polynomials-02.pdf

Take the Chapter 4 Prep Test to find out if you are ready tolearn to:

• Multiply and divide monomials• Add, subtract, multiply, and divide polynomials• Write a number in scientific notation

SECTION 4.1

A To add polynomialsB To subtract polynomials

SECTION 4.2

A To multiply monomialsB To simplify powers of monomials

SECTION 4.3

A To multiply a polynomial by amonomial

B To multiply two polynomialsC To multiply two binomials using the

FOIL methodD To multiply binomials that have

special productsE To solve application problems

SECTION 4.4

A To divide monomialsB To write a number in scientific

notation

SECTION 4.5

A To divide a polynomial by amonomial

B To divide polynomials

191

PREP TEST

Do these exercises to prepare for Chapter 4.

ARE YOU READY?OBJECTIVES

1. Subtract:1 [1.2B]

3. Simplify:

[1.6A]

5. If is a fraction in simplest

form, what number is not a possible value of b?0 [1.6A]

7. Simplify:

[2.2A]

9. Simplify:[2.2C]�6x � 24

�312x � 82

5x 2 � 9x � 63x2 � 4x � 1 � 2x2 � 5x � 7

a

b

23

�24

�36

�2 � 1�32

CHAPTER

4CHAPTER

4PolynomialsPolynomials

2. Multiply:[1.3A]

4. Evaluate when 48 [2.1A]

6. Are and 2x like terms?No [2.2A]

8. Simplify: �4y � 4y0 [2.2A]

10. Simplify:3xy � 4y � 2(5xy � 7y)

[2.2D]�7xy � 10y

2x2

n � �2.3n4

�18�3162

Tim

Fitz

harri

s/M

inde

n Pi

ctur

es/F

irst L

ight

46043_04_Ch04_0191-0234.qxd 10/27/09 12:48 PM 91

A monomial is a number, a variable, or a product of numbers and variables. For instance,

7

A number A variable A product of a A product of anumber and a variable number and variables

A polynomial is a variable expression in which the terms are monomials.

A polynomial of one term is a monomial. is a monomial.A polynomial of two terms is a binomial. is a binomial.A polynomial of three terms is a trinomial. is a trinomial.

The degree of a polynomial in one variable is the greatest exponent on a variable. Thedegree of is 3; the degree of is 4. The degree of anonzero constant is zero. For instance, the degree of 7 is zero.

The terms of a polynomial in one variableare usually arranged so that the exponents on the variable decrease from left to right.This is called descending order.

Polynomials can be added, using either a horizontal or a vertical format, by combininglike terms.

Add . Use a horizontal format.

Add . Use a vertical format.

2x3 � 4x2 � 2x � 13

2x3 � 8x � 12 2x3�4x2 � 6x � 19

1�4x2 � 6x � 92 � 112 � 8x � 2x32

� 3x3 � 7x2 � 5x � 5

� 3x3 � 7x2 � 1�7x � 2x2 � 12 � 7213x3 � 7x � 22 � 17x2 � 2x � 72

13x3 � 7x � 22 � 17x2 � 2x � 72

2y4 � y2 � 14x3 � 5x2 � 7x � 8

7x2 � 5x � 74x � 2�7x2

12xy2a2

3b

S E C T I O N

Addition and Subtraction of Polynomials

192 CHAPTER 4 • Polynomials

4.1OBJECTIVE A To add polynomials

Take NoteThe expression is not a monomial because cannot be written as aproduct of variables.

The expression is not a

monomial because it is aquotient of variables.

2xy

�x3�x

Instructor Note

An analogy may help studentsunderstand these terms.Polynomial is similar to theword car. Chevrolet and Fordare types of cars. Monomialsand binomials are types ofpolynomials.

As a class exercise, askstudents to identifymonomials. For instance,

which of , y, 6x,

abxy, and are monomials?

Then ask students to identifypolynomials. Here are somepossible examples: ,

, and .x 4 � �x 2 � 7�2x � 1

x

� 1,2x

�5

x 2� 1,

x

2�

x 2

5

x � 7

3z

6 � x,2

3

1

2

HOW TO • 1

HOW TO • 2

2y4 � y3 � 2y2 � 4y � 1

7z4 � 4z3 � z � 6

5x3 � 4x2 � 6x � 1

• Use the Commutative andAssociative Properties of Additionto rearrange and group like terms.

• Then combine like terms.

• Combine the terms in each column.

• Arrange the terms of each polynomial in descendingorder, with like terms in the same column.

EXAMPLE • 1 YOU TRY IT • 1Use a horizontal format to add

.

Solution

Use a horizontal format to add.

Your solution

8x2 � 7x � 3

1�4x3 � 2x2 � 82 � 14x3 � 6x2 � 7x � 52

� 10x2 � 5x � 18 � 18x2 � 2x22 � 1�4x � 9x2 � 1�9 � 92

18x2 � 4x � 92 � 12x2 � 9x � 92

18x2 � 4x � 92 � 12x2 � 9x � 92

Solution on p. S10

In-Class Examples

1. Use a horizontal format to add .

12x2 � 5x � 10

2. Use a vertical format to add .

� 5x2 � 4x � 14

��9x2 � 6x � 8� � �4x2 � 10x � 6�

�4x2 � 5x � 7� � �8x2 � 10x � 3�

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 192

SECTION 4.1 • Addition and Subtraction of Polynomials 193

The opposite of the polynomial is .

To simplify the opposite of a polynomial,change the sign of each term to its opposite.

Polynomials can be subtracted using either a horizontal or a vertical format. To subtract,add the opposite of the second polynomial to the first.

Subtract . Use a horizontal format.

Subtract . Use a vertical format.

The opposite of is .

3y3 � 2y2 � 30

� 2y2 � 4y � 21 3y3 � 4y � 9

�2y2 � 4y � 212y2 � 4y � 21

19 � 4y � 3y32 � 12y2 � 4y � 212

� �2y3 � 4y2 � y � 11 � �2y3 � 4y2 � 1�6y � 5y2 � 17 � 42 � 14y2 � 6y � 72 � 1�2y3 � 5y � 42

14y2 � 6y � 72 � 12y3 � 5y � 42

14y2 � 6y � 72 � 12y3 � 5y � 42

�13x2 � 7x � 82 � �3x2 � 7x � 8

�13x2 � 7x � 8213x2 � 7x � 82

OBJECTIVE B To subtract polynomials

EXAMPLE • 2 YOU TRY IT • 2

Use a vertical format to add.

Solution

Use a vertical format to add.

Your solution

�3x 3 � 2x 2 � 10x

16x3 � 2x � 82 � 1�9x3 � 2x2 � 12x � 82

�3x3 � 4x2 � 2x � 2

2x3 � 5x � 11 �5x3 � 4x2 � 7x � 9

1�5x3 � 4x2 � 7x � 92 � 12x3 � 5x � 112

Solution on p. S10

Take NoteThis is the same definitionused for subtraction ofintegers: Subtraction isaddition of the opposite.

EXAMPLE • 3 YOU TRY IT • 3Use a horizontal format to subtract

.

Solution

Use a horizontal format to subtract.

