
sta
nd
ard
not
atio
n.
21.(
5 &
10$
2 )(2
.3 &
1012
)22
.(2.
5 &
10$
3 )(6
&10
15)
1.15
'10
11;1
15,0
00,0
00,0
001.
5 '
1013
;15,
000,
000,
000,
000
23.(
3.9
&10
3 )(4
.2 &
10$
11)
24.(
4.6
&10
$4 )
(3.1
&10
$1 )
1.63
8 '
10!
7 ;0.
0000
0016
381.
426
'10
!4 ;
0.00
0142
6
25.
26.
27.
2.0
'10
6 ;2,
000,
000
1.6
'10
!5 ;
0.00
0016
2.34
'10
2 ;23
4
28.
29.
30.
2.0
'10
!3 ;
0.00
22.
0 '
107 ;
20,0
00,0
006.
5 '
10!
6 ;0.
0000
065
31.B
IOLO
GY
A c
ubic
mill
imet
er o
f hu
man
blo
od c
onta
ins
abou
t 5
&10
6re
d bl
ood
cells
.An
adul
t hu
man
bod
y m
ay c
onta
in a
bout
5 &
106
cubi
c m
illim
eter
s of
blo
od.A
bout
how
man
yre
d bl
ood
cells
doe
s su
ch a
hum
an b
ody
cont
ain?
abou
t 2.
5 '
1013
or 2
5 tr
illio
n
32.P
OPU
LATI
ON
The
pop
ulat
ion
of A
rizo
na is
abo
ut 4
.778
&10
6pe
ople
.The
land
are
a is
abou
t 1.
14 &
105
squa
re m
iles.
Wha
t is
the
pop
ulat
ion
dens
ity
per
squa
re m
ile?
abou
t 42
peo
ple
per
squa
re m
ile
2.01
5 &
10$
3%
%3.
1 &
102
1.68
&10
4%
%8.
4 &
10$
41.
82 &
105
%%
9.1
&10
7
1.17
&10
2%
%5
&10
$1
6.72
&10
3%
%4.
2 &
108
3.12
&10
3%
%1.
56 &
10$
3
Pra
ctic
e (A
vera
ge)
Sci
entif
ic N
otat
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-3
8-3
Answers (Lesson 8-3)
© Glencoe/McGraw-Hill A10 Glencoe Algebra 1
Rea
din
g t
o L
earn
Math
emati
csS
cien
tific
Not
atio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-3
8-3
©G
lenc
oe/M
cGra
w-H
ill47
1G
lenc
oe A
lgeb
ra 1
Lesson 8-3
Pre-
Act
ivit
yW
hy
is s
cien
tifi
c n
otat
ion
im
por
tan
t in
ast
ron
omy?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
3 at
the
top
of
page
425
in y
our
text
book
.
In t
he t
able
,eac
h m
ass
is w
ritt
en a
s th
e pr
oduc
t of
a n
umbe
r an
d a
pow
erof
10.
Loo
k at
the
fir
st f
acto
r in
eac
h pr
oduc
t.H
ow a
re t
hese
fac
tors
alik
e?
They
are
all
grea
ter
than
1 a
nd le
ss t
han
10.
Read
ing
the
Less
on
1.Is
the
num
ber
0.05
43 &
104
in s
cien
tifi
c no
tati
on?
Exp
lain
.
No;
the
first
fac
tor
is le
ss t
han
1.
2.C
ompl
ete
each
sen
tenc
e to
cha
nge
from
sci
enti
fic
nota
tion
to
stan
dard
not
atio
n.
a.To
exp
ress
3.6
4 &
106
in s
tand
ard
nota
tion
,mov
e th
e de
cim
al p
oint
plac
es t
o th
e .
b.To
exp
ress
7.8
25 &
10$
3in
sta
ndar
d no
tati
on,m
ove
the
deci
mal
poi
nt
plac
es t
o th
e .
3.C
ompl
ete
each
sen
tenc
e to
cha
nge
from
sta
ndar
d no
tati
on t
o sc
ient
ific
not
atio
n.
a.To
exp
ress
0.0
0078
65 in
sci
enti
fic
nota
tion
,mov
e th
e de
cim
al p
oint
pl
aces
to t
he r
ight
and
wri
te
.
b.To
exp
ress
54,
000,
000,
000
in s
cien
tifi
c no
tati
on,m
ove
the
deci
mal
poi
nt
plac
es t
o th
e le
ft a
nd w
rite
.
4.W
rite
pos
itiv
eor
neg
ativ
eto
com
plet
e ea
ch s
ente
nce.
a.po
wer
s of
10
are
used
to
expr
ess
very
larg
e nu
mbe
rs in
scie
ntif
ic n
otat
ion.
b.po
wer
s of
10
are
used
to
expr
ess
very
sm
all n
umbe
rs in
scie
ntif
ic n
otat
ion.
Hel
ping
You
Rem
embe
r
5.D
escr
ibe
the
met
hod
you
wou
ld u
se t
o es
tim
ate
how
man
y ti
mes
gre
ater
the
mas
s of
Satu
rn is
tha
n th
e m
ass
of P
luto
.
Div
ide
5.69
'10
26by
1.2
7 '
1022
.Sin
ce 5
.69
(1.
27 #
4.48
and
10
26(
1022
is 1
04,t
he m
ass
of M
ars
is a
bout
4.4
8 '
104
times
the
m
ass
of P
luto
.
Neg
ativ
e
Pos
itive
5.4
'10
1010
7.86
5 '
10!
44
left
3ri
ght
6
©G
lenc
oe/M
cGra
w-H
ill47
2G
lenc
oe A
lgeb
ra 1
Con
vert
ing
Met
ric
Uni
tsSc
ient
ific
not
atio
n is
con
veni
ent
to u
se f
or u
nit
conv
ersi
ons
in t
he m
etri
c sy
stem
.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-3
8-3
How
man
y k
ilom
eter
sar
e th
ere
in 4
,300
,000
met
ers?
Div
ide
the
mea
sure
by
the
num
ber
ofm
eter
s (1
000)
in o
ne k
ilom
eter
.Exp
ress
both
num
bers
in s
cien
tifi
c no
tati
on.
"4.
3 &
103
The
ans
wer
is 4
.3 3
103
km
.
4.3
&10
6%
%1
&10
3
Con
vert
370
0 gr
ams
into
mil
ligr
ams.
Mul
tipl
y by
the
num
ber
of m
illig
ram
s(1
000)
in 1
gra
m.
(3.7
&10
3 )(1
&10
3 ) "
3.7
&10
6
The
re a
re 3
.7 &
106
mg
in 3
700
g.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Com
ple
te t
he
foll
owin
g.E
xpre
ss e
ach
an
swer
in
sci
enti
fic
not
atio
n.
1.25
0,00
0 m
"km
2.37
5 km
"m
3.24
7 m
"cm
4.50
00 m
"m
m
5.0.
0004
km
"m
6.0.
01 m
m "
m
7.60
00 m
"m
m8.
340
cm "
km
9.52
,000
mg
"g
10.4
20 k
L "
L
Sol
ve.
11.T
he p
lane
t M
ars
has
a di
amet
er o
f 6.
76 &
103
km.W
hat
is t
hedi
amet
er o
f M
ars
in m
eter
s? E
xpre
ss t
he a
nsw
er in
bot
h sc
ient
ific
and
deci
mal
not
atio
n.6,
760,
000
m;6
.76
'10
6m
12.T
he d
ista
nce
from
ear
th t
o th
e su
n is
149
,590
,000
km
.Lig
httr
avel
s 3.
0 &
108
met
ers
per
seco
nd.H
ow lo
ng d
oes
it t
ake
light
from
the
sun
to
reac
h th
e ea
rth
in m
inut
es?
Rou
nd t
o th
e ne
ares
t hu
ndre
dth.
8.31
min
13.A
ligh
t-ye
ar is
the
dis
tanc
e th
at li
ght
trav
els
in o
ne y
ear.
(See
Exe
rcis
e 12
.) H
ow f
ar is
a li
ght
year
in k
ilom
eter
s? E
xpre
ss y
our
answ
er in
sci
enti
fic
nota
tion
.Rou
nd t
o th
e ne
ares
t hu
ndre
dth.
9.46
'10
12km
4.2
!10
55.
2 !
101
3.4
!10
"3
6 !
106
1 !
10"
54
!10
"1
5 !
106
2.47
!10
4
3.75
!10
52.
5 !
102
Answers (Lesson 8-3)
© Glencoe/McGraw-Hill A11 Glencoe Algebra 1
Ans
wer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
6-4
8-4
©G
lenc
oe/M
cGra
w-H
ill47
3G
lenc
oe A
lgeb
ra 1
Lesson 8-4
Deg
ree
of a
Pol
ynom
ial
A p
olyn
omia
lis
a m
onom
ial o
r a
sum
of
mon
omia
ls.A
bin
omia
lis
the
sum
of
two
mon
omia
ls,a
nd a
tri
nom
ial
is t
he s
um o
f th
ree
mon
omia
ls.
Poly
nom
ials
wit
h m
ore
than
thr
ee t
erm
s ha
ve n
o sp
ecia
l nam
e.T
he d
egre
eof
a m
onom
ial
is t
he s
um o
f th
e ex
pone
nts
of a
ll it
s va
riab
les.
The
deg
ree
of t
he
pol
ynom
ial
is t
he s
ame
as t
he d
egre
e of
the
mon
omia
l ter
m w
ith
the
high
est
degr
ee.
Sta
te w
het
her
eac
h e
xpre
ssio
n i
s a
pol
ynom
ial.
If t
he
exp
ress
ion
is
a p
olyn
omia
l,id
enti
fy i
t as
a m
onom
ial,
bin
omia
l,or
tri
nom
ial.
Th
en g
ive
the
deg
ree
of t
he
pol
ynom
ial.
Exp
ress
ion
Pol
ynom
ial?
Mon
omia
l,B
inom
ial,
Deg
ree
of t
he
or T
rino
mia
l?P
olyn
omia
l
3x$
7xyz
Yes.
3x
$7x
yz"
3x#
($7x
yz),
bino
mia
l3
whi
ch is
the
sum
of t
wo
mon
omia
ls
$25
Yes.
$25
is a
rea
l num
ber.
mon
omia
l0
7n3
#3n
$4
No.
3n$
4"
, whi
ch is
not
no
ne o
f the
se—
a m
onom
ial
Yes.
The
exp
ress
ion
sim
plifi
es to
9 x3
#4x
#x
#4
#2x
9x3
#7x
#4,
whi
ch is
the
sum
tr
inom
ial
3of
thre
e m
onom
ials
Sta
te w
het
her
eac
h e
xpre
ssio
n i
s a
pol
ynom
ial.
If t
he
exp
ress
ion
is
a p
olyn
omia
l,id
enti
fy i
t as
a m
onom
ial,
bin
omia
l,or
tri
nom
ial.
1.36
yes;
mon
omia
l2.
#5
no
3.7x
$x
#5
yes;
bino
mia
l4.
8g2 h
$7g
h#
2ye
s;tr
inom
ial
5.#
5y$
8no
6.6x
#x2
yes;
bino
mia
l
Fin
d t
he
deg
ree
of e
ach
pol
ynom
ial.
7.4x
2 y3 z
68.
$2a
bc3
9.15
m1
10.s
#5t
111
.22
012
.18x
2#
4yz
$10
y2
13.x
4$
6x2
$2x
3$
104
14.2
x3y2
$4x
y35
15.$
2r8 s
4#
7r2 s
$4r
7 s6
13
16.9
x2#
yz8
917
.8b
#bc
56
18.4
x4y
$8z
x2#
2x5
5
19.4
x2$
12
20.9
abc
#bc
$d
55
21.h
3 m#
6h4 m
2$
76
1% 4y
2
3 % q2
3 % n4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill47
4G
lenc
oe A
lgeb
ra 1
Wri
te P
olyn
omia
ls in
Ord
erT
he t
erm
s of
a p
olyn
omia
l are
usu
ally
arr
ange
d so
tha
t th
e po
wer
s of
one
var
iabl
e ar
e in
asc
end
ing
(inc
reas
ing)
ord
er o
r d
esce
nd
ing
(dec
reas
ing)
ord
er.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-4
8-4
Arr
ange
th
e te
rms
ofea
ch p
olyn
omia
l so
th
at t
he
pow
ers
ofx
are
in a
scen
din
g or
der
.
a.x4
!x2
)5x
3
$x2
#5x
3#
x4
b.8x
3 y!
y2)
6x2 y
)xy
2
$y2
#xy
2#
6x2 y
#8x
3 y
Arr
ange
th
e te
rms
ofea
ch p
olyn
omia
l so
th
at t
he
pow
ers
ofx
are
in d
esce
nd
ing
ord
er.
a.x4
)4x
5!
x24x
5#
x4$
x2
b.!
