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Integrated Algebra

Chapter 7: Exponents and Polynomials

Name:______________________________

Teacher:____________________________

Pd: _______

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Table of Contents

o Chapter 7-1: SWBAT: Evaluate and simplify expressions containing zero and integer exponents.

Pgs: 3-5

HW: Page 6 (Evens Only)

o Chapter 7-2: SWBAT: Convert between standard notation and scientific notation.

Pgs: 7-10

HW: Page 11

o Lesson 7-3: SWBAT: Use Multiplication Properties of Exponents to Evaluate and Simplify

Expressions.

Pgs: 12-15

HW: Page 16

o Lesson 7-4: SWBAT: Use Division Properties of Exponents to Evaluate and Simplify Expressions

Pgs: 17-19

HW: Page 20

o Half Period Quiz Lessons: 7-1 to7-4

o Lesson 12-6: SWBAT: Divide a polynomial by a monomial.

Pgs: 21-22

HW: Page 23

o Lesson 7-6: SWBAT: Add and Subtract Polynomials.

Pgs: 24-26

HW: Page 27

o Lesson 7-7: SWBAT: Multiply Polynomials by a monomial.

Pgs: 28-31

HW: Page 32

o Lesson 7-7: SWBAT: Multiply two binomials.

Pgs: 33-36

HW: Page 37

o Full Period Quiz Lessons: 12-6, 7-5 to7-7

o Practice Test: Review using E-Clickers

Pgs: 38-40

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Chapter 7 – 1

SWBAT: Evaluate and simplify expressions containing zero and integer exponents.

Warm – Up

Evaluate each expression for the given values of the variables.

You have seen positive exponents. Recall that to simplify 32, use 3 as a factor 2 times: 3

2 = 3 * 3 = 9.

But what does it mean for an exponent to be negative or 0? You can use a table and look for a pattern to

figure it out.

Example 1: Simplify these expressions.

Practice: Simplify these expressions.

1. x3y2 for x = –1 and y = 10 2. y2

x3 for x = –1 and y = 10

a. 2–4

b. 70

c. (-5)–3

d. -5–3

a. 3–3 c. (-2)–3

b. 150 d. -2–3

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Example 2: Evaluate the expression for the given value of the variables.

a. x–2 for x = 4

Practice: Evaluate the expression for the given value of the variables.

a. p–3 for p = 4

Example 3: Simplifying Expressions with Zero and Negative Numbers

a. 7w–4

Practice: Simplifying Expressions with Zero and Negative Numbers

a. 2r0m-3

b. –2a0b-4 for a = 5 and b = –3

b. 8a-2b0 for a = -2 and b = 6

b. -5

k–2c.

a0b-2

c-3d6

b. r–3

7c.

g4

h-6

d. 2a-5

b-6e.

x-5

3y12 f. 20p-1

5q-3

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Challenge Problem: Simplify

[2-2 + (6 + 2)0]

Summary:

Exit Ticket:

6

Chapter 7-1 Homework (EVENS ONLY)

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Chapter 7 – 2

SWBAT: Convert between standard notation and scientific notation.

Warm – Up

Simplify:

a3c-4x5

a3b-2

c-1

Scientific Notation

…

…

…

(1)

(2)

(3)

3.2 x 1013? 23.6 x 10-8?

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Example 1: Scientific Notation to Standard Form

The exponent represents the number of places the decimal needs to be moved.

Move decimal point: to the ________ for ____________ exponents of 10

OR to the ________ for ____________ exponents of 10.

1.015 x 10-8

5.024 x 103

Practice Problems: Express in standard form:

1) 5.02 x 103 3) 603 x 10-4

2) 52.8 x 106 4) 0.03 x 10-2

Example 2:Standard Form to Scientific Notation

Remember, the decimal is at the end of the final zero.

The decimal must be moved to ensure that the _____________ is between _____________.

How many places did it move? This number will be the _____________.

Express in scientific notation:

4,750,000

0.000789

Practice Problems: Express in scientific notation:

1) 4500 3) 6,560,000

2) 0.00002 4) 0.00203

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Example 3:Using Scientific Notation to compute products and quotients of numbers.

Multiplying and Scientific Notation

Example:

To find the answer to this problem, we can multiply 9.3 and 6.2 and get 57.66.

We can also multiply to get .

So putting these two pieces together we get:

But remember that to be in scientific notation…

_____________________________________________________________________________

1) What is the product of and written in scientific notation?

1) 2) 3) 4)

2) What is the product of , , and expressed in scientific notation?

1) 2) 3) 4)

Dividing and Scientific Notation

Example:

3) What is the quotient of and ?

1) 2) 3) 4)

4) The quotient of and expressed in scientific notation is

1) 4,000

2) 40,000

3) 4)

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Challenge Problem:

Summary:

Exit Ticket:

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Chapter 7-2 Homework

Part A: Express each of the following in standard form.