Your solution

�7w 3 � 4w 2 � 10w � 7

1�4w3 � 8w � 82 � 13w3 � 4w2 � 2w � 12

� �2c2 � 14c � 4 � 17c2 � 9c � 122 � 1�9c2 � 5c � 82

17c2 � 9c � 122 � 19c2 � 5c � 82

17c2 � 9c � 122 � 19c2 � 5c � 82

Solutions on p. S10

EXAMPLE • 4 YOU TRY IT • 4Use a vertical format to subtract

.

Solution

Use a vertical format to subtract.

Your solution

13y 3 � 4y 2 � 2

113y3 � 6y � 72 � 14y2 � 6y � 92

�k3 � 2k � 9 �k3 � 3k2 � 6k � 8 �k3 � 3k2 � 4k � 1

13k2 � 4k � 12 � 1k3 � 3k2 � 6k � 82In-Class Examples

1. Use a horizontal format to subtract.

14x2 � x � 1

2. Use a vertical format to subtract.

�4y3 � 2y2 � 13y � 4

�7y 2 � 8y � 2� � �4y 3 � 9y 2 � 5y � 6�

�6x 2 � 2x � 4� � ��8x 2 � x � 3�

• Add the opposite of the secondpolynomial to the first.

• Combine like terms.

• Arrange the terms of each polynomial in descending order,with like terms in the same column.

• Note that , but 0 is not written.4y � 4y � 0

HOW TO • 3

HOW TO • 4

• Add the opposite ofto the

first polynomial.�k3 � 3k2 � 6k � 8�

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 193

For Exercises 1 to 8, state whether the expression is a monomial.

1. 17 Yes � 2. Yes � 3. No 4. xyz Yes

5. Yes � 6. No 7. Yes 8. �x Yes

For Exercises 9 to 16, state whether the expression is a monomial, a binomial, atrinomial, or none of these.

9. 10. 11. � 12.Binomial None of these Trinomial Binomial

13. 14. 15. � 16.

None of these Monomial Binomial Trinomial

For Exercises 17 to 26, add. Use a horizontal format.

17. � 18.

19. � 20.

21. 22.

23. � 24.

25. 26.

For Exercises 27 to 36, add. Use a vertical format.

27. � 28.

29. 30.

31. � 32.

33. � 34.

35. � 36.5r 3 � 5r 2 � r � 32a3 � 3a2 � 11a � 215r3 � 6r2 � 3r2 � 1�3 � 2r � r2212a3 � 7a � 12 � 11 � 4a � 3a2 2

�y 3 � y 2 � 6y � 2x 3 � 2x 2 � 6x � 61y2 � 3y3 � 12 � 1�4y3 � 6y � 321�7x � x3 � 42 � 12x2 � x � 102

4x 2 � 9x � 95x 2 � 7x � 201x2 � x � 52 � 13x2 � 10x � 4212x2 � 6x � 122 � 13x2 � x � 82

3x 2 � 15x � 24y 2 � 813x2 � 9x2 � 16x � 2421y2 � 4y2 � 1�4y � 82

8y 2 � 4y�2x 2 � 3x13y2 � 2y2 � 15y2 � 6y21x2 � 7x2 � 1�3x2 � 4x2

3y 3 � 4y 2 � 4y � 353r 3 � 2r 2 � 11r � 7114 � 4y � 3y32 � 1�4y2 � 21217 � 5r � 2r2 2 � 13r3 � 6r2

3y 3 � 2y 2 � 8y � 125x 3 � 10x 2 � x � 414y � 3y3 � 92 � 12y2 � 4y � 21217x � 5x3 � 72 � 110x2 � 8x � 32

�3x 2 � 8x � 63a2 � 3a � 171�6x2 � 7x � 32 � 13x2 � x � 3212a2 � 7a � 102 � 1a2 � 4a � 72

8x 2 � 2xy7x 2 � xy � 4y 212x2 � 4y22 � 16x2 � 2xy � 4y2214x2 � 5xy2 � 13x2 � 6xy � 4y22

y 2 � 7y5x 2 � 8x1�3y2 � y2 � 14y2 � 6y214x2 � 2x2 � 1x2 � 6x2

12a4 � 3a � 26x2 � 7xab

4

2x

� 3

x2 � y29x2 � x � 12y � 3�y3x � 5

�5 xxy

z2

3y

17

�x3x4

4.1 EXERCISES


OBJECTIVE A To add polynomials

Quick Quiz

1. Is a monomial? Yes

2. Is a monomial, a binomial, atrinomial, or none of these? Binomial

3. Add:7x2 � 10x � 3

�2x 2 � 3x � 4� � �5x 2 � 7x � 1�

3y 2 � 8

x 2y

3

Suggested Assignment

Exercises 1–35, oddsExercises 39–57, oddsMore challenging problems: Exercises 37, 59, 60, 61

�Selected exercises available online at www.webassign.net/brookscole.

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 194

SECTION 4.1 • Addition and Subtraction of Polynomials 195

For Exercises 39 to 48, subtract. Use a horizontal format.

39. � 40.

41. 42.

43. � 44.

45. � 46.

47. � 48.

For Exercises 49 to 58, subtract. Use a vertical format.

49. � 50.

51. � 52.

53. 54.

55. � 56.

57. � 58.

59. What polynomial must be added to so that the sum is ?

Applying the Concepts

60. Is it possible to subtract two polynomials, each of degree 3, and have the differencebe a polynomial of degree 2? If so, give an example. If not, explain why not.

61. Is it possible to add two polynomials, each of degree 3, and have the sum be a polynomial of degree 2? If so, give an example. If not, explain why not.

x2 � 9x � 114x2 � 3x � 23x2 � 6x � 9

x 3 � 2x 2 � 4x � 6y 3 � 5y 2 � 2y � 414 � x � 2x2 2 � 1�2 � 3x � x3 21�2y � 6y2 � 2y3 2 � 14 � y2 � y3 2

2y 3 � 5y 2 � 4y � 54x 3 � 3x 2 � 3x � 115y2 � y � 22 � 1�3 � 3y � 2y3214x3 � 5x � 22 � 11 � 2x � 3x22

�2x 2 � 7x � 8�7x � 713x2 � 2x � 22 � 15x2 � 5x � 621x2 � 2x � 12 � 1x2 � 5x � 82

�7a2 � 2a � 43y 2 � 4y � 21�3a2 � 2a2 � 14a2 � 4212y2 � 4y2 � 1�y2 � 22

�6y4x1y2 � 4y2 � 1y2 � 10y21x2 � 6x2 � 1x2 � 10x2

�2x 3 � 5x 2 � 2x � 74y 3 � 2y 2 � 2y � 41�3 � 2x � 3x2 2 � 14 � 2x2 � 2x3 21�1 � y � 4y3 2 � 13 � 3y � 2y2 2

�4b 3 � b 2 � b � 153a 3 � 217 � 8b � b2 2 � 14b3 � 7b � 8211 � 2a � 4a3 2 � 1a3 � 2a � 32

�3x 3 � 2x 2 � 3x � 2�2x 3 � x 2 � 212x2 � 5x � 32 � 13x3 � 2x � 521�2x3 � x � 12 � 1�x2 � x � 32

8y 2 � y � 32x 2 � 3x � 115y2 � 2y � 12 � 1�y � 2 � 3y2 213x2 � x � 32 � 14x � x2 � 22

3x 2 � 4xy�y 2 � 13xy1x2 � 3xy2 � 1�2x2 � xy21y2 � 10xy2 � 12y2 � 3xy2

OBJECTIVE B To subtract polynomials

Quick Quiz

1. Subtract using a vertical format.�5x2 � 5x � 2

2. Subtract using a horizontal format.�13x2 � 15x � 16

��7x 2 � 3x � 8� � �6x 2 � 12x � 8�

�4x 2 � 7x � 6� � �9x 2 � 12x � 8�

For Exercises 37 and 38, use the polynomials shown at the right. Assume that a, b, c,and d are all positive numbers. Choose the correct answer from this list:(i) P � Q (ii) Q � R (iii) P � R (iv) None of the above

37. Which sum will be a trinomial? 38. Which sum will be zero?(iv) (i)

P � ax3 � bx2 � cx � dQ � �ax3 � bx2 � cx � dR � �ax3 � bx2 � cx � d

For answers to the Writing exercises, please see the Appendix in the Instructor’s Resource Binder that accompanies this textbook.