6xy
)y3
!x2
y2)
x4y2
x4y2
$ x
2 y2
$6x
y#
y3
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Arr
ange
th
e te
rms
of e
ach
pol
ynom
ial
so t
hat
th
e p
ower
s of
xar
e in
as
cen
din
g or
der
.
1.5x
#x2
#6
2.6x
#9
$4x
23.
4xy
#2y
#6x
2
6 )
5x)
x2
9 )
6x!
4x2
2y)
4xy
)6x
2
4.6y
2 x$
6x2 y
#2
5.x4
#x3
#x2
6.2x
3$
x#
3x7
2 )
6y2 x
!6x
2 yx
2)
x3 )
x4
!x
)2x
3 )3x
7
7.$
5cx
#10
c2x3
#15
cx2
8.$
4nx
$5n
3 x3 #
59.
4xy
#2y
#5x
2
!5c
x)
15cx
2)
10c
2 x3
5 !
4nx
!5n
3 x3
2y)
4xy
)5x
2
Arr
ange
th
e te
rms
of e
ach
pol
ynom
ial
so t
hat
th
e p
ower
s of
xar
e in
d
esce
nd
ing
ord
er.
10.2
x#
x2$
511
.20x
$10
x2#
5x3
12.x
2#
4yx$
10x5
x2)
2x!
55x
3!
10x2
)20
x!
10x5
)x2
)4y
x
13.9
bx#
3bx2
$6x
314
.x3
#x5
$x2
15.a
x2#
8a2 x
5$
4
!6x
3)
3bx
2)
9bx
x5
)x
3!
x2
8a2 x
5)
ax2
!4
16.3
x3y
$4x
y2$
x4y2
#y5
17.x
4#
4x3
$7x
5#
1
!x
4 y2
)3x
3 y!
4xy
2)
y5
!7x
5)
x4
)4x
3)
1
18.$
3x6
$x5
#2x
819
.$15
cx2
#8c
2 x5
#cx
2x8
!3x
6!
x58c
2 x5
!15
cx2 )
cx
20.2
4x2 y
$12
x3y2
#6x
421
.$15
x3#
10x4
y2#
7xy2
6x4
!12
x3 y
2)
24x
2 y10
x4 y
2!
15x
3)
7xy
2
Answers (Lesson 8-4)
© Glencoe/McGraw-Hill A12 Glencoe Algebra 1
Skil
ls P
ract
ice
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-4
8-4
©G
lenc
oe/M
cGra
w-H
ill47
5G
lenc
oe A
lgeb
ra 1
Lesson 8-4
Sta
te w
het
her
eac
h e
xpre
ssio
n i
s a
pol
ynom
ial.
If t
he
exp
ress
ion
is
a p
olyn
omia
l,id
enti
fy i
t as
a m
onom
ial,
a bi
nom
ial,
or a
tri
nom
ial.
1.5m
n#
n22.
4by
#2b
$by
3.$
32
yes;
bino
mia
lye
s;bi
nom
ial
yes;
mon
omia
l
4.5.
5x2
$3x
$4
6.2c
2#
8c#
9 $
3
yes;
mon
omia
lno
yes;
trin
omia
l
GEO
MET
RYW
rite
a p
olyn
omia
l to
rep
rese
nt
the
area
of
each
sh
aded
reg
ion
.
7.ab
!xy
8.4r
2!
"r2
Fin
d t
he
deg
ree
of e
ach
pol
ynom
ial.
9.12
010
.3r4
411
.b#
61
12.4
a3$
2a3
13.5
abc
$2b
2#
13
14.8
x5y4
$2x
89
Arr
ange
th
e te
rms
of e
ach
pol
ynom
ial
so t
hat
th
e p
ower
s of
xar
e in
asc
end
ing
ord
er.
15.3
x#
1 #
2x2
1 )
3x)
2x2
16.5
x$
6 #
3x2
!6
)5x
)3x
2
17.9
x2#
2 #
x3#
x2
)x
)9x
2)
x3
18.$
3 #
3x3
$x2
#4x
!3
)4x
!x
2)
3x3
19.7
r5x
#21
r4$
r2x2
$15
x320
.3a2
x4#
14a2
$10
x3#
ax2
21r4
)7r
5 x!
r2x
2!
15x3
14a2
)ax
2!
10x3
)3a
2 x4
Arr
ange
th
e te
rms
of e
ach
pol
ynom
ial
so t
hat
th
e p
ower
s of
xar
e in
des
cen
din
gor
der
.
21.x
2#
3x3
#27
$x
22.2
5 $
x3#
x
3x3
)x2
!x
)27
!x3
)x
)25
23.x
$3x
2#
4 #
5x3
24.x
2#
64 $
x#
7x3
5x3
!3x
2)
x)
47x
3)
x2!
x)
64
25.2
cx#
32 $
c3x2
#6x
326
.13
$x3
y3#
x2y2
#x
6x3
!c
3 x2
)2c
x)
32!
x3y
3)
x2y
2)
x)
13
r
ax
by
3x % 7
©G
lenc
oe/M
cGra
w-H
ill47
6G
lenc
oe A
lgeb
ra 1
Sta
te w
het
her
eac
h e
xpre
ssio
n i
s a
pol
ynom
ial.
If t
he
exp
ress
ion
is
a p
olyn
omia
l,id
enti
fy i
t as
a m
onom
ial,
a bi
nom
ial,
or a
tri
nom
ial.
1.7a
2 b#
3b2
$a2
b2.
y3#
y2$
93.
6g2 h
3 k
yes;
bino
mia
lye
s;tr
inom
ial
yes;
mon
omia
l
GEO
MET
RYW
rite
a p
olyn
omia
l to
rep
rese
nt
the
area
of
each
sh
aded
reg
ion
.
4.ab
!b
25.
d2
!"
d2
Fin
d t
he
deg
ree
of e
ach
pol
ynom
ial.
6.x
#3x
4$
21x2
#x3
47.
3g2 h
3#
g3h
5
8.$
2x2 y
#3x
y3#
x24
9.5n
3 m$
2m3
#n2
m4
#n2
6
10.a
3 b2 c
#2a
5 c#
b3c2
611
.10s
2 t2
#4s
t2$
5s3 t
25
Arr
ange
th
e te
rms
of e
ach
pol
ynom
ial
so t
hat
th
e p
ower
s of
xar
e in
asc
end
ing
ord
er.
12.8
x2$
15 #
5x5
13.1
0bx
$7b
2#
x4#
4b2 x
3
!15
)8x
2)
5x5
!7b
2)
10bx
)4b
2 x3
)x
4
14.$
3x3 y
#8y
2#
xy4
15.7
ax$
12 #
3ax3
#a2
x2
8y2
)xy
4!
3x3 y
!12
)7a
x)
a2x2
)3a
x3
Arr
ange
th
e te
rms
of e
ach
pol
ynom
ial
so t
hat
th
e p
ower
s of
xar
e in
des
cen
din
gor
der
.
16.1
3x2
$5
#6x
3$
x17
.4x
#2x
5$
6x3
#2
6x3
)13
x2!
x!
52x
5!
6x3
)4x
)2
18.g
2 x$
3gx3
#7g
3#
4x2
19.$
11x2
y3#
6y$
2xy
#2x
4
!3g
x3
)4x
2)
g2 x
)7g
32x
4!
11x
2 y3
!2x
y)
6y
20.7
a2x2
#17
$a3
x3#
2ax
21.1
2rx3
#9r
6#
r2x
#8x
6
!a
3 x3
)7a
2 x2
)2a
x)
178x
6)
12rx
3)
r2x
)9r
6
22.M
ON
EYW
rite
a p
olyn
omia
l to
repr
esen
t th
e va
lue
of t
ten-
dolla
r bi
lls,f
fift
y-do
llar
bills
,and
hon
e-hu
ndre
d-do
llar
bills
.10
t)50
f)10
0h
23.G
RA
VIT
YT
he h
eigh
t ab
ove
the
grou
nd o
f a
ball
thro
wn
up w
ith
a ve
loci
ty o
f 96
fee
t pe
rse
cond
fro
m a
hei
ght
of 6
fee
t is
6 #
96t
$16
t2fe
et,w
here
tis
the
tim
e in
sec
onds
.A
ccor
ding
to
this
mod
el,h
ow h
igh
is t
he b
all a
fter
7 s
econ
ds?
Exp
lain
.!
106
ft;T
he h
eigh
t is
neg
ativ
e be
caus
e th
e m
odel
doe
s no
t ac
coun
t fo
r th
e ba
ll hi
ttin
g th
e gr
ound
whe
n th
e he
ight
is 0
fee
t.1 # 4
dab
b
1 % 5
Pra
ctic
e (A
vera
ge)
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-4
8-4
Answers (Lesson 8-4)
© Glencoe/McGraw-Hill A13 Glencoe Algebra 1
Ans
wer
s
Rea
din
g t
o L
earn
Math
emati
csPo
lyno
mia
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-4
8-4
©G
lenc
oe/M
cGra
w-H
ill47
7G
lenc
oe A
lgeb
ra 1
Lesson 8-4
Pre-
Act
ivit
yH
ow a
re p
olyn
omia
ls u
sefu
l in
mod
elin
g d
ata?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
4 at
the
top
of
page
432
in y
our
text
book
.
•H
ow m
any
term
s do
es t
4$
9t3
#26
t$
18t
#76
hav
e?
five
•W
hat
coul
d yo
u ca
ll a
poly
nom
ial w
ith
just
one
ter
m?
a m
onom
ial
Read
ing
the
Less
on
1.W
hat
is t
he m
eani
ng o
f th
e pr
efix
es m
ono-
,bi-
,and
tri
-?
Mon
o- m
eans
one
,bi-
mea
ns t
wo,
and
tri-
mea
ns t
hree
.
2.W
rite
exa
mpl
es o
f w
ords
tha
t be
gin
wit
h th
e pr
efix
es m
ono-
,bi-
,and
tri
-.
Sam
ple
answ
er:m
onoc
ycle
(on
e w
heel
),bi
cycl
e (t
wo
whe
els)
,tri
cycl
e(t
hree
whe
els)
3.C
ompl
ete
the
tabl
e. mon
omia
lbi
nom
ial
trin
omia
lpo
lyno
mia
l with
mor
e th
an t
hree
ter
ms
Exa
mpl
e3r
2 t2x
2#
3x5x
2#
3x#
27s
2#
s4#
2s3
$s
#5
Num
ber
of T
erm
s1
23
5
4.W
hat
is t
he d
egre
e of
the
mon
omia
l 3xy
2 z?
4
5.W
hat
is t
he d
egre
e of
the
pol
ynom
ial 4
x4#
2x3 y
3#
y2#
14?
Exp
lain
how
you
fou
ndyo
ur a
nsw
er.
6;S
ince
0 )
4 %
4 ,4
x4
has
degr
ee 4
;sin
ce 0
)3
)3
%6,
2x3 y
3ha
sde
gree
6;y
2ha
s de
gree
2;a
nd 1
4 ha
s de
gree
0.T
he h
ighe
st d
egre
e of
thes
e te
rms
is 6
.