1) 5.2 x 103 2) 3.6 x 10

1 3) 9.65 x 10

–4

4) 6.452 x 102 5) 8.5 x 10

–2 6) 8.77 x 10

–1

Part B: Express each of the following in scientific notation.

7) 78,000 8) 16 9) 0.00053

10) 0.0043 11) 250000000000 12) 0.875

Part C: Multiplying and Dividing Scientific Notation

13) (6.02 x 1023

) (8.65 x 104) 14) (3.2 x 10

–7) (8.6 x 10

9) 15) (5.4 x 10

4) (2.2 x 10

7)

16) (1.6 x 105) (2.4 x 10

15) 17) (7.0 x 10

28) (–3.2 x 10

–20) (–6.4 x 10

35)

18) If is divided by , the result is

1) 1

2) 0.01

3) 4)

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Chapter 7 – 3

SWBAT: Use Multiplication Properties of Exponents to Evaluate and Simplify Expressions.

Warm – Up Use a calculator to find the value of each expression.

a. 53 55 =

c. 51 57 =

b. 56 52 =

d. 54 54 =

Example 1: Products of Powers

Notice that when we multiply powers with the same base, you ___________ the exponents.

For any nonzero number a and any integer m and n, ______________________.

Simplify.

a) c4 d3 c2

b) 5x 2y4 3x8

Rearrange factors.

Multiply coefficients.

Add exponents of powers

with the same base.

Solution

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Practice: Simplify.

Example 2: Raise a Power to a Power Simplify.

1) (x3)4

Notice that when we raise a power to a power, you ____________ the exponents.

For any nonzero number a and any integer m and n, ______________________.

Practice: Simplify.

Example 3: Raise a Product to a Power

We can use repeated multiplication to simplify expressions like (5y)³.

Simplify.

Method 1 Method 2

(5y)³ (5y)³

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Notice that when we raise a product to a power, you _____________________________ and ______________.

For any nonzero, real numbers a and b and any integer n, ___________________.

Practice: Simplify.

Additional Examples: Simplify.

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Challenge Problem

Summary

Exit Ticket

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Chapter 7-3 Homework:

Simplify.

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Chapter 7 – 4

SWBAT: Use Division Properties of Exponents to Evaluate and Simplify Expressions

Warm – Up Simplify.

4(3c3)3

Example 1: Dividing Powers with the Same Base

When we divide powers with the same base, you ______________ the exponents.

For any number a and any integers m and n, ______________________.

Simplify. Use only positive exponents.

Practice: Simplify. Use only positive exponents.

1)

45

5 2)

4

3

4

3

x

x 3)

6 3

3

a b

ab

4)

10

2

z

z 5)

9

3

x

x 6)

8 4

3 12 3

r t

r s t

7)

3 5

6 3

4

2

x y

x y 8) 3

5

15

ab

a

9)

4 2 3

5 2

18

3

a b c

a bc

x6

x2

x2

x6

a4b-2

a2b3

10x3y5

10x5y2

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Example 2: Finding Positive Powers of a Quotient

For any nonzero real numbers a and b and positive integer n, ______________________.

Simplify. Use only positive exponents.

3

5 4

Practice: Simplify. Use only positive exponents.

Example 3: Finding Negative Powers of a Quotient

For any nonzero real numbers a and b and positive integer n, ______________________.

Simplify. Use only positive exponents.

2

3 -4

Practice: Simplify. Use only positive exponents.

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Challenge Problem: Simplify. Use only positive exponents.

Summary

Exit Ticket

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Chapter 7-4 Homework:

Simplify the following using the rules of exponents.

1)

12

4

h

h 2)

10

6

c

c 3)

5

4

3

2

x

x

4)

15 6

5 8 3

r t

r s t 5)

9 4

9

a b

a b 6) 2

39

z

xz

7) 32

23

7

28

qp

qp 8) 3

614

x

yx 9)

413

9320

yx

zyx

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Chapter 12 – 6

SWBAT: Divide a polynomial by a monomial.

Warm – Up

****Half Period Quiz****

Example Divide.

(6x3 + 8x2 - 4x) 2x

Practice Problems Divide.

(8x3 - 4x2 + 12x) (-4x2)(15x3 - 20x2 + 5x) 5x

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Challenge Problem:

Divide and Simplify.

Summary:

Exit Ticket:

When is divided by , the quotient is

1)

2)

3)

4)

45a4b3 - 90a3b

15a2b

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Chapter 12 – 6 Homework Directions: Simplify each expression.