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 195

Recall that in an exponential expression such as , is the base and 6 is the exponent.The exponent indicates the number of times the base occurs as a factor.

The product of exponential expressionswith the same base can be simplified by writing each expression in factoredform and then writing the result with an exponent.

Note that adding the exponents resultsin the same product.

Simplify:

Simplify:

� �6a5b7

� �61a4�12 1b3�42

1�3a4b32 12ab42 � 1�3 � 22 1a4 � a2 1b3 � b42

1�3a4b32 12ab42

� y8 y4 � y � y3 � y4�1�3

y4 � y � y3

xx6

S E C T I O N

Multiplication of Monomials


4.2OBJECTIVE A To multiply monomials

• The bases are the same. Add the exponents.Recall that .y � y1

• Use the Commutative and AssociativeProperties of Multiplication torearrange and group factors.

• To multiply expressions with thesame base, add the exponents.

• Simplify.

⎫ ⎪ ⎬ ⎪ ⎭ ⎫ ⎬ ⎭

⎫⎪⎪⎪⎬⎪⎪⎪⎭

• Multiply coefficients. Addexponents with same base.

HOW TO • 1

HOW TO • 2

Take NoteThe Rule for MultiplyingExponential Expressionsrequires that the bases be thesame. The expression cannot be simplified.

a5b7

3 factors 2 factors

5 factors

x3 � x2 � x3�2 � x5

� x5

x3 � x2 � 1x � x � x2 � 1x � x2

Rule for Multiplying Exponential Expressions

If and are positive integers, then .x m � x n � x m�nnm


Simplify:

Solution

Simplify:

Your solution

�24m 3n 6

18m3n2 1�3n52

� �20a6b3� 1�5 � 42 1a � a52b3

1�5ab32 14a52

1�5ab32 14a52

Solutions on p. S10


Simplify:

Solution

Simplify:

Your solution

�36p 9q 5

112p4q32 1�3p5q22

� 24x7y7� 16 � 42 1x3 � x42 1y2 � y52

16x3y22 14x4y52

16x3y22 14x4y52

In-Class Examples

Simplify.1. �24x4y7

2. �35x8y13�7x 6y 4� ��5x 2y 9��6x 4y � ��4y 6�• Multiply coefficients. Add

exponents with same base.

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 196

SECTION 4.2 • Multiplication of Monomials 197

The power of a monomial can be simplified by writing the power in factored form andthen using the Rule for Multiplying Exponential Expressions.

Note that multiplying each exponent inside the parentheses by the exponent outside theparentheses results in the same product.

Simplify:

• Evaluate 53. � 125x6y9

� 53x6y9

• Use the Rule for Simplifying thePower of a Product. Note that 5 � 51.

15x2y323 � 51 �3x2 �3y3 �3

15x2y323

• Multiply each exponent inside the parentheses by the exponentoutside the parentheses.

1a2b322 � a2 �2b3 �2 � a4b61x423 � x4 �3 � x12

� a4b6� x12

• Use the Rule for MultiplyingExponential Expressions.

� a2�2b3�3� x4�4�4

• Write in factored form.1a2b322 � 1a2b32 1a2b321x423 � x4 � x4 � x4

OBJECTIVE B To simplify powers of monomials

Point of InterestOne of the first symbolicrepresentations of powers was given by Diophantus (c. 250 A.D.) in his bookArithmetica. He used �Y for x2 and �Y for x3. The symbol�Y was the first two letters ofthe Greek word dunamis,which means “power”; �Y wasfrom the Greek word kubos,which means “cube.” He alsocombined these symbols todenote higher powers. Forinstance, ��Y was the symbolfor x5.


Simplify:

Solution

Simplify:

Your solution

�27a12b 3c 6

1�3a4bc223

� 1�224p12r4 � 16p12r4

1�2p3r24 � 1�221 �4p3 �4r1 �4

1�2p3r24

Solutions on p. S10


Simplify:

Solution

Simplify:

Your solution

�4x 7y 8

1�xy42 1�2x3y222

� 12a2b2 18a9b62 � 16a11b7

� 12a2b2 123a9b62

� 12a2b2 121 �3a3 �3b2 �32

12a2b2 12a3b223

12a2b2 12a3b223

In-Class Examples

Simplify.1.

2. 32x5y15

3. �125x3y12z9

4. �24a7b15�3ab 3� ��2a 2b 4�3��5xy 4z 3�3�4x 2y 3� �2xy 4�3�3x 4�2

• Use the Rule forSimplifying thePower of a Product.

• Use the Rule forSimplifying thePower of a Product.

Rule for Simplifying the Power of an Exponential Expression

If and are positive integers, then .1x m2n � x mnnm

Rule for Simplifying the Power of a Product

If , , and are positive integers, then .1x my n2 p � x mpy nppnm

HOW TO • 3

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 197

For Exercises 3 to 35, simplify.

3. � 4. 5. 6.

7. � 8. 9. � 10.

11. � 12. 13. 14.

15. � 16. 17. � 18.

19. � 20. 21. 22.

23. � 24. 25. � 26.

27. 28. � 29.

30. 31. 32.

33. 34. 35.�36a3b2c3�12a10b 730x 6y 813a2b2 1�6bc2 12ac2214a2b2 1�3a3b42 1a5b221�2x2y32 13xy2 1�5x3y42

12a7b7c68x 7yz 6�30x 5y 5z 51�a3b42 1�3a4c22 14b3c4214x4z2 1�yz32 1�2x3z221�2x3y22 1�3x2z22 1�5y3z32

�24a3b3c3x 3y 5z 3x 3y 3z 213ab22 1�2abc2 14ac221xy2z2 1x2y2 1z2y221x2y2 1yz2 1xyz2

�12x 3y 340y10z 68a3b56a5b13x2y2 1�4xy221�5y4z2 1�8y6z521�2a2b3 2 1�4ab2 21�6a32 1�a2b2

�14x 2y 7�30a5b8x 5y 7za3b5c417xy42 1�2xy321�5a2b22 16a3b621�x2y3z2 1�x3y421�a2b32 1�ab2c42

�a3b7cx 4y 5z6a3b4�6x 3y 51�ab2c2 1a2b521x2yz2 1x2y421�3a2b2 1�2ab3212xy2 1�3x2y42

24a3b712x 7y 8�6a5b4�10x 9y1�6a2b42 1�4ab321�4x2y42 1�3x5y421�3a32 12a2b421�2x42 15x5y2

x 3y11x 3y 410a119a71x2y42 1xy721x22 1xy421�5a62 1�2a521�3a32 1�3a42

�40z13�42c6�8y 430x 31�8z52 15z8217c22 1�6c421�4y32 12y216x22 15x2

4.2 EXERCISES


OBJECTIVE A To multiply monomials

Quick Quiz

Simplify.1. (4a3b2)(5ab) 20a4b3

2. (�4x2y3z2)(�3x4yz3) 12x6y4z5

3. (2ab2c)(3a2b3)(5a3bc2) 30a6b6c3


Exercises 1–67, oddsMore challenging problems:

Exercises 69–75, odds;Exercises 77, 78

For Exercises 1 and 2, state whether the expression can be simplified using the Rule forMultiplying Exponential Expressions.