Hel
ping
You
Rem
embe
r
6.U
se a
dic
tion
ary
to f
ind
the
mea
ning
of
the
term
s as
cend
ing
and
desc
endi
ng.W
rite
the
irm
eani
ngs
and
then
des
crib
e a
situ
atio
n in
you
r ev
eryd
ay li
fe t
hat
rela
tes
to t
hem
.
asce
ndin
g:go
ing,
grow
ing,
or m
ovin
g up
war
d;de
scen
ding
:mov
ing
from
a hi
gher
to
a lo
wer
pla
ce;S
ampl
e an
swer
:clim
bing
sta
irs,
hiki
ng
©G
lenc
oe/M
cGra
w-H
ill47
8G
lenc
oe A
lgeb
ra 1
Poly
nom
ial F
unct
ions
Supp
ose
a lin
ear
equa
tion
suc
h as
23x
#y
"4
is
sol
ved
for
y.T
hen
an e
quiv
alen
t eq
uati
on,
y"
3x#
4,is
fou
nd.E
xpre
ssed
in t
his
way
,yis
a
func
tion
of x
,or
f(x)
"3x
#4.
Not
ice
that
the
ri
ght
side
of
the
equa
tion
is a
bin
omia
l of
degr
ee 1
.
Hig
her-
degr
ee p
olyn
omia
ls in
xm
ay a
lso
form
fu
ncti
ons.
An
exam
ple
is f
(x) "
x3#
1,w
hich
is
a p
olyn
omia
l fun
ctio
n of
deg
ree
3.Yo
u ca
n gr
aph
this
fun
ctio
n us
ing
a ta
ble
of o
rder
ed
pair
s,as
sho
wn
at t
he r
ight
.
For
eac
h o
f th
e fo
llow
ing
pol
ynom
ial
fun
ctio
ns,
mak
e a
tabl
eof
val
ues
for
xan
d y
%f(
x).T
hen
dra
w t
he
grap
h o
n t
he
grid
.
1.f(
x) "
1 $
x22.
f(x)
"x2
$5
3.f(
x) "
x2#
4x$
14.
f(x)
"x3
x
y
Ox
y
O
x
y
Ox
y
O
x
y
O
xy
$1 %
1 2%$
2 %3 8%
$1
0
01
12
1 %1 2%
4%3 8%
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-4
8-4
Answers (Lesson 8-4)
© Glencoe/McGraw-Hill A14 Glencoe Algebra 1
Stu
dy
Gu
ide
and I
nte
rven
tion
Add
ing
and
Sub
trac
ting
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
6-5
8-5
©G
lenc
oe/M
cGra
w-H
ill47
9G
lenc
oe A
lgeb
ra 1
Lesson 8-5
Add
Pol
ynom
ials
To a
dd p
olyn
omia
ls,y
ou c
an g
roup
like
ter
ms
hori
zont
ally
or
wri
teth
em in
col
umn
form
,alig
ning
like
ter
ms
vert
ical
ly.L
ike
term
sar
e m
onom
ial t
erm
s th
atar
e ei
ther
iden
tica
l or
diff
er o
nly
in t
heir
coe
ffic
ient
s,su
ch a
s 3p
and
$5p
or 2
x2y
and
8x2 y
.
Fin
d (
2x2
)x
!8)
)(3
x!
4x2
)2)
.
Hor
izon
tal
Met
hod
Gro
up li
ke t
erm
s.
(2x2
#x
$8)
#(3
x$
4x2
#2)
"[(
2x2
#($
4x2 )
] #
(x#
3x) #
[($
8) #
2)]
"$
2x2
#4x
$6.
The
sum
is $
2x2
#4x
$6.
Fin
d (
3x2
)5x
y) )
(xy
)2x
2 ).
Ver
tica
l M
eth
odA
lign
like
term
s in
col
umns
and
add
.
3x2
#5x
y(#
)2x
2#
xyP
ut th
e te
rms
in d
esce
ndin
g or
der.
5x2
#6x
y
The
sum
is 5
x2#
6xy.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d e
ach
su
m.
1.(4
a$
5) #
(3a
#6)
2.(6
x#
9) #
(4x2
$7)
7a)
14x
2)
6x)
2
3.(6
xy#
2y#
6x) #
(4xy
$x)
4.(x
2#
y2) #
($x2
#y2
)
10xy
)5x
)2y
2y2
5.(3
p2$
2p#
3) #
(p2
$7p
#7)
6.(2
x2#
5xy
#4y
2 ) #
($xy
$6x
2#
2y2 )
4p2
!9p
)10
!4x
2)
4xy
)6y
2
7.(5
p#
2q) #
(2p2
$8q
#1)
8.(4
x2$
x#
4) #
(5x
#2x
2#
2)
2p2
)5p
!6q
)1
6x2
)4x
)6
9.(6
x2#
3x) #
(x2
$4x
$3)
10.(
x2#
2xy
#y2
) #(x
2$
xy$
2y2 )
7x2
!x
!3
2x2
)xy
!y2
11.(
2a$
4b$
c) #
($2a
$b
$4c
)12
.(6x
y2#
4xy)
#(2
xy$
10xy
2#
y2)
!5b
!5c
!4x
y2)
6xy
)y2
13.(
2p$
5q) #
(3p
#6q
) #(p
$q)
14.(
2x2
$6)
#(5
x2#
2) #
($x2
$7)
6p6x
2!
11
15.(
3z2
#5z
) #(z
2#
2z) #
(z$
4)16
.(8x
2#
4x#
3y2
#y)
#(6
x2$
x#
4y)
4z2 )
8z!
414
x2)
3x)
3y2 )
5y
©G
lenc
oe/M
cGra
w-H
ill48
0G
lenc
oe A
lgeb
ra 1
Subt
ract
Pol
ynom
ials
You
can
subt
ract
a p
olyn
omia
l by
addi
ng it
s ad
diti
ve in
vers
e.To
fin
d th
e ad
diti
ve in
vers
e of
a p
olyn
omia
l,re
plac
e ea
ch t
erm
wit
h it
s ad
diti
ve in
vers
e or
opp
osit
e.
Fin
d (
3x2
)2x
!6)
!(2
x)
x2)
3).
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Add
ing
and
Sub
trac
ting
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-5
8-5
Exam
ple
Exam
ple
Hor
izon
tal
Met
hod
Use
add
itiv
e in
vers
es t
o re
wri
te a
s ad
diti
on.
The
n gr
oup
like
term
s.
(3x2
#2x
$6)
$(2
x#
x2#
3)"
(3x2
#2x
$6)
#[(
$2x
)#($
x2) #
($3)
]"
[3x2
#($
x2)]
#[2
x#
($2x
)] #
[$6
#($
3)]
"2x
2#
($9)
"2x
2$
9
The
dif
fere
nce
is 2
x2$
9.
Ver
tica
l M
eth
od
Alig
n lik
e te
rms
in c
olum
ns a
ndsu
btra
ct b
y ad
ding
the
add
itiv
e in
vers
e.
3x2
#2x
$6
($)
x2#
2x#
3
3x2
#2x
$6
(#)$
x2$
2x$
3
2x2
$9
The
dif
fere
nce
is 2
x2$
9.
Exer
cises
Exer
cises
Fin
d e
ach
dif
fere
nce
.
1.(3
a$
5) $
(5a
#1)
2.(9
x#
2) $
($3x
2$
5)
!2a
!6
3x2
)9x
)7
3.(9
xy#
y$
2x) $
(6xy
$2x
)4.
(x2
#y2
) $($
x2#
y2)
3xy
) y
2x2
5.(6
p2#
4p#
5) $
(2p2
$5p
#1)
6.(6
x2#
5xy
$2y
2 ) $
($xy
$2x
2$
4y2 )
4p2
)9p
)4
8x2
)6x
y)
2y2
7.(8
p$
5q) $
($6p
2#
6q$
3)8.
(8x2
$4x
$3)
$($
2x$
x2#
5)
6p2
)8p
!11
q)
39x
2!
2x!
8
9.(3
x2$
2x) $
(3x2
#5x
$1)
10.(
4x2
#6x
y#
2y2 )
$($
x2#
2xy
$5y
2 )
!7x
)1
5x2 )
4xy
)7y
2
11.(
2h$
6j$
2k) $
($7h
$5j
$4k
)12
.(9x
y2#
5xy)
$($
2xy
$8x
y2)
9h!
j)2k
17xy
2)
7xy
13.(
2a$
8b) $
($3a
#5b
)14
.(2x
2$
8) $
($2x
2$
6)
5a!
13b
4x2
!2
15.(
6z2
#4z
#2)
$(4
z2#
z)16
.(6x
2$
5x#
1) $
($7x
2$
2x#
4)
2z2
)3z
)2
13x2
!3x
!3
Answers (Lesson 8-5)
© Glencoe/McGraw-Hill A15 Glencoe Algebra 1
Ans
wer
s
Skil
ls P
ract
ice
Add
ing
and
Sub
trac
ting
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-5
8-5
©G
lenc
oe/M
cGra
w-H
ill48
1G
lenc
oe A
lgeb
ra 1
Lesson 8-5
Fin
d e
ach
su
m o
r d
iffe
ren
ce.
1.(2
x#
3y) #
(4x
#9y
) 6x
)12
y2.
(6s
#5t
) #(4
t#
8s) 1
4s)
9t
3.(5
a#
9b) $
(2a
#4b
) 3a
)5b
4.(1
1m$
7n) $
(2m
#6n
) 9m
!13
n
5.(m
2$
m) #
(2m
#m
2 ) 2
m2
)m
6.(x
2$
3x) $
(2x2
#5x
) !x2
!8x
7.(d
2$
d#
5) $
(2d
#5)
d2
!3d
8.(2
e2$
5e) #
(7e
$3e
2 ) !
e2)
2e
9.(5
f#
g$
2) #
($2f
#3)
10.(
6k2
#2k
#9)
#(4
k2$
5k)
3f)
g)
110
k2
!3k
)9
11.(
x3$
x#
1) $
(3x
$1)
12.(
b2#
ab$
2) $
(2b2
#2a
b)
x3
!4x
)2
!b
2!
ab!
2
13.(
7z2
#4
$z)
$($
5 #
3z2 )
14.(
5 #
4n#
2m) #
($6m
$8)
4z2
!z
)9
!3
)4n
!4m
15.(
4t2
#2)
#($
4 #
2t)
16.(
3g3
#7g
) $(4
g#
8g3 )
4t2
)2t
!2
!5g
3)
3g
17.(
2a2
#8a
#4)
$(a
2$
3)18
.(3x
2$
7x#
5) $
($x2
#4x
)
a2)
8a)
74x
2!
11x
)5
19.(
7z2
#z
#1)
$($
4z#
3z2
$3)
20.(
2c2
#7c
#4)
#(c
2#
1 $
9c)
4z2
)5z
)4
3c2
!2c
)5
21.(
n2#
3n#
2) $
(2n2
$6n
$2)
22.(
a2#
ab$
3b2 )
#(b
2#
4a2
$ab
)
!n
2)
9n)
45a
2!
2b2
23.(
!2$
5!$
6) #
(2!2
#5
#!)
24.(
2m2
#5m
#1)
$(4
m2
$3m
$3)
3!2
!4!
!1
!2m
2)
8m)
4
25.(
x2$
6x#
2) $
($5x
2#
7x$
4)26
.(5b
2$
9b$
5) #
(b2
$6
#2b
)
6x2
!13
x)
66b
2!
7b!
11
27.(
2x2
$6x
$2)
#(x
2#
4x) #
(3x2
#x
#5)
6x2
!x
)3
©G
lenc
oe/M
cGra
w-H
ill48
2G
lenc
oe A
lgeb
ra 1
Fin
d e
ach
su
m o
r d
iffe
ren
ce.
1.(4
y#
5) #
($7y
$1)
2.($
x2#
3x) $
(5x
#2x
2 )!
3y)
4!
3x2
!2x
3.(4
k2#
8k#
2) $
(2k
#3)
4.(2
m2
#6m
) #(m
2$
5m#
7)4k
2)
6k!
13m
2)
m)
7
5.(2
w2
$3w
#1)
#(4
w$
7)6.
(g3
#2g
2 ) $
(6g
$4g
2#
2g3 )
2w2
)w
!6
!g
3)
6g2
!6g
7.(5
a2#
6a#
2) $
(7a2
$7a
#5)
8.($
4p2
$p
#9)
#(p
2#
3p$
1)!