1)

x

xxx 234 63 2)

2

256

8

8408

x

xxx

3)

2

234

9

18369

x

xxx 4)

m

mmm

2

222 234

5)

x

xxx

5

2550100 58

6)

x

xxx

2

10412 23

7)

2

24

10

1030

r

rr 8)

2

34

4

1648

x

xx

9) xxxx 6)61236( 23 10) * 52 3)3612( xxx

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SWBAT: Add and Subtract Polynomials.

Chapter 7 – 6

Warm – Up

Subtract )8( 2aa from )834( 2aa

Remember!

___________________________ are:

- constants

- terms with the SAME variable(s) raised to the SAME powers.

Example 1: Adding Polynomials

Practice: Adding Polynomials

1. 2.

3. )227()12( 22 xxx 4. )76()48( 22 nnnn

5. )32()33( 232 mmm 6. )25()227( 334 bbb

Find (2x2 - 3x + 4) + (3x2 + 2x - 3).

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7.

Example 2: Subtracting Polynomials

Practice Problems: Subtracting Polynomials

Find each difference. Write your answer in standard form.

8. 9.

10. 11.

12. )46()354( 22 bbb 13.

Find (7x2 - 3x + 1) - (x2 + 4x - 2).

a. (3x3 + 5x - 2) - (x3 - 4x - 3) b. (2a3 + 9a - 2) - (a2 + 4a - 7)

c. (y2 - 7y) - (-4y2 + 3y - 1) d. (x2 + 2x + 5) - (2x2 - 4x - 5)

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Challenge Problems Find the perimeter of the figure below.

Summary

Exit Ticket

What is the sum of and ?

1)

2)

3)

4)

5x + 2

6x - 10

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Chapter 7-6 - Homework:

Add or subtract.

From )462( 2bb subtract )554( 2 bb

Find the difference when )942( 2 xx is subtracted from )185( 2 xx

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Chapter 7 – 7 (Day 1) SWBAT: Multiply polynomials by a monomial.

Warm – Up Use the distributive property to simplify each expression.

a. 3(x + 2)

b. 2(x + y)

c. (m - 7)2

d. -(b - c)

Example 1: Product of Monomials Step 1: Group factors with like bases together.

Step 2: Multiply.

Multiply 3x2 and 2x3

Multiply 4a2b3 and 3a3b

Practice Problems: Product of Monomials

1. 2. 3.

Example 2: Product of a Polynomial and a Monomial Step 1: Distribute.

Step 2: Group factors with like bases together.

Step 3: Multiply.

Multiply 3x and (2x + 1)

Multiply 2x2y and (3x - y)

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Practice Problems: Product of a Polynomial and a Monomial

Practice Find the product.

1) )5(3 x 2) 7( 5 8)x

3) 2( 2)x x 4) 2 ( 12)k k

5) 2 3(3 )a a a 6)

2 53 (9 4 )x x x

7) 4 62 ( 3 )y y y 8)

2 9 25 (2 3 )s s s

9) )142(3 2 xx 10) )422(5 2 xxx

Example 3: Product of a Polynomial and a Monomial Step 1: Distribute.

Step 2: Group factors with like bases together.

Step 3: Multiply.

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Practice Problems: Product of a Polynomial and a Monomial Step 1: Distribute.

Step 2: Group factors with like bases together.

Step 3: Multiply.

1) 2 6 2 98 (2 4 ) 3 ( 2 9 )x y x y xy x y 2) 2 25 (4 2 ) 3 ( 2 4 )a a a a a a

3) 2 22 (5 6) 5 ( 3 4) 7( 5)a a a a a a 4) 2 25 ( 7 3) 2 (2 19 2 )w w w w w w

5) 26 (2 3) 5(2 9 3)t t t t 6) 3 4 2 2 38 ( 2 ) 4 (1 6 )b b b c b b b c b

7) 2 2 3 34 ( 8 3 ) 7 (8 1)x x x x x xy 8)

3 2 4 5 5 7 88 (3 2 ) 2 (5 7 6)x y xy x y x x y xy

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Challenge Problem:

Summary:

Exit Ticket:

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Chapter 7-7 (Day 1) Homework: Find the product.

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Chapter 7 – 7 (Day 2)

SWBAT: Multiply two binomials.

Warm – Up

1) )12(4 2 xx 2) )32(3 22 baab 3) )3(2 223 abbaba

Method 1: Use the Distributive Property Step 1: Distribute the first polynomial over the terms of the second. Step 2: Multiply.

Step 3: Combine like terms.

(2x + 1)(x - 5)

Method 2: Use a Box Step 1: Write the product of the monomials in each row or column.

Step 2: Add the terms inside the rectangle.

(2x + 1)(x - 5)

(2x + 1)(x - 5)

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Method 3: Use FOIL Step 1: Multiply the FIRST terms.

Step 2: Multiply the OUTER terms.

Step 3: Multiply the INNER terms.