1. a. x4 � x5 b. x4x5 2. a. x4y4 b. x4 � x4

a. No b. Yes a. No b. No

OBJECTIVE B To simplify powers of monomials

For Exercises 36 and 37, state whether the expression can be simplified using the Rulefor Simplifying the Power of a Product.

36. a. (xy)3 b. (x � y)3 37. a. (a3 � b4)2 b. (a3b4)2

a. Yes b. No a. No b. Yes

�

�


46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 198

SECTION 4.2 • Multiplication of Monomials 199


38. � 39. � 40. 41. � 42.

43. � 44. 45. � 46. � 47.

48. � 49. � 50. 51. � 52.

53. � 54. 55. � 56.

57. 58. 59. 60.

61. � 62. 63. � 64.

65. � 66. 67. � 68.



69. � 70. 71. 72.

73. � 74. 75. 76.0 0

77. Evaluate . Are the results the same? If not, which expression has the larger value?No. is larger.

78. If n is a positive integer and , does ? Explain your answer.x � yxn � yn

12322 � 26 � 64, 2(32) � 29� 5122(32)

12322 and 2(32)

13a617x 4y 813a322 � 4a6 � 12a2231x2y422 � 12xy2244y2z4 � 12yz22212a3b223 � 8a9b6

2x 6y 62x 6y 2 � 9x 4y 2�12x 212x 21x2y223 � 1x3y3222x6y2 � 13x2y224x2 � 14x223x2 � 13x22

�24a3b 8�54a 9b 39a 4b10�8a 7b 51�3b22 12ab2231�2a32 13a2b231a2b22 1�3ab4221ab22 1�2a2b23

24x 6y 4�18x 3y 4192x 6y 1064x12y 31�3y2 1�2x2y231�2x2 1�3xy2221�3y2 1�4x2y3231�2x231�2x3y23

�8x 13y 9a 4b 6a 9b 524x 8y 71�x2y3221�2x3y231ab2221ab221a3b221ab2313x2y2 12x2y223

�54y13�8x 748a12b 2�243x15y1012y2 1�3y4231�2x2 12x32213b22 12a3241�3x3y225

16a4b129x 4y 232x 15y 20x 4y 6a9b121�2ab32413x2y2212x3y4251x2y3221a3b423

�8x 6�27y 3y12�x 6x 81�2x2231�3y231�y3241�x2231�x224

�y15x 14y 8x15z121�y5231x7221y4221x3251z423


Quick Quiz

Simplify.1. (x4)3 x12

2. (3a3bc5)2 9a6b2c10

3. (5ab3c4)(�2a2bc3)4 80a9b7c16

�

�

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 199

Multiplication of two polynomials requires the repeated application of the DistributiveProperty.

A convenient method for multiplying two polynomials is to use a vertical format similarto that used for multiplication of whole numbers.

• Multiply by 2.

• Multiply by y.

• Add the terms in each column. y3 � 2y2 � 14y � 12

y3 � 4y2 � 6y � 1 y2 � 4y � 62y

2y2 � 8y � 12 � 1 y2 � 4y � 622

y � 2

y2 � 4y � 6

� y3 � 2y2 � 14y � 12

� (y3 � 4y2 � 6y) � (2y2 � 8y � 12)

1y2 � 4y � 62 1y � 22 � 1y2 � 4y � 62y � 1y2 � 4y � 622

To multiply a polynomial by a monomial, use the Distributive Property and the Rule forMultiplying Exponential Expressions.

Multiply:

� �12a3 � 15a2 � 18a

�3a14a2 � 5a � 62 � �3a14a22 � 1�3a2 15a2 � 1�3a2 162

�3a14a2 � 5a � 62

S E C T I O N

Multiplication of Polynomials


4.3OBJECTIVE A To multiply a polynomial by a monomial


Multiply:

Solution

Multiply:

Your solution

8y2 � 12y

1�2y � 32 1�4y2

15x � 42 1�2x2 � 5x(�2x) � 4(�2x) � �10x2 � 8x

15x � 42 1�2x2

Solutions on p. S10


Multiply:

Solution

Multiply:

Your solution

�3a4 � 2a3 � 7a2

�a213a2 � 2a � 72

� 8a4b � 4a3b2 � 2a2b3� 2a2b(4a2) � 2a2b(2ab) � 2a2b(b2)

2a2b14a2 � 2ab � b22

2a2b14a2 � 2ab � b22

In-Class Examples

Multiply.1. (6a � 5)(�4a) �24a2 � 20a

2. 3x2y(5x2 � 4xy � 2y2)15x4y � 12x3y2 � 6x2y3

3. (2x � 4)3x 6x2 � 12x

HOW TO • 1

Instructor Note

Before doing an examplesimilar to the one at the right,show students that theprocedure for multiplicationcan be related to multiplyingwhole numbers, such as473 � 28.

• Use the DistributiveProperty.

OBJECTIVE B To multiply two polynomials

4. 4xy(3x2 � 2xy � 4y2)12x3y � 8x2y2 � 16xy3

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 200

It is frequently necessary to find the product of two binomials. The product can be foundusing a method called FOIL, which is based on the Distributive Property. The letters ofFOIL stand for First, Outer, Inner, and Last. To find the product of two binomials, addthe products of the First terms, the Outer terms, the Inner terms, and the Last terms.

Multiply:

Multiply the First terms.

Multiply the Outer terms.

Multiply the Inner terms.

Multiply the Last terms.

F O I LAdd the products.

Combine like terms.

Multiply:

Multiply:

� 3x2 � 10xy � 8y2

� 3x2 � 12xy � 2xy � 8y2

13x � 2y21x � 4y2 � 3x1x2 � 3x14y2 � 1�2y21x2 � 1�2y214y2

13x � 2y2 1x � 4y2

� 12x2 � 17x � 6

� 12x2 � 8x � 9x � 6

14x � 32 13x � 22 � 4x13x2 � 4x1�22 � 1�32 13x2 � 1�32 1�22

14x � 32 13x � 22

� 2x2 � 13x � 15

� 2x2 � 10x � 3x � 1512x � 32 1x � 52

3 � 5 � 1512x � 32 1x � 52

3 � x � 3x12x � 32 1x � 52

2x � 5 � 10x12x � 32 1x � 52

2x � x � 2x212x � 32 1x � 52

12x � 32 1x � 52

SECTION 4.3 • Multiplication of Polynomials 201

Multiply:

2a4 � 10a3 � a2 � 2a � 15 2a4 � 10a3 � a2 � 3a � 15 2a4 � 10a3 � a2 � 5a � 15

2a4 � 10a3 � a2 � a � 15 2a4 � 12a3 � a2 � 3a � 13

12a3 � a � 32 1a � 52

OBJECTIVE C To multiply two binomials using the FOIL method

HOW TO • 2

HOW TO • 4

HOW TO • 5

Take NoteFOIL is not really a differentway of multiplying. It is basedon the Distributive Property.