2a2
)13
a!
3!
3p2
)2p
)8
9.(x
3$
3x#
1) $
(x3
#7
$12
x)10
.(6c
2$
c#
1) $
($4
#2c
2#
8c)
9x!
64c
2!
9c)
5
11.(
$b3
#8b
c2#
5) $
(7bc
2$
2 #
b3)
12.(
5n2
$3n
#2)
#($
n#
2n2
$4)
!2b
3)
bc2
)7
7n2
!4n
!2
13.(
4y2
#2y
$8)
$(7
y2#
4 $
y)14
.(w
2$
4w$
1) #
($5
#5w
2$
3w)
!3y
2)
3y!
126w
2!
7w!
6
15.(
4u2
$2u
$3)
#(3
u2$
u#
4)16
.(5b
2$
8 #
2b) $
(b#
9b2
#5)
7u2
!3u
)1
!4b
2)
b!
13
17.(
4d2
#2d
#2)
#(5
d2
$2
$d)
18.(
8x2
#x
$6)
$($
x2#
2x$
3)9d
2)
d9x
2!
x!
3
19.(
3h2
#7h
$1)
$(4
h#
8h2
#1)
20.(
4m2
$3m
#10
) #
(m2
#m
$2)
!5h
2)
3h!
25m
2!
2m)
8
21.(
x2#
y2$
6) $
(5x2
$y2
$5)
22.(
7t2
#2
$t)
#(t
2$
7 $
2t)
!4x
2)
2y2
!1
8t2
!3t
!5
23.(
k3$
2k2
#4k
#6)
$($
4k#
k2$
3)24
.(9j
2#
j#jk
) #($
3j2
$jk
$4j
)k
3!
3k2
)8k
)9
6j2
!3j
25.(
2x#
6y$
3z) #
(4x
#6z
$8y
) #(x
$3y
#z)
7x!
5y)
4z
26.(
6f2
$7f
$3)
$(5
f2$
1 #
2f) $
(2f2
$3
#f)
!f2
!10
f)1
27.B
USI
NES
ST
he p
olyn
omia
l s3
$70
s2#
1500
s$
10,8
00 m
odel
s th
e pr
ofit
a c
ompa
nym
akes
on
selli
ng a
n it
em a
t a
pric
e s.
A s
econ
d it
em s
old
at t
he s
ame
pric
e br
ings
in a
prof
it o
f s3
$30
s2#
450s
$50
00.W
rite
a p
olyn
omia
l tha
t ex
pres
ses
the
tota
l pro
fit
from
the
sal
e of
bot
h it
ems.
2s3
!10
0s2
)19
50s
!15
,800
28.G
EOM
ETRY
The
mea
sure
s of
tw
o si
des
of a
tri
angl
e ar
e gi
ven.
If P
is t
he p
erim
eter
,and
P"
10x
#5y
,fin
d th
e m
easu
re o
f th
e th
ird
side
.2x
)2y
3x )
4y
5x !
y
Pra
ctic
e (A
vera
ge)
Add
ing
and
Sub
trac
ting
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-5
8-5
Answers (Lesson 8-5)
© Glencoe/McGraw-Hill A16 Glencoe Algebra 1
Rea
din
g t
o L
earn
Math
emati
csA
ddin
g an
d S
ubtr
actin
g Po
lyno
mia
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-5
8-5
©G
lenc
oe/M
cGra
w-H
ill48
3G
lenc
oe A
lgeb
ra 1
Lesson 8-5
Pre-
Act
ivit
yH
ow c
an a
dd
ing
pol
ynom
ials
hel
p y
ou m
odel
sal
es?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
5 at
the
top
of
page
439
in y
our
text
book
.
Wha
t op
erat
ion
wou
ld y
ou u
se t
o fi
nd h
ow m
uch
mor
e th
e tr
adit
iona
l toy
sale
s R
wer
e th
an t
he v
ideo
gam
es s
ales
V?
subt
ract
ion
Read
ing
the
Less
on
1.U
se t
he e
xam
ple
($3x
3#
4x2
#5x
#1)
#($
5x3
$2x
2#
2x$
7).
a.Sh
ow w
hat
is m
eant
by
grou
ping
like
ter
ms
hori
zont
ally
.
[!3x
3)
(!5x
3 )]
)[(
4x2
)(!
2x2 )
] )
(5x
)2x
) )
[1 )
(!7)
]
b.Sh
ow w
hat
is m
eant
by
alig
ning
like
ter
ms
vert
ical
ly.
!3x
3)
4x2
)5x
)1
())
!5x
3!
2x2
)2x
!7
c.C
hoos
e on
e m
etho
d,th
en a
dd t
he p
olyn
omia
ls.
!8x
3)
2x2
)7x
!6
2.H
ow is
sub
trac
ting
a p
olyn
omia
l lik
e su
btra
ctin
g a
rati
onal
num
ber?
You
subt
ract
by
addi
ng t
he a
dditi
ve in
vers
e.
3.A
n al
gebr
a st
uden
t go
t th
e fo
llow
ing
exer
cise
wro
ng o
n hi
s ho
mew
ork.
Wha
t w
as
his
erro
r?
(3x5
$3x
4#
2x3
$4x
2#
5) $
(2x5
$x3
#2x
2$
4)"
[3x5
#($
2x5 )
] #($
3x4 )
#[2
x3#
($x3
)] #
[$4x
2#
($2x
2 )] #
(5 #
4)"
x5$
3x4
#x3
$6x
2#
9
He
did
not
add
the
addi
tive
inve
rse
of !
x3 .
Hel
ping
You
Rem
embe
r
4.H
ow is
add
ing
and
subt
ract
ing
poly
nom
ials
ver
tica
lly li
ke a
ddin
g an
d su
btra
ctin
gde
cim
als
vert
ical
ly?
Alig
ning
like
ter
ms
whe
n ad
ding
or
subt
ract
ing
poly
nom
ials
is li
ke u
sing
plac
e va
lue
to a
lign
digi
ts w
hen
you
add
or s
ubtr
act
deci
mal
s.
©G
lenc
oe/M
cGra
w-H
ill48
4G
lenc
oe A
lgeb
ra 1
Cir
cula
r A
reas
and
Vol
umes
Are
a of
Cir
cle
Vol
um
e of
Cyl
inde
rV
olu
me
of C
one
A"
!r2
V"
!r2
hV
"!
r2h
Wri
te a
n a
lgeb
raic
exp
ress
ion
for
eac
h s
had
ed a
rea.
(Rec
all
that
th
e d
iam
eter
of
a ci
rcle
is
twic
e it
s ra
diu
s.)
1.2.
3.
!x2
!!!
"2%
!x2
(y2
)2x
y)!
x2
Wri
te a
n a
lgeb
raic
exp
ress
ion
for
th
e to
tal
volu
me
of e
ach
fig
ure
.
4.5.
5!
x3
[13x
3)
(4a
)9b
)x2 ]
Eac
h f
igu
re h
as a
cyl
ind
rica
l h
ole
wit
h a
rad
ius
of 2
in
ches
an
d
a h
eigh
t of
5 i
nch
es.F
ind
eac
h v
olu
me.
6.7.
x3
!20
!in
33!
x3
!20
!in
317
5!#
4
3x
4x
5x
7x
! # 122 # 3
3x — 2
x )
a
x
x )
b
x
2x5x
19 # 2! # 2
x # 2
3x2x
2xx
yx
xy
xx
r
h
r
h
r
1 % 3
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-5
8-5
Answers (Lesson 8-5)
© Glencoe/McGraw-Hill A17 Glencoe Algebra 1
Ans
wer
s
Stu
dy
Gu
ide
an
d I
nte
rven
tion
Mul
tiply
ing
a Po
lyno
mia
l by
a M
onom
ial
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-6
8-6
©G
lenc
oe/M
cGra
w-H
ill48
5G
lenc
oe A
lgeb
ra 1
Lesson 8-6
Prod
uct
of M
onom
ial a
nd P
olyn
omia
lT
he D
istr
ibut
ive
Pro
pert
y ca
n be
use
d to
mul
tipl
y a
poly
nom
ial b
y a
mon
omia
l.Yo
u ca
n m
ulti
ply
hori
zont
ally
or
vert
ical
ly.S
omet
imes
mul
tipl
ying
res
ults
in li
ke t
erm
s.T
he p
rodu
cts
can
be s
impl
ifie
d by
com
bini
ng li
ke t
erm
s.
Fin
d !
3x2 (
4x2
"6x
!8)
.
Hor
izon
tal
Met
hod
!3x
2 (4x
2"
6x!
8)#
!3x
2 (4x
2 ) "
(!3x
2 )(6
x) !
(!3x
2 )(8
)#
!12
x4"
(!18
x3) !
(!24
x2)
#!
12x4
!18
x3"
24x2
Ver
tica
l M
eth
od4x
2"
6x!
8($
) !
3x2
!12
x4!
18x3
"24
x2
The
pro
duct
is !
12x4
!18
x3"
24x2
.
Sim
pli
fy !
2(4x
2"
5x)
!x(
x2"
6x).
!2(
4x2
"5x
) !x(
x2
"6x
)#
!2(
4x2 )
"(!
2)(5
x) "
(!x)
(x2 )
"(!
x)(6
x)#
!8x
2"
(!10
x) "
(!x3
) "(!
6x2 )
#(!
x3) "
[!8x
2"
(!6x
2 )] "
(!10
x)#
!x3
!14
x2!
10x
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d e
ach
pro
du
ct.
1.x(
5x"
x2)
2.x(
4x2
"3x
"2)
3.!
2xy(
2y"
4x2 )
5x2 "
x3
4x3 "
3x2 "
2x!
4xy
2!
8x3 y
4.!
2g(g
2!
2g"
2)5.
3x(x
4"
x3"
x2)
6.!
4x(2
x3!
2x"
3)
!2g
3"
4g2
!4g
3x5
"3x
4"
3x3
!8x
4 "8x
2 !12
x
7.!
4cx(
10 "
3x)
8.3y
(!4x
!6x
3 !2y
)9.
2x2 y
2 (3x
y"
2y"
5x)
!40
cx!
12cx
2!
12xy
!18
x3 y
!6y
26x
3 y3
"4x
2 y3
"10
x3 y
2
Sim
pli
fy.
10.x
(3x
!4)
!5x
11.!
x(2x
2!
4x) !
6x2
3x2
!9x
!2x
3!
2x2
12.6
a(2a
!b)
"2a
(!4a
"5b
)13
.4r(
2r2
!3r
"5)
"6r
(4r2
"2r
"8)
4a2
"4a
b32
r3"
68r
14.4
n(3n
2"
n!
4) !
n(3
!n)
15.2
b(b2
"4b
"8)
!3b
(3b2
"9b
!18
)
12n
3"
5n2
!19
n!
7b3
!19
b2
"70
b
16.!
2z(4
z2!
3z"
1) !
z(3z
2"
2z!
1)17
.2(4
x2!
2x) !
3(!
6x2
"4)
"2x
(x!
1)
!11
z3
"4z
2!
z28
x2!
6x!
12
©G
lenc
oe/M
cGra
w-H
ill48
6G
lenc
oe A
lgeb
ra 1
Solv
e Eq
uati
ons
wit
h Po
lyno
mia
l Exp
ress
ions
Man
y eq
uati
ons
cont
ain
poly
nom
ials
tha
t m
ust
be a
dded
,sub
trac
ted,
or m
ulti
plie
d be
fore
the
equ
atio
n ca
n be
sol
ved.
Sol
ve 4
(n!
2) "
5n#
6(3
!n
) "
19.
4(n
!2)
"5n
#6(
3 !
n) "
19O
rigin
al e
quat
ion
4n!
8 "
5n#
18 !
6n"
19D
istr
ibut
ive
Pro
pert
y
9n!
8 #
37 !
6nC
ombi
ne li
ke te
rms.
15n
!8
#37
Add
6n
to b
oth
side
s.
15n
#45
Add
8 to
bot
h si
des.
n#
3D
ivid
e ea
ch s
ide
by 1
5.
The
sol
utio
n is
3.
Sol
ve e
ach
equ
atio
n.
1.2(
a!