Step 4: Multiply the LAST terms.

Step 5: Combine like terms.

First

Outer

Inner

Last

Practice Problems Directions: Simplify each expression by using the distribute property, box method or FOIL method.

1) ( 9)( 9)b b 2) 2)2( x

3) (3 4)(4 7)x x 4) (2 3)( 4)x x

5) The length of a rectangle is 53 x and its width is 2x . Express the area of the rectangle.

(2x + 1)(x - 5)

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Example 2: Multiplying Polynomials

Multiply.

)4)(32( yxyx 2 2 2 2( )( )c d c d

Practice Problems: Multiplying Polynomials

Multiply.

1) (4 )(5 2 )x y x y 2) )6)(2( 2233 yxyx

3) )4)(2( 22 baba 4) )5)(2( 2222 yxyx

5) (2x3 – 5)

2 6) (x

3 – y

5)

2

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Challenge Problem

Multiply and Simplify.

)3()52( 2 xx 22 )15()93( xx

Summary

Exit Ticket

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Chapter 7-7 (Day 2) Homework

Find the product.

The length of a rectangle is 2 5x and its width is 7x . Express the area of the rectangle.

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Chapter 7 Review SWBAT: Assess their mastery of concepts and skills.

7-2: Scientific Notation

1. Which is the following is the standard form of 4 x 10-6?

A. 0.000004 B. 0.0000004 C. 40,000 D. 400,000

2. In 2000, the population of Thomasville, North Carolina, was about 20,000 people. Write this population in scientific notation.

A. 0.2 x 105 B. 2 x 104 C. 2 x 105 D. 20 x 103

7-3: Multiplication Properties of Exponents

3. Simplify 86 ∙ 83 . A. 82 B. 83 C. 89 D. 818

4. Simplify (a3b)2.

A. a3b2 B. a6b C. a6b2 D. a9b2

5. Simplify (23)3.

A. 20 B. 21 C. 26 D. 29

6. Which of the following is equivalent to 6-4? A. (-6)4 B. 1

64 C. -24 D. 24

7. Evaluate x-3 for x = 2. A. -8 B. -6 C. 1

9 D. 1

8

8. Evaluate 4w2 for x = 2. A. 103 B. 112 C. 403 D. 529

7-4: Division Properties of Exponents

9. Simplify 310 . 32

A. 35 B. 38 C. 312 D. 320

10. Simplify . A. 9

7 B. 6

14 C. 9

49 D. 7 9

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11. Simplify A. -25

81 B. 81

25 C. 10

18 D. 18

10

12. Simplify . A. B. C. D. 81x12

12-6: Dividing Polynomials

13. Divide (12x2 + 6x – 3) ÷ 3. A. 12x2 + 6x - 1 B. 12x2 + 6x C. 4x2 + 2x - 1 D. 12x2 + 2x - 1

7-5: Describing Polynomials

14. Which polynomial is written in standard form? A. -5x3 + 2x + 9x2 B. -5x3 + 9x2 + 2x C. 2x + 9x2 - 5x3 D. 9x2 + 2x - 5x3

7-6: Adding Polynomials

15. Add (2x3 – 5) + (x3+ 3). A. 3x3 - 2 B. 3x6 - 2 C. 3x6 - 15 D. 3x3 + 8

7-6: Subtracting Polynomials

16. Subtract (6a3 + 3a) - (4a2 + 2a). A. 2a2 + a B. 2a2 + 5a C. 3 D. 3a3

17. Subtract (7x + 2) - (x - 4). A. 6x - 6 B. 6x + 6 C. 6x2 + 6 D. 6x2 - 6

3x20

x2

3x20

x8

81x20

x8

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7-7: Multiplication by a Monomial

18. Multiply (4r3)(2r5). A. 8r8 B. 8r15 C. 2048r8 D. 2048r15

19. Multiply (3x3)(2y)2(4x4).

A. 48x12y2 B. 28x12y2 C. 28x7y2 D. 48x7y2

7-7: Multiplying Binomial 22. Multiply (b + 3)(b2 + 2b).

A. b3 + 2b3 + 3b B. b3 + 5b2 + 6b C. b3 + 6b2 + 3b D. 4b5 + 8b3 + 3b

23. Multiply (x - 4)(x2 + 9x).

A. 2x2 + 5x2 – 39x B. x3 + 5x2 – 36x C. 2x3 + 5x2 + 33x D. x3 + 5x2 – 39x

7-7: Multiplying two Binomials

20. Multiply (x + 2)(x + 3). A. x2 + 6 B. x2 + 5x + 5 C. 2x + 5 D. x2 + 5x + 6

21. Multiply (2x - 3)2.

A. 2x2 - 3 B. 2x2 - 12x + 9 C. 4x2 - 3 D. 4x2 - 12x + 9