F O I L

� 2x 2 � 13x � 15� 2x 2 � 10x � 3x � 15

� 2x�x � 5� � 3�x � 5��2x � 3� �x � 5�


Multiply:

Solution

� 3(2b3 � b � 1)� 2b(2b3 � b � 1)

Multiply:

Your solution

6y 4 � 4y 3 � 2y 2 � 9y � 3

12y3 � 2y2 � 32 13y � 12

4b4 � 6b3 � 2b2 � b � 3

4b4 � � 2b2 � 2b 6b3 � 3b � 3

2b � 3 2b3 � b � 1

12b3 � b � 12 12b � 32

Solution on p. S10

In-Class Examples

Multiply.1. (2z2 � 4z � 5)(4z � 2) 8z3 � 20z2 � 28z � 10

2. (6a3 � 4a2 � 3a)(3a � 2) 18a4 � 17a2 � 6a

3. (3x3 � 2x � 5)(7x � 4) 21x4 � 12x3 � 14x2 � 27x � 20

• Note that spaces are provided in eachproduct so that like terms are in thesame column.

• Add the terms in each column.

HOW TO • 3

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 201


Using FOIL, it is possible to find a pattern for the product of the sum and difference oftwo terms and for the square of a binomial.

Multiply:

Expand:

� 9x2 � 12x � 4• This is the square of a

binomial. 13x � 222 � 13x22 � 213x2 1�22 � 1�222

13x � 222

� 4x2 � 9• This is the product of the sum and

difference of the same terms. 12x � 32 12x � 32 � 12x22 � 32

12x � 32 12x � 32

OBJECTIVE D To multiply binomials that have special products


Multiply:

Solution

Multiply:

Your solution

8y 2 � 22y � 15

14y � 52 12y � 32

� 6a2 � 7a � 2

12a � 12 13a � 22 � 6a2 � 4a � 3a � 2

12a � 12 13a � 22

Solutions on p. S10


Multiply:

Solution

Multiply:

Your solution

9b2 � 9b � 10

13b � 22 13b � 52

� 12x2 � x � 6

13x � 22 14x � 32 � 12x2 � 9x � 8x � 6

13x � 22 14x � 32

In-Class Examples

Multiply.1. (7x � 3)(5x � 6) 35x2 � 57x � 18

2. (8y � 7)(3y � 4) 24y2 � 11y � 28

3. (6w � 4)(5w � 2) 30w2 � 8w � 8

Instructor Note

Binomials that are otherwiseidentical except that one is asum and one is a differenceare called conjugates ofeach other.

Instructor Note

Remind students that the rulethat applies to (ab)2 isdifferent from the rule thatapplies to (a � b)2.

Take NoteThe word expand is usedfrequently to mean “multiplyout a power.”

Product of the Sum and Difference of the Same Terms

Square of the first termSquare of the second term

� a 2 � b 2

1a � b2 1a � b2 � a 2 � ab � ab � b 2

Square of a Binomial

Square of the first termTwice the product of the two termsSquare of the last term

� a 2 � 2ab � b 2

1a � b22 � 1a � b2 1a � b2 � a 2 � ab � ab � b 2

HOW TO • 6

HOW TO • 7

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 202


OBJECTIVE E To solve application problems


Multiply:

Solution

Multiply:

Your solution

4a 2 � 25c 2

12a � 5c2 12a � 5c2

14z � 2w2 14z � 2w2 � 16z2 � 4w2

14z � 2w2 14z � 2w2

Solutions on p. S10


Expand:

Solution

Expand:

Your solution

9x2 � 12xy � 4y2

13x � 2y22

12r � 3s22 � 4r 2 � 12rs � 9s2

12r � 3s22 In-Class Examples

1. Multiply:100x2 � 9

2. Expand:16x2 � 24xy � 9y2

�4x � 3y �2

�10x � 3� �10x � 3�


The length of a rectangle is m.The width is m. Find the area of therectangle in terms of the variable .

Strategy

To find the area, replace the variables L and W in theequation A � L � W by the given values and solve for A.

Solution

The area is m2.

The radius of a circle is ft. Use the equation, where is the radius, to find the area of the

circle in terms of . Leave the answer in terms of .

Your strategy

Your solution

ft21�x2 � 8�x � 16�2

�xrA � �r 2

1x � 42

1x2 � 3x � 282

A � x2 � 3x � 28 A � x2 � 4x � 7x � 28 A � 1x � 72 1x � 42 A � L � W

x1 x � 42

1x � 72

Solution on p. S10

In-Class Examples

1. The radius of a circle is ft.

Use the equation A � �r2, wherer is the radius, to find the area ofthe circle in terms of x. Leave theanswer in terms of �.(�x2 � 10�x � 25�) ft2

�x � 5�

x − 4

x + 7

x − 4

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 203

For Exercises 1 to 32, multiply.

1. � 2. 3. 4.

5. � 6. 7. � 8.

9. 10. 11. � 12.

13. 14. 15. � 16.

17. 18. 19. � 20.

21. � 22. 23.

24. 25. � 26.

27. � 28. 29.

30. 31. � 32.

33. Which of the following expressions are equivalent to 4x � x(3x � 1)?(i) 4x � 3x2 � x (ii) �3x2 � 5x (iii) 4x � 3x2 � x (iv) 9x2 � 3x (v) 3x(3x � 1)(ii) and (iii)

2a 3b � 4a 2b 2 � 6ab 3x 3y � 3x 2y 2 � xy 3�15x 4 � 15x 3 � 35x 2ab 12a2 � 4ab � 6b22xy 1x2 � 3xy � y22�5x2 13x2 � 3x � 72

6y 4 � 3y 3 � 6y 2�5b 4 � 10b 2 � 10b�2a 3 � 6a 2 � 8a�3y2 1�2y2 � y � 221b3 � 2b � 22 1�5b21a2 � 3a � 42 1�2a2

12x 4 � 8x 3 � 24x 2�6y 4 � 12y 3 � 14y 2�4y 6 � 6y 4 � 7y 34x2 13x2 � 2x � 622y2 1�3y2 � 6y � 72y3 1�4y3 � 6y � 72

3x 6 � 3x 4 � 2x 2�5b 3 � 7b 2 � 35b2a3 � 3a2 � 2ax2 13x4 � 3x2 � 22�b 15b2 � 7b � 352�a 1�2a2 � 3a � 22

�3y 3 � 2y 2 � 6y2x 4 � 3x 2 � 2x�2x 3y 2 � x 2y 3�x 3y � xy 3y 1�3y2 � 2y � 62x 12x3 � 3x � 22�x2y 12xy � y22�xy 1x2 � y22

4x 2 � 2x3x 2 � 4x3y 2 � 2y6x 2 � 12x12x � 122x13x � 42x13y � 22y12x � 423x

�3y 3 � 12y 212x 3 � 6x 2y 10 � 2y 6�3x 5 � 7x 33y 14y � y222x 16x2 � 3x2�y4 12y2 � y62�x3 13x2 � 72