3) #
3(!
2a"
6)3
2.3(
x"
5) !
6 #
183
3.3x
(x!
5) !
3x2
#!
302
4.6(
x2"
2x) #
2(3x
2"
12)
2
5.4(
2p"
1) !
12p
#2(
8p"
12)
!1
6.2(
6x"
4) "
2 #
4(x
!4)
!3
7.!
2(4y
!3)
!8y
"6
#4(
y!
2)1
8.c(
c"
2) !
c(c
!6)
#10
c!
126
9.3(
x2!
2x) #
3x2
"5x
!11
110
.2(4
x"
3) "
2 #
!4(
x"
1)!
1
11.3
(2h
!6)
!(2
h"
1) #
97
12.3
(y"
5) !
(4y
!8)
#!
2y"
10!
13
13.3
(2a
!6)
!(!
3a!
1) #
4a!
22
14.5
(2x2
!1)
!(1
0x2
!6)
#!
(x"
2)!
3
15.3
(x"
2) "
2(x
"1)
#!
5(x
!3)
16.4
(3p2
"2p
) !12
p2#
2(8p
"6)
!3 $ 2
7 $ 10
1 $
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Mul
tiply
ing
a P
olyn
omia
l by
a M
onom
ial
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-6
8-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
3 ! 5
Answers (Lesson 8-6)
© Glencoe/McGraw-Hill A18 Glencoe Algebra 1
Skil
ls P
ract
ice
Mul
tiply
ing
a Po
lyno
mia
l by
a M
onom
ial
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-6
8-6
©G
lenc
oe/M
cGra
w-H
ill48
7G
lenc
oe A
lgeb
ra 1
Lesson 8-6
Fin
d e
ach
pro
du
ct.
1.a(
4a#
3)2.
$c(
11c
#4)
4a2
)3a
!11
c2
!4c
3.x(
2x$
5)4.
2y(y
$4)
2x2
!5x
2y2
!8y
5.$
3n(n
2#
2n)
6.4h
(3h
$5)
!3n
3!
6n2
12h
2!
20h
7.3x
(5x2
$x
#4)
8.7c
(5 $
2c2
#c3
)
15x3
!3x
2)
12x
35c
!14
c3)
7c4
9.$
4b(1
$9b
$2b
2 )10
.6y(
$5
$y
#4y
2 )
!4b
)36
b2
)8b
3!
30y
!6y
2)
24y
3
11.2
m2 (
2m2
#3m
$5)
12.$
3n2 (
$2n
2#
3n#
4)
4m4
)6m
3!
10m
26n
4!
9n3
!12
n2
Sim
pli
fy.
13.w
(3w
#2)
#5w
14.f
(5f
$3)
$2f
3w2
)7w
5f2
!5f
15.$
p(2p
$8)
$5p
16.y
2 ($
4y#
5) $
6y2
!2p
2)
3p!
4y3
!y
2
17.2
x(3x
2#
4) $
3x3
18.4
a(5a
2$
4) #
9a
3x3
)8x
20a3
!7a
19.4
b($
5b$
3) $
2(b2
$7b
$4)
20.3
m(3
m#
6) $
3(m
2#
4m#
1)
!22
b2
)2b
)8
6m2
)6m
!3
Sol
ve e
ach
equ
atio
n.
21.3
(a#
2) #
5 "
2a#
4!
722
.2(4
x#
2) $
8 "
4(x
#3)
4
23.5
(y#
1) #
2 "
4(y
#2)
$6
!5
24.4
(b#
6) "
2(b
#5)
#2
!6
25.6
(m$
2) #
14 "
3(m
#2)
$10
!2
26.3
(c#
5) $
2 "
2(c
#6)
#2
1
©G
lenc
oe/M
cGra
w-H
ill48
8G
lenc
oe A
lgeb
ra 1
Fin
d e
ach
pro
du
ct.
1.2h
($7h
2$
4h)
2.6p
q(3p
2#
4q)
3.$
2u2 n
(4u
$2n
)!
14h
3!
8h2
18p
3 q)
24pq
2!
8u3 n
)4u
2 n2
4.5j
k(3j
k#
2k)
5.$
3rs(
$2s
2#
3r)
6.4m
g2(2
mg
#4g
)15
j2k
2)
10jk
26r
s3!
9r2 s
8m2 g
3)
16m
g3
7.$
m(8
m2
#m
$7)
8.$
n2($
9n2
#3n
#6)
!2m
3!
m2
)m
6n4
!2n
3!
4n2
Sim
pli
fy.
9.$
2!(3
!$
4) #
7!10
.5w
($7w
#3)
#2w
($2w
2#
19w
#2)
!6!
2)
15!
!4w
3)
3w2
)19
w
11.6
t(2t
$3)
$5(
2t2
#9t
$3)
12.$
2(3m
3#
5m#
6) #
3m(2
m2
#3m
#1)
2t2
!63
t)15
9m2
!7m
!12
13.$
3g(7
g$
2) #
3(g2
#2g
#1)
$3g
($5g
#3)
!3g
2)
3g)
3
14.4
z2(z
$7)
$5z
(z2
$2z
$2)
#3z
(4z
$2)
!z3
!6z
2)
4z
Sol
ve e
ach
equ
atio
n.
15.5
(2s
$1)
#3
"3(
3s#
2)8
16.3
(3u
#2)
#5
"2(
2u$
2)!
3
17.4
(8n
#3)
$5
"2(
6n#
8) #
118
.8(3
b#
1) "
4(b
#3)
$9
!
19.h
(h$
3) $
2h"
h(h
$2)
$12
420
.w(w
#6)
#4w
"$
7 #
w(w
#9)
!7
21.t
(t#
4) $
1 "
t(t
#2)
#2
22.u
(u$
5) #
8u"
u(u
#2)
$4
!4
23.N
UM
BER
TH
EORY
Let
xbe
an
inte
ger.
Wha
t is
the
pro
duct
of
twic
e th
e in
tege
r ad
ded
to t
hree
tim
es t
he n
ext
cons
ecut
ive
inte
ger?
5x)
3
INV
ESTM
ENTS
For
Exe
rcis
es 2
4–26
,use
th
e fo
llow
ing
info
rmat
ion
.K
ent
inve
sted
$5,
000
in a
ret
irem
ent
plan
.He
allo
cate
d x
dolla
rs o
f th
e m
oney
to
a bo
ndac
coun
t th
at e
arns
4%
inte
rest
per
yea
r an
d th
e re
st t
o a
trad
itio
nal a
ccou
nt t
hat
earn
s 5%
inte
rest
per
yea
r.
24.W
rite
an
expr
essi
on t
hat
repr
esen
ts t
he a
mou
nt o
f m
oney
inve
sted
in t
he t
radi
tion
alac
coun
t.5,
000
!x
25.W
rite
a p
olyn
omia
l mod
el in
sim
ples
t fo
rm f
or t
he t
otal
am
ount
of
mon
ey T
Ken
t ha
sin
vest
ed a
fter
one
yea
r.(H
int:
Eac
h ac
coun
t ha
s A
#IA
dolla
rs,w
here
Ais
the
ori
gina
lam
ount
in t
he a
ccou
nt a
nd I
is it
s in
tere
st r
ate.
)T
%5,
250
!0.
01x
26.I
f K
ent
put
$500
in t
he b
ond
acco
unt,
how
muc
h m
oney
doe
s he
hav
e in
his
ret
irem
ent
plan
aft
er o
ne y
ear?
$5,2
45
3 # 2
1 # 41 # 2
7 # 41 # 4
2 % 31 % 4
Pra
ctic
e (A
vera
ge)
Mul
tiply
ing
a Po
lyno
mia
l by
a M
onom
ial
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-6
8-6
Answers (Lesson 8-6)
© Glencoe/McGraw-Hill A19 Glencoe Algebra 1
Ans
wer
s
Rea
din
g t
o L
earn
Math
emati
csM
ultip
lyin
g a
Poly
nom
ial b
y a
Mon
omia
l
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-6
8-6
©G
lenc
oe/M
cGra
w-H
ill48
9G
lenc
oe A
lgeb
ra 1
Lesson 8-6
Pre-
Act
ivit
yH
ow i
s fi
nd
ing
the
pro
du
ct o
f a
mon
omia
l an
d a
pol
ynom
ial
rela
ted
to f
ind
ing
the
area
of
a re
ctan
gle?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
6 at
the
top
of
page
444
in y
our
text
book
.
You
may
rec
all t
hat
the
form
ula
for
the
area
of
a re
ctan
gle
is A
"!w
.In
this
rec
tang
le,!
"an
d w
".H
ow w
ould
you
subs
titu
te t
hese
val
ues
in t
he a
rea
form
ula?
A%
(x)
3)(2
x)
Read
ing
the
Less
on
1.R
efer
to
Les
son
8-6.
a.H
ow is
the
Dis
trib
utiv
e P
rope
rty
used
to
mul
tipl
y a
poly
nom
ial b
y a
mon
omia
l?
The
mon
omia
l is
mul
tiplie
d by
eac
h te
rm in
the
pol
ynom
ial.
b.U
se t
he D
istr
ibut
ive
Pro
pert
y to
com
plet
e th
e fo
llow
ing.
2y2 (
3y2
#2y
$7)
"2y
2 ()
#2y
2 ()
$2y
2 ()
"#
$
$3x
3 (x3
$2x
2#
3)"
$#
"#
$
2.W
hat
is t
he d
iffe
renc
e be
twee
n si
mpl
ifyi
ng a
n ex
pres
sion
and
sol
ving
an
equa
tion
?
Sim
plify
ing
an e
xpre
ssio
n is
com
bini
ng li
ke t
erm
s.S
olvi
ng a
n eq
uatio
nis
fin
ding
the
val
ue o
f th
e va
riab
le t
hat
mak
es t
he e
quat
ion
true
.
Hel
ping
You
Rem
embe
r
3.U
se t
he e
quat
ion
2x(x
$5)
#3x
(x#
3) "
5x(x
#7)
$9
to s
how
how
you
wou
ld e
xpla
inth
e pr
oces
s of
sol
ving
equ
atio
ns w
ith
poly
nom
ial e
xpre
ssio
ns t
o an
othe
r al
gebr
a st
uden
t.
Use
the
Dis
trib
utiv
e P
rope
rty.
2x2
!10
x)
3x2
)9x
%5x
2)
35x
!9
Com
bine
like
ter
ms.
5 x2
!x
%5x
2)
35x
!9
Sub
trac
t 5 x
2fr
om b
oth
side
s.!
x%
35x
!9
Sub
trac
t 35
xfr
om b
oth
side
s.!
36x
%!
9
Div
ide
each
sid
e by
!36
.x
%0.
25
9x3
6x5
!3x
6
(!3x
3 )(3
)(!
3x3 )
(2x
2 )!
3x3 (
x3 )
14y
24y
36y
4
72y
3y2
2xx
)3
©G
lenc
oe/M
cGra
w-H
ill49
0G
lenc
oe A
lgeb
ra 1
Figu
rate
Num
bers
The
num
bers
bel
ow a
re c
alle
d pe
nta
gon
al n
um
bers
.The
y ar
e th
e nu
mbe
rs o
f dot
s or
dis
ksth
at c
an b
e ar
rang
ed a
s pe
ntag
ons.
1.F
ind
the
prod
uct
n(3n
$1)
.!
2.E
valu
ate
the
prod
uct
in E
xerc
ise
1 fo
r va
lues
of n
from
1 t
hrou
gh 4
.1,
5,12
,22
3.W
hat
do y
ou n
otic
e?Th
ey a
re t
he f
irst
four
pen
tago
nal n
umbe
rs.
4.F
ind
the
next
six
pen
tago
nal n
umbe
rs.
35,5
1,70
,92,
117,
145
5.F
ind
the
prod
uct
n(n
#1)
.)
6.E
valu
ate
the
prod
uct
in E
xerc
ise
5 fo
r va
lues
of n
from
1 t
hrou
gh 5
.On
anot
her
shee
t of
pape
r,m
ake
draw
ings
to
show
why
the
se n
umbe
rs a
re c
alle
d th
e tr
iang
ular
num
bers
.1,
3,6,
10,1
5
7.F
ind
the
prod
uct
n(2n
$1)
.2n
2!
n
8.E
valu
ate
the
prod
uct
in E
xerc
ise
7 fo
r va
lues
of n
from
1 t
hrou
gh 5
.Dra
w t
hese
hexa
gona
l num
bers
.1,
6,15
,28,
45
9.F
ind
the
firs
t 5
squa
re n
umbe
rs.A
lso,
wri
te t
he g
ener
al e
xpre
ssio
n fo
r an
y sq
uare
num
ber.
1,4,
9,16
,25;
n2
The
num
bers
you
hav
e ex
plor
ed a
bove
are
cal
led
the
plan
e fi
gura
te n
umbe
rs b
ecau
se t
hey
can
be a
rran
ged
to m
ake
geom
etri
c fi
gure
s.Yo
u ca
n al
so c
reat
e so
lid f
igur
ate
num
bers
.
10.I
f yo
u pi
le 1
0 or
ange
s in
to a
pyr
amid
wit
h a
tria
ngle
as
a ba
se,y
ou g
et o
ne o
f th
ete
trah
edra
l num
bers
.How
man
y la
yers
are
the
re in
the
pyr
amid
? H
ow m
any
oran
ges
are
ther
e in
the
bot
tom
laye
rs?
3 la
yers
;6
11.E
valu
ate
the
expr
essi
on
n3#
n2#
nfo
r va
lues
of n
from
1 t
hrou
gh 5
to
find
the
firs
t fi
ve t
etra
hedr
al n
umbe
rs.
1,4,
10,2
0,35
1 % 31 % 2
1 % 6
n # 2n2# 2
1 % 2
n # 23 n
2#
21 % 2
●●
2212
51
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-6
8-6
Answers (Lesson 8-6)
© Glencoe/McGraw-Hill A20 Glencoe Algebra 1
Stu
dy
Gu
ide
and I
nte
rven
tion
Mul
tiply
ing
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-7
8-7
©G
lenc
oe/M
cGra
w-H
ill49
1G
lenc
oe A
lgeb
ra 1
Lesson 8-7
Mul
tipl
y Bi
nom
ials
To m
ulti
ply
two
bino
mia
ls,y
ou c
an a
pply
the
Dis
trib
utiv
e P
rope
rty
twic
e.A
use
ful w
ay t
o ke
ep t
rack
of
term
s in
the
pro
duct
is t
o us
e th
e F
OIL
met
hod
asill
ustr
ated
in E
xam
ple
2.
Fin
d (
x)
3)(x
!4)
.
Hor
izon
tal
Met
hod
(x#
3)(x
$4)
"x(
x$
4) #
3(x
$4)
"(x
)(x)
#x(
$4)
#3(
x)#
3($
4)"
x2$
4x#
3x$
12"
x2$
x$
12
Ver
tica
l M
eth
od
x#
3(&
) x
$4
$4x
$12
x2#
3xx2
$x
$12
The
pro
duct
is x
2$
x$
12.
Fin
d (
x!
2)(x
)5)
usi
ng
the
FO
IL m
eth
od.
(x$
2)(x
#5)
Firs
t O
uter
In
ner
Last
"(x
)(x)
#(x
)(5)
#($
2)(x
) #($
2)(5
)"
x2#
5x#
($2x
) $10
"x2
#3x
$10
The
pro
duct
is x
2#
3x$
10.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d e
ach
pro
du
ct.
1.(x
#2)
(x#
3)2.
(x$
4)(x
#1)
3.(x
$6)
(x$
2)
x2)
5x)
6x2
!3x
!4
x2!
8x)
12
4.(p
$4)
(p#
2)5.
(y#
5)(y
#2)
6.(2
x$
1)(x
#5)
p2
!2p
!8
y2)
7y)
102x
2)
9x!
5
7.(3
n$
4)(3
n$
4)8.
(8m
$2)
(8m
#2)
9.(k
#4)
(5k
$1)
9n2
!24
n)
1664
m2
!4
5k2
)19
k!
4
10.(
3x#
1)(4
x#
3)11
.(x
$8)
($3x
#1)
12.(
5t#
4)(2
t$
6)
12x2
)13
x)
3!
3x2
)25
x!
810
t2!
22t!
24
13.(
5m$
3n)(
4m$
2n)
14.(
a$
3b)(
2a$
5b)
15.(
8x$
5)(8
x#
5)
20m
2!
22m
n)
6n2
2a2
!11
ab)
15b
264
x2!
25
16.(
2n$
4)(2
n#
5)17
.(4m
$3)
(5m
$5)
18.(
7g$
4)(7
g#
4)
4n2
)2n
!20
20m
2!
35m
)15
49g
2!
16
©G
lenc
oe/M
cGra
w-H
ill49
2G
lenc
oe A
lgeb
ra 1
Mul
tipl
y Po
lyno
mia
lsT
he D
istr
ibut
ive
Pro
pert
y ca
n be
use
d to
mul
tipl
y an
y tw
o po
lyno
mia
ls. F
ind
(3x
)2)
(2x2
!4x
)5)
.
(3x
#2)
(2x2
$4x
#5)
"3x
(2x2
$4x
#5)
#2(
2x2
$4x
#5)
Dis
trib
utiv
e P
rope
rty
"6x
3$
12x2
#15
x#
4x2
$8x
#10
Dis
trib
utiv
e P
rope
rty
"6x
3$
8x2
#7x
#10
Com
bine
like
term
s.
The
pro
duct
is 6
x3$
8x2
#7x
#10
.
Fin
d e
ach
pro
du
ct.
1.(x
#2)
(x2
$2x
#1)
2.(x
#3)
(2x2
#x
$3)
x3
!3x
)2
2x3
)7x
2!
9
3.(2
x$
1)(x
2$
x#
2)4.
(p$
3)(p
2$
4p#
2)
2x3
!3x
2)
5x!
2p
3!
7p2
)14
p!
6
5.(3
k#
2)(k
2#
k$
4)6.
(2t
#1)
(10t
2$
2t$
4)
3k3
)5k
2!
10k
!8
20t3
)6t
2!
10t!
4
7.(3
n$
4)(n
2#
5n$
4)8.
(8x
$2)
(3x2
#2x
$1)
3n3
)11
n2
!32
n)
1624
x3
)10
x2!
12x
)2
9.(2
a#
4)(2
a2$
8a#
3)10
.(3x
$4)
(2x2
#3x
#3)
4a3
!8a
2!
26a
)12
6x3
)x2
!3x
!12
11.(
n2#
2n$
1)(n
2#
n#
2)12
.(t2
#4t
$1)
(2t2
$t
$3)
n4
)3n
3)
3n2
)3n
!2
2t4
)7t
3!
9t2
!11
t)3
13.(
y2$
5y#
3)(2
y2#
7y$
4)14
.(3b
2$
2b#
1)(2
b2$
3b$
4)
2y4
!3y
3!
33y
2)
41y
!12
6b4
!13
b3
!4b
2)
5b!
4
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Mul
tiply
ing
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-7
8-7
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 8-7)
© Glencoe/McGraw-Hill A21 Glencoe Algebra 1
Ans
wer
s
Skil
ls P
ract
ice
Mul
tiply
ing
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-7
8-7
©G
lenc
oe/M
cGra
w-H
ill49
3G
lenc
oe A
lgeb
ra 1
Lesson 8-7
Fin
d e
ach
pro
du
ct.
1.(m
#4)
(m#
1)2.
(x#
2)(x
#2)
m2
)5m
)4
x2
)4x
)4
3.(b
#3)
(b#
4)4.
(t#
4)(t
$3)
b2
)7b
)12
t2)
t!12
5.(r
#1)
(r$
2)6.
(z$
5)(z
#1)
r2!
r!
2z2
!4z
!5
7.(3
c#
1)(c
$2)
8.(2
x$
6)(x
#3)
3c2
!5c
!2
2x2
!18
9.(d
$1)
(5d
$4)
10.(
2!#
5)(!
$4)
5d2
!9d
)4
2!2
!3!
!20
11.(
3n$
7)(n
#3)
12.(
q#
5)(5
q$
1)
3n2
)2n
!21
5q2
)24
q!
5
13.(
3b#
3)(3
b$
2)14
.(2m
#2)
(3m
$3)
9b2
)3b
!6
6m2
!6
15.(
4c#
1)(2
c#
1)16
.(5a
$2)
(2a
$3)
8c2
)6c
)1
10a2
!19
a)
6
17.(
4h$
2)(4
h$
1)18
.(x
$y)
(2x
$y)
16h
2!
12h
)2
2x2
!3x
y)
y2
19.(
e#
4)(e
2#
3e$
6)20
.(t
#1)
(t2
#2t
#4)
e3)
7e2
)6e
!24
t3)
3t2
)6t
)4
21.(
k#
4)(k
2#
3k$
6)22
.(m
#3)
(m2
#3m
#5)
k3
)7k
2)
6k!
24m
3)
6m2
)14
m)
15
GE
OM
ET
RY
Wri
te a
n e
xpre
ssio
n t
o re
pre
sen
t th
e ar
ea o
f ea
ch f
igu
re.
23.
24.
8x2
)18
x!
5 un
its2
(x2
)4x
)4)
"un
its2
2x )
4
2x )
5
4x !
1
©G
lenc
oe/M
cGra
w-H
ill49
4G
lenc
oe A
lgeb
ra 1
Fin
d e
ach
pro
du
ct.
1.(q
#6)
(q#
5)2.
(x#
7)(x
#4)
3.(s
#5)
(s$
6)q
2)
11q
)30
x2)
11x
)28
s2!
s!
30
4.(n
$4)
(n$
6)5.
(a$
5)(a
$8)
6.(w
$6)
(w$
9)n
2!
10n
)24
a2!
13a
)40
w2
!15
w)
54
7.(4
c#
6)(c
$4)
8.(2
x$
9)(2
x#
4)9.
(4d
$5)
(2d
$3)
4c2
!10
c!
244x
2!
10x
!36
8d2
!22
d)
15
10.(
4b#
3)(3
b$
4)11
.(4m
#2)
(4m
$3)
12.(
5c$
5)(7
c#
9)12
b2
!7b
!12
16m
2!
4m!
635
c2
)10
c!
45
13.(
6a$
3)(7
a$
4)14
.(6h
$3)
(4h
$2)
15.(
2x$
2)(5
x$
4)42
a2
!45
a)
1224
h2
!24
h)
610
x2!
18x
)8
16.(
3a$
b)(2
a$
b)17
.(4g
#3h
)(2g
#3h
)18
.(4x
#y)
(4x
#y)
6a2
!5a
b)
b2
8g2
)18
gh)
9h2
16x2
)8x
y)
y2
19.(
m#
5)(m
2#
4m$
8)20
.(t
#3)
(t2
#4t
#7)
m3
)9m
2)
12m
!40
t3)
7t2
)19
t)21
21.(
2h#
3)(2
h2#
3h#
4)22
.(3d
#3)
(2d
2#
5d$
2)4h
3)
12h
2)
17h
)12
6d3
)21
d2
)9d
!6
23.(
3q#
2)(9
q2$
12q
#4)
24.(
3r#
2)(9
r2#
6r#
4)27
q3
!18
q2
!12
q)
827
r3)
36r2
)24
r)
8
25.(
3c2
#2c
$1)
(2c2
#c
#9)
26.(
2!2
#!
#3)
(4!2
#2!
$2)
6c4
)7c
3)
27c2
)17
c!
98!
4)
8!3
)10
!2)
4!!
6
27.(
2x2
$2x
$3)
(2x2
$4x
#3)
28.(
3y2
#2y
#2)
(3y2
$4y
$5)
4x4
!12
x3
)8x
2)
6x!
99y
4!
6y3
!17
y2
!18
y!
10
GEO
MET
RYW
rite
an
exp
ress
ion
to
rep
rese
nt
the
area
of
each
fig
ure
.
29.
4x2
!2x
!2
units
230
.4x
2)
3x!
1 un
its2
31.N
UM
BER
TH
EORY
Let
xbe
an
even
inte
ger.