�12y 4 � 6y 3�5x 4 � 5x 34b 3 � 32b 23a 3 � 6a 2�6y2 1y � 2y22�5x2 1x2 � x24b2 1b � 823a2 1a � 22

y 2 � 7y�x 2 � 7x�y 2 � 3yx 2 � 2x�y 17 � y2�x 1x � 72y 13 � y2x 1x � 22

4.3 EXERCISES


OBJECTIVE A To multiply a polynomial by a monomial


Exercises 1–83, every other oddExercises 87–101, oddsExercises 105–111, oddsMore challenging problems: Exercises 113, 114

OBJECTIVE B To multiply two polynomials


34. 35. 36.a3 � 6a2 � 13a � 12x 3 � 4x 2 � 11x � 14x 3 � 4x 2 � 5x � 21a2 � 3a � 42 1a � 321x2 � 2x � 721x � 221x2 � 3x � 22 1x � 12

Quick Quiz

Multiply.1. y(y � 5) y2 � 5y

2. (a � 3)2a2 2a3 � 6a2

3. �5b3(2b2 � 3b � 6) �10b5 � 15b4 � 30b3

�


�

�

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 204


37. 38. � 39.

40. � 41. 42.

43. 44. � 45.

46. 47. 48.

49. 50. � 51.

52. If a polynomial of degree 3 is multiplied by a polynomial of degree 2, what is thedegree of the resulting polynomial?5

4a 4 � 12a 3 � 13a 2 � 8a � 3y 4 � 4y 3 � y 2 � 5y � 218b 4 � 33b 3 � 5b 2 � 42b � 712a3 � 3a2 � 2a � 12 12a � 321y3 � 2y2 � 3y � 12 1y � 2213b3 � 5b2 � 72 16b � 12

5a 4 � 20a 3 � 5a 2 � 22a � 812y 3 � 3y 2 � 29y � 1515y 3 � 16y 2 � 70y � 1615a3 � 5a � 22 1a � 4213y2 � 3y � 52 14y � 3215y2 � 8y � 22 13y � 82

2y 4 � 7y 3 � 4y 2 � 16y � 8x 4 � 4x 3 � 3x 2 � 14x � 82y 3 � y 2 � 10y1y3 � 4y2 � 82 12y � 121x3 � 3x � 22 1x � 421y2 � 2y2 12y � 52

x 3 � 3x 2 � 5x � 15�2a 3 � 3a 2 � 8a � 3�6x 3 � 31x 2 � 41x � 101x2 � 52 1x � 321�a2 � 2a � 32 12a � 121�2x2 � 7x � 22 13x � 52

�2a 3 � 7a 2 � 7a � 2�2b 3 � 7b 2 � 19b � 202x 3 � 9x 2 � 19x � 151�a2 � 3a � 22 12a � 121�2b2 � 3b � 42 1b � 521x2 � 3x � 52 12x � 32

OBJECTIVE C To multiply two binomials using the FOIL method


53. � 54. 55. � 56.

57. � 58. 59. 60.

61. � 62. � 63. � 64.

65. � 66. 67. 68.

69. � 70. 71. � 72.15a 2 � 71a � 8421a 2 � 83a � 8030a 2 � 61a � 309x 2 � 54x � 7715a � 122 13a � 7217a � 162 13a � 5215a � 62 16a � 5213x � 72 13x � 112

5y 2 � 16y � 453y 2 � 2y � 168x 2 � 26x � 214x 2 � 31x � 2115y � 92 1y � 5213y � 82 1y � 2212x � 32 14x � 7214x � 32 1x � 72

7x 2 � 26x � 83x 2 � 11x � 45y 2 � 11y � 22x 2 � 15x � 717x � 22 1x � 4213x � 12 1x � 421y � 22 15y � 1212x � 12 1x � 72

a 2 � 17a � 72y 2 � 10y � 21x 2 � 5x � 50y 2 � 5y � 241a � 82 1a � 921y � 72 1y � 321x � 102 1x � 521y � 32 1y � 82

b 2 � 3b � 18a2 � a � 12y 2 � 7y � 10x 2 � 4x � 31b � 62 1b � 321a � 32 1a � 421y � 22 1y � 521x � 12 1x � 32

Quick Quiz

Multiply.1. x3 � 2x2 � 13x � 10

2. y3 � y2 � 4y � 4

3. 6a4 � 19a3 � 12a2 � 17a � 30�3a3 � 2a2 � a � 6� �2a � 5��y2 � 4��y � 1��x2 � 3x � 2� �x � 5�

�

�

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 205

73. 74. � 75.

76. 77. � 78.

79. � 80. � 81.

82. � 83. � 84.

85. What polynomial has quotient 3x � 4 when divided by 4x � 5?12x2 � x � 20

16x 2 � 78xy � 27y 256x 2 � 61xy � 15y 210x 2 � 21xy � 10y 212x � 9y2 18x � 3y218x � 3y2 17x � 5y215x � 2y2 12x � 5y2

14x 2 � 97xy � 60y 233x 2 � 83xy � 14y 215x 2 � 56xy � 48y 212x � 15y2 17x � 4y2111x � 2y2 13x � 7y215x � 12y2 13x � 4y2

36a 2 � 63ab � 20b 2100a 2 � 100ab � 21b 214a 2 � 31ab � 10b 2112a � 5b2 13a � 4b2110a � 3b2 110a � 7b212a � 5b2 17a � 2b2

2a 2 � 11ab � 63b 235a 2 � 12ab � b 26a 2 � 25ab � 14b 21a � 9b2 12a � 7b215a � b2 17a � b213a � 2b2 12a � 7b2


OBJECTIVE D To multiply binomials that have special products

Quick Quiz

Multiply.1. (x � 1)(x � 5) x2 � 6x � 5 2. (2x � 3)(3x � 4) 6x2 � 17x � 12

3. (6x � 7)(4x � 3) 24x2 � 10x � 21

OBJECTIVE E To solve application problems


86. � 87. � 88. � 89.

90. 91. 92. � 93.

For Exercises 94 to 101, expand.

94. � 95. 96. � 97.

98. 99. 100. 101.4a2 � 36ab � 81b 225x 2 � 20xy � 4y 2x 2 � 4xy � 4y 2x 2 � 6xy � 9y 212a � 9b2215x � 2y221x � 2y221x � 3y22

36x 2 � 60x � 259a 2 � 30a � 25y 2 � 6y � 9x 2 � 2x � 116x � 52213a � 5221y � 3221x � 122

16x 2 � 81y 216 � 9y 281x 2 � 49x 2 � 4914x � 9y2 14x � 9y214 � 3y2 14 � 3y219x � 22 19x � 2213x � 72 13x � 72

16x 2 � 494x 2 � 9y 2 � 36y 2 � 2514x � 72 14x � 7212x � 32 12x � 321y � 62 1y � 621y � 52 1y � 52

Quick Quiz

1. Multiply: (5y � 3)(5y � 3) 25y2 � 9

2. Expand: (x � 9)2 x2 � 18x � 81

3. Expand: (3x � 2y)2 9x2 � 12xy � 4y2

104. Geometry The length of a rectangle is ft. The width is ft. Find the area of the rectangle in terms of the variable x.110x 2 � 35x2 ft2

12x � 7215x22x − 7

5x

�

�

�

�

102. Simplify: 103. Expand:a 3 � 9a 2 � 27a � 27

�a � 3�3

4ab�a � b�2 � �a � b�2

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 206


105. Geometry The width of a rectangle is in. The length of the rectangle istwice the width. Find the area of the rectangle in terms of the variable x.