Wha
t is
the
pro
duct
of
the
next
tw
oco
nsec
utiv
e ev
en in
tege
rs?
x2)
6x)
8
32.G
EOM
ETRY
The
vol
ume
of a
rec
tang
ular
pyr
amid
is o
ne t
hird
the
pro
duct
of
the
area
of it
s ba
se a
nd it
s he
ight
.Fin
d an
exp
ress
ion
for
the
volu
me
of a
rec
tang
ular
pyr
amid
who
se b
ase
has
an a
rea
of 3
x2#
12x
#9
squa
re f
eet
and
who
se h
eigh
t is
x#
3 fe
et.
x3)
7x2
)15
x)
9 fe
et3
3x )
2
5x !
4
x )
1
4x )
2
2x !
2
Pra
ctic
e (A
vera
ge)
Mul
tiply
ing
Poly
nom
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-7
8-7
Answers (Lesson 8-7)
© Glencoe/McGraw-Hill A22 Glencoe Algebra 1
Rea
din
g t
o L
earn
Math
emati
csM
ultip
lyin
g Po
lyno
mia
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-7
8-7
©G
lenc
oe/M
cGra
w-H
ill49
5G
lenc
oe A
lgeb
ra 1
Lesson 8-7
Pre-
Act
ivit
yH
ow i
s m
ult
iply
ing
bin
omia
ls s
imil
ar t
o m
ult
iply
ing
two-
dig
itn
um
bers
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
7 at
the
top
of
page
452
in y
our
text
book
.
In y
our
own
wor
ds,e
xpla
in h
ow t
he d
istr
ibut
ive
prop
erty
is u
sed
twic
e to
mul
tipl
y tw
o-di
git
num
bers
.
The
ones
of
the
first
fac
tor
are
mul
tiplie
d by
the
ten
s an
d th
eon
es o
f th
e ot
her
fact
or.T
hen
the
tens
of
the
first
fac
tor
are
mul
tiplie
d by
the
ten
s an
d on
es o
f th
e ot
her
fact
or.
Read
ing
the
Less
on
1.H
ow is
mul
tipl
ying
bin
omia
ls s
imila
r to
mul
tipl
ying
tw
o-di
git
num
bers
?
Bin
omia
ls h
ave
two
term
s an
d ea
ch t
erm
of
one
bino
mia
l is
mul
tiplie
d by
each
ter
m o
f th
e ot
her
bino
mia
l.
2.C
ompl
ete
the
tabl
e us
ing
the
FO
IL m
etho
d.
Pro
duct
of
)P
rodu
ct o
f)
Pro
duct
of
)P
rodu
ct o
f Fi
rst T
erm
sO
uter
Ter
ms
Inne
r Ter
ms
Last
Ter
ms
(x#
5)(x
$3)
"(x
)(x)
#(x
)(!
3)#
(5)(
x)#
(5)(
!3)
"x2
#!
3x#
5x#
!15
"x2
#2x
$15
(3y
#6)
(y$
2)"
(3y)
(y)
#(3
y)(!
2)#
(6)(
y)#
(6)(
!2)
"3y
2#
!6y
#6y
#!
12
"3y
2$
12
Hel
ping
You
Rem
embe
r
3.T
hink
of
a m
etho
d fo
r re
mem
beri
ng a
ll th
e pr
oduc
t co
mbi
nati
ons
used
in t
he F
OIL
met
hod
for
mul
tipl
ying
tw
o bi
nom
ials
.Des
crib
e yo
ur m
etho
d us
ing
wor
ds o
r a
diag
ram
.S
ampl
e an
swer
:Im
agin
e th
at t
he t
wo
bino
mia
ls a
re w
ritt
en o
n th
e flo
or.
For
FOIL
,thi
nk o
f al
l the
pos
sibl
e w
ays
you
coul
d ha
ve y
our
left
foot
on
a te
rm o
f th
e fir
st b
inom
ial a
nd y
our
righ
t fo
ot o
n a
term
of
the
seco
ndbi
nom
ial.
Your
fee
t co
uld
be o
n th
e fir
st t
erm
s,th
e ou
ter
term
s,th
e in
ner
term
s,or
the
last
ter
ms.
©G
lenc
oe/M
cGra
w-H
ill49
6G
lenc
oe A
lgeb
ra 1
Pasc
al’s
Tri
angl
eT
his
arr
ange
men
t of
nu
mbe
rs i
s ca
lled
Pas
cal’s
Tri
angl
e.It
was
fir
st p
ubl
ish
ed i
n 1
665,
but
was
kn
own
hu
nd
red
s of
yea
rs e
arli
er.
1.E
ach
num
ber
in t
he t
rian
gle
is f
ound
by
addi
ng t
wo
num
bers
.W
hat
two
num
bers
wer
e ad
ded
to g
et t
he 6
in t
he 5
th r
ow?
3 an
d 3
2.D
escr
ibe
how
to
crea
te t
he 6
th r
ow o
f Pa
scal
’s T
rian
gle.
The
first
and
last
num
bers
are
1.E
valu
ate
1 )
4,4
)6,
6 )
4,an
d 4
)1
to f
ind
the
othe
r nu
mbe
rs.
3.W
rite
the
num
bers
for
row
s 6
thro
ugh
10 o
f th
e tr
iang
le.
Row
6:
15
1010
51
Row
7:
16
1520
156
1R
ow 8
:1
721
3535
217
1R
ow 9
:1
828
5670
5628
81
Row
10:
19
3684
126
126
8436
91
Mu
ltip
ly t
o fi
nd
th
e ex
pan
ded
for
m o
f ea
ch p
rod
uct
.
4.(a
#b)
2a2
)2a
b)
b2
5.(a
#b)
3a3
)3a
2 b)
3ab
2)
b3
6.(a
#b)
4a4
)4a
3 b)
6a2 b
2)
4ab
3)
b4
Now
com
par
e th
e co
effi
cien
ts o
f th
e th
ree
pro
du
cts
in E
xerc
ises
4–6
wit
h
Pas
cal’s
Tri
angl
e.
7.D
escr
ibe
the
rela
tion
ship
bet
wee
n th
e ex
pand
ed f
orm
of
(a#
b)n
and
Pasc
al’s
Tri
angl
e.
The
coef
ficie
nts
of t
he e
xpan
ded
form
are
foun
d in
row
n)
1 of
Pas
cal’s
Tria
ngle
.
8.U
se P
asca
l’s T
rian
gle
to w
rite
the
exp
ande
d fo
rm o
f (a
#b)
6 .
a6)
6a5 b
)15
a4b
2)
20a3
b3
)15
a2b
4)
6ab
5)
b6
11
11
21
13
31
14
64
1
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-7
8-7
Answers (Lesson 8-7)
© Glencoe/McGraw-Hill A23 Glencoe Algebra 1
Ans
wer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Spe
cial
Pro
duct
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-8
8-8
©G
lenc
oe/M
cGra
w-H
ill49
7G
lenc
oe A
lgeb
ra 1
Lesson 8-8
Squa
res
of S
ums
and
Dif
fere
nces
Som
e pa
irs
of b
inom
ials
hav
e pr
oduc
ts t
hat
follo
w s
peci
fic
patt
erns
.One
suc
h pa
tter
n is
cal
led
the
squa
re o
f a
sum
.Ano
ther
is c
alle
d th
esq
uare
of
a di
ffer
ence
.
Squ
are
of a
sum
(a#
b)2
"(a
#b)
(a#
b) "
a2#
2ab
#b2
Squ
are
of a
diff
eren
ce(a
$b)
2"
(a$
b)(a
$b)
"a2
$2a
b#
b2
Fin
d (
3a)
4)(3
a)
4).
Use
the
squ
are
of a
sum
pat
tern
,wit
h a
"3a
and
b"
4.
(3a
#4)
(3a
#4)
"(3
a)2
#2(
3a)(
4) #
(4)2
"9a
2#
24a
#16
The
pro
duct
is 9
a2#
24a
#16
.
Fin
d (
2z!
9)(2
z!
9).
Use
the
squ
are
of a
dif
fere
nce
patt
ern
wit
ha
"2z
and
b"
9.
(2z
$9)
(2z
$9)
"(2
z)2
$2(
2z)(
9) #
(9)(
9)"
4z2
$36
z#
81
The
pro
duct
is 4
z2$
36z
#81
.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d e
ach
pro
du
ct.
1.(x
$6)
22.
(3p
#4)
23.
(4x
$5)
2
x2
!12
x)
369p
2)
24p
)16
16x2
!40
x)
25
4.(2
x$
1)2
5.(2
h#
3)2
6.(m
#5)
2
4x2
!4x
)1
4h2
)12
h)
9m
2)
10m
)25
7.(c
#3)
28.
(3 $
p)2
9.(x
$5y
)2
c2
)6c
)9
9 !
6p)
p2
x2!
10xy
)25
y2
10.(
8y#
4)2
11.(
8 #
x)2
12.(
3a$
2b)2
64y
2)
64y
)16
64 )
16x
)x2
9a2
!12
ab)
4b2
13.(
2x$
8)2
14.(
x2#
1)2
15.(
m2
$2)
2
4x2
!32
x)
64x
4)
2x2
)1
m4
!4m
2)
4
16.(
x3$
1)2
17.(
2h2
$k2
)218
. !x
#3 "2
x6
!2x
3 )1
4h4
!4h
2 k2
)k
4x2
)x
)9
19.(
x$
4y2 )
220
.(2p
#4q
)221
. !x
$2 "2
x2!
8xy
2)
16y
44p
2)
16pq
)16
q2
x2!
x)
48 # 3
4 # 92 % 3
3 # 21 # 161 % 4
©G
lenc
oe/M
cGra
w-H
ill49
8G
lenc
oe A
lgeb
ra 1
Prod
uct
of a
Sum
and
a D
iffe
renc
eT
here
is a
lso
a pa
tter
n fo
r th
e pr
oduc
t of
asu
m a
nd a
dif
fere
nce
of t
he s
ame
two
term
s,(a
#b)
(a$
b).T
he p
rodu
ct is
cal
led
the
dif
fere
nce
of
squ
ares
.
Pro
duct
of
a S
um a
nd a
Diff
eren
ce(a
#b)
(a$
b) "
a2$
b2
Fin
d (
5x)
3y)(
5x!
3y).
(a#
b)(a
$b)
"a2
$b2
Pro
duct
of a
Sum
and
a D
iffer
ence
(5x
#3y
)(5x
$3y
) "(5
x)2
$(3
y)2
a"
5xan
d b
"3y
"25
x2$
9y2
Sim
plify
.
The
pro
duct
is 2
5x2
$9y
2 .
Fin
d e
ach
pro
du
ct.
1.(x
$4)
(x#
4)2.
(p#
2)(p
$2)
3.(4
x$
5)(4
x#
5)
x2!
16p
2!
416
x2!
25
4.(2
x$
1)(2
x#
1)5.
(h#
7)(h
$7)
6.(m
$5)
(m#
5)
4x2
!1
h2
!49
m2
!25
7.(2
c$
3)(2
c#
3)8.
(3 $
5q)(
3 #
5q)
9.(x
$y)
(x#
y)
4c2
!9
9 !
25q
2x2
!y2
10.(
y$
4x)(
y#
4x)
11.(
8 #
4x)(
8 $
4x)
12.(
3a$
2b)(
3a#
2b)
y2!
16x2
64 !
16x2
9a2
!4b
2
13.(
3y$
8)(3
y#
8)14
.(x2
$1)
(x2
#1)
15.(
m2
$5)
(m2
#5)
9y2
!64
x4
!1
m4
!25
16.(
x3$
2)(x
3#
2)17
.(h2
$k2
)(h2
#k2
)18
. !x
#2 "!
x$
2 "x
6!
4h
4!
k4
x2!
4
19.(
3x$
2y2 )
(3x
#2y
2 )20
.(2p
$5s
)(2p
#5s
)21
. !x
$2y
"!x
#2y
"9x
2!