106. Geometry The length of a side of a square is km. Find the area of thesquare in terms of the variable x.

107. Geometry The radius of a circle is cm. Find the area of the circle interms of the variable x. Leave the answer in terms of �.

108. Geometry The base of a triangle is m and the height is m. Find thearea of the triangle in terms of the variable x.

109. Sports A softball diamond has dimensions 45 ft by 45 ft. A base-path border x feet wide lies on both the first-base side and the third-base side of the diamond.Express the total area of the softball diamond and the base paths in terms of thevariable x.

110. Sports An athletic field has dimensions 30 yd by 100 yd. An end zone that is w yards wide borders each end of the field. Express the total area of the field andthe end zones in terms of the variable w.

111. The Olympics See the news clipping at theright. The Water Cube is not actually a cubebecause its height is not equal to its length andwidth. The width of a wall of the Water Cube is22 ft more than five times the height. (Source:Structurae)a. Express the width of a wall of the Water

Cube in terms of the height h. (5h � 22) ftb. Express the area of one wall of the Water

Cube in terms of the height h. (5h2 � 22h) ft2

112. The expression w(3w � 1) cm2 represents the area of a rectangle of width w.Describe in words the relationship between the length and width of the rectangle. The length is 1 cm less than three times the width.


113. Add to the product of .

114. Subtract from the product of .x 3 � 7x 2 � 7

x2 � x � 3 and x � 44x2 � x � 5

7x 2 � 11x � 82x � 5 and 3x � 1x2 � 2x � 3

160w � 30002 yd2

190x � 20252 ft2

14x 2 � 10x2 m2

12x � 5214x2

1�x 2 � 8�x � 16�2 cm2

1x � 42

14x 2 � 4x � 12 km2

12x � 12

118x 2 � 12x � 22 in2

13x � 12

Quick Quiz

1. The width of a rectangle isft. The length of the

rectangle is ft. Findthe area of the rectangle interms of the variable x.(12x2 � 5x � 2) ft2

�4x � 1��3x � 2�

2x + 1

2x + 5

4x

45

x

45

x

30

w w100

The Water Cube

�

�

�

�

Olympic WaterCube CompletedThe National AquaticsCenter, also known as the Water Cube, wascompleted on the morningof December 26, 2006.Built in Beijing, China, forthe 2008 Olympics, theWater Cube is designed tolook like a “cube” of watermolecules.

In the News

Source: Structurae

Chris

tian

Kobe

r/Ro

bert

Hard

ing

Wor

ld Im

ager

y/Ge

tty Im

ages

�

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 207

The quotient of two exponential expressions withthe same base can be simplified by writing eachexpression in factored form, dividing by the common factors, and then writing the result withan exponent.

Note that subtracting the exponents gives the same result.

To divide two monomials with the same base, subtract the exponents of the like bases.

Simplify:

Simplify:

Simplify:

Because the bases are not the same, is already in simplest form.

Consider the expression , . This expression can be simplified, as shown below, by

subtracting exponents or by dividing by common factors.

The equations and suggest the following definition of .x0� 1x4

x4� x0x4

x4

x4

x4�

x � x � x � xx � x � x � x

� 1x4

x4� x4�4 � x0

x � 0x4

x4

p7

z4

p7

z4

� rt5

• Subtract the exponents of the like bases. r8t6

r7t� r 8�7t6�1

r8t6

r7t

� a4

• The bases are the same. Subtract the exponents. a7

a3� a7�3

a7

a3

S E C T I O N

Integer Exponents and Scientific Notation


4.4OBJECTIVE A To divide monomials

Instructor Note

Here we are just verbalizingthe rule for division ofpolynomials. The formaldefinition comes after wedefine negative exponents.Have students write this rule,even just copy it onto a pieceof paper, and then practice afew exercises, such as

and

It may also help to give

them the expression to

emphasize that the basesmust be the same.

a9

b5

y 8

ya4

a2

Definition of Zero as an Exponent

If , then . The expression is not defined.00x 0 � 1x � 0

x5

x2� x5�2 � x3

x5

x2�

x � x � x � x � xx � x

� x3

1 1

1 1

1 1 1 1

1 1 1 1

HOW TO • 1

HOW TO • 2

HOW TO • 3

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 208

SECTION 4.4 • Integer Exponents and Scientific Notation 209

Simplify: ,

Simplify:

Consider the expression , . This expression can be simplified, as shown below, by

subtracting exponents or by dividing by common factors.

The equations and suggest that .

An exponential expression is in simplest form when it is written with only positive exponents.

Evaluate .

Simplify: ,

Simplify:

2

5a�4�

2

5�

1

a�4�

2

5� a4 �

2a4

5

2

5a�4

3n�5 � 3 �1

n5�

3

n5

n � 03n�5

• Evaluate the expression. �1

16

• Use the Definition of a Negative Exponent. 2�4 �1

24

2�4

1

x2x�2 �

1

x2�

x4

x6� x�2x4

x6

x4

x6�

x � x � x � xx � x � x � x � x � x

�1

x2

x4

x6� x4�6 � x�2

x � 0x4

x6

�14x3y720 � �112 � �1

�14x3y720

• Any nonzero expression to the zero power is 1.112a320 � 1

a � 0112a320Take NoteIn the example at the right,we indicate that If wetry to evaluate when

we have

However, is not defined.Therefore, we must assumethat To avoid statingthis for every example orexercise, we will assume thatvariables do not take onvalues that result in theexpression .00

a � 0.

00

�12�0�3�0 � �12�0��0 � 00

a � 0,�12a3�0

a � 0.

Take NoteNote from the example at the right that is apositive number. A negativeexponent does not changethe sign of a number.

2�4

Take NoteFor the expression , theexponent on n is (negat ive 5). The iswritten in the denominator as

. The exponent on 3 is 1(positive 1). The 3 remains inthe numerator. Also, weindicated that . This isdone because division byzero is not defined. In thistextbook, we will assume thatvalues of the variables arechosen so that division byzero does not occur.

n � 0

n 5

n�5

�53n�5

Point of InterestIn the 15th century, theexpression was used tomean . The use ofreflected an Italian influence.In Italy, was used for minusand was used for plus. Itwas understood that 2referred to an unnamedvariable. Issac Newton, in the17th century, advocated thenegative exponent notationthat we currently use.

mp

m

m12x�2

122m

HOW TO • 4

HOW TO • 5

Definition of a Negative Exponent

If and is a positive integer, then

and1

x�n � x nx�n �1x n

nx � 0

HOW TO • 6

HOW TO • 7

HOW TO • 8

• Use the Definition of a NegativeExponent to rewrite the expressionwith a positive exponent.

• Use the Definition of a Negative Exponent torewrite the expression with a positive exponent.

1 1 1 1

1 1 1 1

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 209


The expression , , can be simplified by squaring or by multiplying each

exponent in the quotient by the exponent outside the parentheses.

Simplify:

The example above suggests the following rule.

Now that zero as an exponent and negative exponents have been defined, a rule fordividing exponential expressions can be stated.

Evaluate .