4y4
4p2
!25
s2x2
!4y
216 # 9
4 % 34 % 31 # 16
1 % 41 % 4
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Spe
cial
Pro
duct
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-8
8-8
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 8-8)
© Glencoe/McGraw-Hill A24 Glencoe Algebra 1
Skil
ls P
ract
ice
Spe
cial
Pro
duct
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-8
8-8
©G
lenc
oe/M
cGra
w-H
ill49
9G
lenc
oe A
lgeb
ra 1
Lesson 8-8
Fin
d e
ach
pro
du
ct.
1.(n
#3)
22.
(x#
4)(x
#4)
n2
)6n
)9
x2)
8x)
16
3.(y
$7)
24.
(t$
3)(t
$3)
y2
!14
y)
49t2
!6t
)9
5.(b
#1)
(b$
1)6.
(a$
5)(a
#5)
b2
!1
a2!
25
7.(p
$4)
28.
(z#
3)(z
$3)
p2
!8p
)16
z2!
9
9.(!
#2)
(!#
2)10
.(r
$1)
(r$
1)
!2)
4!)
4r2
!2r
)1
11.(
3g#
2)(3
g$
2)12
.(2m
$3)
(2m
#3)
9g2
!4
4m2
!9
13.(
6 #
u)2
14.(
r#
s)2
36 )
12u
)u
2r2
)2r
s)
s2
15.(
3q#
1)(3
q$
1)16
.(c
$e)
2
9q2
!1
c2!
2ce
)e
2
17.(
2k$
2)2
18.(
w#
3h)2
4k2
!8k
)4
w2
)6w
h)
9h2
19.(
3p$
4)(3
p#
4)20
.(t
#2u
)2
9p2
! 1
6t2
)4t
u)
4u2
21.(
x$
4y)2
22.(
3b#
7)(3
b$
7)
x2!
8xy
)16
y2
9b2
!49
23.(
3y$
3g)(
3y#
3g)
24.(
s2#
r2)2
9y2
!9g
2s4
)2s
2 r2
)r4
25.(
2k#
m2 )
226
.(3u
2$
n)2
4k2
)4k
m2
)m
49u
4!
6u2 n
)n
2
27.G
EOM
ETRY
The
leng
th o
f a
rect
angl
e is
the
sum
of
two
who
le n
umbe
rs.T
he w
idth
of
the
rect
angl
e is
the
dif
fere
nce
of t
he s
ame
two
who
le n
umbe
rs.U
sing
the
se f
acts
,wri
te a
verb
al e
xpre
ssio
n fo
r th
e ar
ea o
f th
e re
ctan
gle.
The
area
is t
he s
quar
e of
the
larg
er n
umbe
r m
inus
the
squ
are
of t
he s
mal
ler
num
ber.
©G
lenc
oe/M
cGra
w-H
ill50
0G
lenc
oe A
lgeb
ra 1
Fin
d e
ach
pro
du
ct.
1.(n
#9)
22.
(q#
8)2
3.(!
$10
)2
n2
)18
n)
81q
2)
16q
)64
!2!
20!
)10
0
4.(r
$11
)25.
(p#
7)2
6.(b
#6)
(b$
6)r2
!22
r)
121
p2
)14
p)
49b
2!
36
7.(z
#13
)(z
$13
)8.
(4e
#2)
29.
(5w
$4)
2
z2!
169
16e
2)
16e
)4
25w
2!
40w
)16
10.(
6h$
1)2
11.(
3s#
4)2
12.(
7v$
2)2
36h
2!
12h
)1
9s2
)24
s)
1649
v2
!28
v)
4
13.(
7k#
3)(7
k$
3)14
.(4d
$7)
(4d
#7)
15.(
3g#
9h)(
3g$
9h)
49k
2!
916
d2
!49
9g2
!81
h2
16.(
4q#
5t)(
4q$
5t)
17.(
a#
6u)2
18.(
5r#
s)2
16q
2!
25t2
a2
)12
au)
36u
225
r2)
10rs
)s2
19.(
6c$
m)2
20.(
k$
6y)2
21.(
u$
7p)2
36c
2!
12cm
)m
2k2
!12
ky)
36y
2u
2!
14up
)49
p2
22.(
4b$
7v)2
23.(
6n#
4p)2
24.(
5q#
6s)2
16b
2!
56bv
)49
v2
36n
2)
48np
)16
p2
25q
2)
60qs
)36
s2
25.(
6a$
7b)(
6a#
7b)
26.(
8h#
3d)(
8h$
3d)
27.(
9x#
2y2 )
2
36a2
!49
b2
64h
2!
9d2
81x2
)36
xy2
)4y
4
28.(
3p3
#2m
)229
.(5a
2$
2b)2
30.(
4m3
$2t
)2
9p6
)12
p3 m
)4m
225
a4!
20a2
b)
4b2
16m
6!
16m
3 t)
4t2
31.(
6e3
$c)
232
.(2b
2$
g)(2
b2#
g)33
.(2v
2#
3e2 )
(2v2
#3e
2 )36
e6!
12e
3 c)
c2
4b4
!g
24v
4)
12v2
e2)
9e4
34.G
EOM
ETRY
Jane
lle w
ants
to
enla
rge
a sq
uare
gra
ph t
hat
she
has
mad
e so
tha
t a
side
of t
he n
ew g
raph
will
be
1 in
ch m
ore
than
tw
ice
the
orig
inal
sid
e s.
Wha
t tr
inom
ial
repr
esen
ts t
he a
rea
of t
he e
nlar
ged
grap
h?4s
2)
4s)
1
GEN
ETIC
SF
or E
xerc
ises
35
and
36,
use
th
e fo
llow
ing
info
rmat
ion
.In
a g
uine
a pi
g,pu
re b
lack
hai
r co
lori
ng B
is d
omin
ant
over
pur
e w
hite
col
orin
g b.
Supp
ose
two
hybr
id B
bgu
inea
pig
s,w
ith
blac
k ha
ir c
olor
ing,
are
bred
.
35.F
ind
an e
xpre
ssio
n fo
r th
e ge
neti
c m
ake-
up o
f the
gui
nea
pig
offs
prin
g.0.
25B
B)
0.50
Bb
)0.
25bb
36.W
hat
is t
he p
roba
bilit
y th
at t
wo
hybr
id g
uine
a pi
gs w
ith
blac
k ha
ir c
olor
ing
will
pro
duce
a gu
inea
pig
wit
h w
hite
hai
r co
lori
ng?
25%
Pra
ctic
e (A
vera
ge)
Spe
cial
Pro
duct
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-8
8-8
Answers (Lesson 8-8)
© Glencoe/McGraw-Hill A25 Glencoe Algebra 1
Ans
wer
s
Rea
din
g t
o L
earn
Math
emati
csS
peci
al P
rodu
cts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
8-8
8-8
©G
lenc
oe/M
cGra
w-H
ill50
1G
lenc
oe A
lgeb
ra 1
Lesson 8-8
Pre-
Act
ivit
yW
hen
is
the
pro
du
ct o
f tw
o bi
nom
ials
als
o a
bin
omia
l?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
8 at
the
top
of
page
458
in y
our
text
ook.
Wha
t is
a m
eant
by
the
term
tri
nom
ial
prod
uct?
a th
ree-
term
pol
ynom
ial a
nsw
er w
hen
mul
tiply
ing
poly
nom
ials
Read
ing
the
Less
on
1.R
efer
to
the
Key
Con
cept
s bo
xes
on p
ages
458
,459
,and
460
.
a.W
hen
mul
tipl
ying
tw
o bi
nom
ials
,the
re a
re t
hree
spe
cial
pro
duct
s.W
hat
are
the
thre
esp
ecia
l pro
duct
s th
at m
ay r
esul
t w
hen
mul
tipl
ying
tw
o bi
nom
ials
?
squa
re o
f a
sum
,squ
are
of a
diff
eren
ce,p
rodu
ct o
f a
sum
and
adi
ffer
ence
b.E
xpla
in w
hat
is m
eant
by
the
nam
e of
eac
h sp
ecia
l pro
duct
.
squa
re o
f a
sum
:squ
arin
g th
e su
m o
f tw
o m
onom
ials
;squ
are
of a
diff
eren
ce:s
quar
ing
the
diff
eren
ce o
f tw
o m
onom
ials
;pro
duct
of
asu
m a
nd a
diff
eren
ce:t
he p
rodu
ct o
f th
e su
m a
nd t
he d
iffer
ence
of
the
sam
e tw
o te
rms
c.U
se t
he e
xam
ples
in t
he K
ey C
once
pts
boxe
s to
com
plet
e th
e ta
ble.
Sym
bols
Pro
duct
Exa
mpl
eP
rodu
ct
Squ
are
of a
Sum
(a!
b)2
a2!
2ab
!b
2(x
!7)
2x2
!14
x!
49
Squ
are
of a
D
iffer
ence
(a"
b)2
a2"
2ab
!b
2(x
"4)
2x2
"8x
!16
Pro
duct
of
a S
um
and
a D
iffer
ence
(a!
b)(a
"b)
a2"
b2
(x!
9)(x
"9)
x2"
81
2.W
hat
is a
noth
er p
hras
e th
at d
escr
ibes
the
pro
duct
of
the
sum
and
dif
fere
nce
of
two
term
s?di
ffer
ence
of
squa
res
Hel
ping
You
Rem
embe
r
3.E
xpla
in h
ow F
OIL
can
hel
p yo
u re
mem
ber
how
man
y te
rms
are
in t
he s
peci
al p
rodu
cts
stud
ied
in t
his
less
on.
For
the
squa
re o
f a
sum
and
the
squ
are
of a
diff
eren
ce,t
he in
ner
and
oute
r pr
oduc
ts a
re e
qual
,so
ther
e ar
e ar
e on
lyth
ree
term
s.Fo
r th
e pr
oduc
t of
the
sum
and
diff
eren
ce o
f tw
o te
rms,
two
of t
he p
rodu
cts
for
FOIL
are
opp
osite
s.Th
at m
eans
tha
t th
e fin
al p
rodu
ctha
s on
ly t
wo
term
s.
©G
lenc
oe/M
cGra
w-H
ill50
2G
lenc
oe A
lgeb
ra 1
Sum
s an
d D
iffer
ence
s of
Cub
es
Rec
all t
he f
orm
ulas
for
fin
ding
som
e sp
ecia
l pro
duct
s:
Perf
ect-
squa
re t
rino
mia
ls:
(a!
b)2
"a2
!2a
b!
b2or
(a
#b)
2"
a2#
2ab
!b2
Dif
fere
nce
of t
wo
squa
res:
(a!
b)(a
#b)
"a2
#b2
A p
atte
rn a
lso
exis
ts f
or f
indi
ng t
he c
ube
of a
sum
(a!
b)3 .
1.F
ind
the
prod
uct
of (a
!b)
(a!
b)(a
!b)
.
a3!
3a2 b
!3a
b2
!b
3
2.U
se t
he p
atte
rn f
rom
Exe
rcis
e 1
to e
valu
ate
(x!
2)3 .
x3
!6x
2!
12x
!8
3.B
ased
on
your
ans
wer
to
Exe
rcis
e 1,
pred
ict
the
patt
ern
for
the
cube
of
a di
ffer
ence
(a#
b)3 .
a3"
3a2 b
!3a
b2
"b
3
4.F
ind
the
prod
uct
of (a
#b)
(a#
b)(a
#b)
and
com
pare
it t
o yo
ur
answ
er f
or E
xerc
ise
3.
a3"
3a2 b
!3a
b2
"b
3
5.U
se t
he p
atte
rn f
rom
Exe
rcis
e 4
to e
valu
ate
(x!
4)3 .
x3
!12
x2!
48x
!64
Fin
d e
ach
pro
du
ct.
6.(x
!6)
37.
(x#
10)3
x3
!18
x2!
108x
!21
6x
3"
30x2
!30
0x"
1000
8.(3
x#
y)3
9.(2
x#
y)3
27x3
"27
x2y
!9x
y2
"y
38x
3"
12x2
y!
6xy
2"
y3
10.(
4x!
3y)3
11.(
5x!
2)3
64x3
!14
4x2 y
!10
8xy
2!
27y
312
5x3
!15
0x2
!60
x!
8
En
rich
men
t
NA
ME
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ATE
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RIO
D__
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8-8
8-8
Answers (Lesson 8-8)