�1

53�

1

125

• Use the Rule for Dividing Exponential Expressions. 5�2

5� 5�2�1 � 5�3

5�2

5

�a�6

b�4�

b4

a6

• Use the Rule for Simplifying the Power of a Quotient. a3

b2�2

�a3(�2)

b2(�2)

a3

b2�2

x4

y32

�x4 �2

y3 �2�

x8

y6x4

y32

� x4

y3x4

y3 �x4 � x4

y3 � y3�

x4�4

y3�3�

x8

y6

x4

y3y � 0x4

y32

Take NoteAs a reminder, although it isnot stated, we are assumingthat and Thisassumption is made to ensurethat we do not have divisionby zero.

b � 0.a � 0

Rule for Simplifying the Power of a Quotient

If , , and p are integers and , then .x mp

y np�x m

y np

y � 0nm

Rule for Negative Exponents on Fractional Expressions

If , , and is a positive integer, then

ab

�n

� ba

n

nb � 0a � 0

Rule for Dividing Exponential Expressions

If and are integers and , then .� x m�nxm

xnx � 0nm

HOW TO • 9

• Use the Definition of a Negative Exponent towrite the expression with positive exponents.

• Use the Definition of a Negative Exponentto rewrite the expression with a positiveexponent. Then evaluate.

HOW TO • 10

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 210


Simplify:

The rules for simplifying exponential expressions and powers of exponential expressionsare true for all integers. These rules are restated here, along with the rules for dividingexponential expressions.

Simplify:

Simplify:

�43m15

33n3�

64m15

27n3

�3�3m15n�3

4�3

� c3m�5n

4d

�3

c6m2n3

8m7n2 d�3

� c3m2�7n3�2

4d

�3

c6m2n3

8m7n2 d�3

� �6b3

a2

� �6a�2b3

�3ab�4��2a�3b7� � 33 � ��2� 4 �a1�(�3)b�4�7�13ab�42 1�2a�3b72

�1

x5

� x�5

x4

x9� x4�9

x4

x9HOW TO • 11

HOW TO • 12

HOW TO • 13

Instructor Note

Examples such as the one at the right are included to review the work onmultiplying monomials and to demonstrate that negativeexponents can be used insimplifying products ofexponential expressions.

Instructor Note

There are a few differentways to simplify theexpression at the right.Students can simplify theexpression by starting asfollows:

This method uses the Rule forNegative Exponents onFractional Expressions first.

6m2n3

8m7n2�3

� 8m7n2

6m2n33

Rules of Exponents

If , , and are integers, then

, , ,

x 0 � 1, x � 0

x � 0x�n �1x ny � 0x m

y np

�x mp

y npx � 0x m

x n � x m�n

1x my n2p � x mpy np1x m2n � x mnx m � x n � x m�n

pnm

• Use the Rule for Dividing Exponential Expressions.

• Subtract the exponents.

• Use the Definition of a Negative Exponentto rewrite the expression with a positiveexponent.

• When multiplying exponentialexpressions, add the exponents onlike bases.

• Simplify inside the brackets.

• Subtract the exponents.

• Use the Rule for Simplifying thePower of a Quotient.

• Use the Definition of a NegativeExponent to rewrite the expression withpositive exponents. Then simplify.

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM 1


Simplify:

�2a�7b3

3�

2b3

3a7

�2a�2�5b5�2

3

4a�2b5

6a5b2�

2a�2b5

3a5b2

4a�2b5

6a5b2HOW TO • 14

• Divide the coefficients by theircommon factor.

• Use the Rule for DividingExponential Expressions.

• Use the Definition of a Negative Exponentto rewrite the expression with positiveexponents.

• Rule for Simplifying thePower of a Product


Simplify:

Solution

Simplify:

Your solution

�2x 8y 8

1�2x22 1x�3y�42�2

� �2x7

27

��2x1�6

33

1�2x2 13x�22�3 � 1�2x2 13�3x62

1�2x2 13x�22�3

Solutions on p. S10


Simplify:

Solution

Simplify:

Your solution

8a 8

3b 7

(6a�2b3)�1

(4a3b�2)�2

�1

8t5

�1

23t5

� 2�3r0t�5

� 2�3r�6�(�6)t3�8

12r2t�12�3

1r�3t422�

2�3r�6t3

r�6t8

12r2t�12�3

1r�3t422


Simplify:

Solution

Simplify:

Your solution

9s4

4

c6r3s�3

9r3s�1 d�2

�27a18

8b15

�2�3a18b�15

3�3

c4a�2b3

6a4b�2 d�3

� c2a�6b5

3d

�3

c4a�2b3

6a4b�2 d�3

• Rule for Simplifying thePower of a Product

• Rule for DividingExponential Expressions

• Write the answer insimplest form.

• Simplify inside brackets.

• Rule for Simplifying thePower of a Quotient

• Write answer insimplest form.

In-Class Examples

Simplify.

1.

2.

3.9a12

4b 4c8 8a�3b�1c2

12a3b�3c�2�2

18y 6

x 4

3x�1y 4

6�1x3y�2

6x

y 6(�3x 4y �5)(�2x �3y �1)

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 212


Very large and very small numbers abound in the natural sciences. For example, themass of an electron is 0.000000000000000000000000000000911 kg. Numbers such asthis are difficult to read, so a more convenient system called scientific notation is used.In scientific notation, a number is expressed as the product of two factors, one a numberbetween 1 and 10, and the other a power of 10.

To express a number in scientific notation, write it in the form , where a is anumber between 1 and 10, and n is an integer.

For numbers greater than or equal to 10, move thedecimal point to the right of the first digit. Theexponent is positive and equal to the number ofplaces the decimal point has been moved.

For numbers less than 1, move the decimal pointto the right of the first nonzero digit. The exponent

is negative. The absolute value of the exponentis equal to the number of places the decimal pointhas been moved.

Changing a number written in scientific notation to decimal notation also requiresmoving the decimal point.

When the exponent is positive, move the deci-mal point to the right the same number of placesas the exponent.

When the exponent is negative, move the decimalpoint to the left the same number of places as theabsolute value of the exponent.

n

n

a � 10n

OBJECTIVE B To write a number in scientific notation

Point of InterestAn electron microscope useswavelengths that areapproximately 4 � 10�12

meter to make images ofviruses.

The human eye can detectwavelengths between 4.3 � 10�7 meter and 6.9 � 10�7 meter. Althoughthese wavelengths are veryshort, they are approximately105 times longer than thewavelengths used in anelectron microscope.

93,000,000 � 9.3 � 107

240,000 � 2.4 � 105

0.0000832 � 8.32 � 10�5

0.0003 � 3 � 10�4

6.34 � 10�7 � 0.000000634

8.1 � 10�3 � 0.0081

2.3 � 108 � 230,000,000

3.45 � 106 � 3,450,000

EXAMPLE • 4 YOU TRY IT • 4Write the number 824,300,000 in scientific notation.

Solution

Write the number 0.000000961 in scientific notation.

Your solution

9.61 � 10�7824,300,000 � 8.243 � 108

Solutions on p. S10


Write the number in decimal notation.

Solution

Write the number in decimal notation.

Your solution

7,329,000

7.329 � 106

6.8 � 10�10 � 0.00000000068

6.8 � 10�10

In-Class Examples

1. Write the number 0.00394 inscientific notation. 3.94 � 10�3

2. Write the number 3.8 � 104 indecimal notation. 38,000

Integrating

TechnologySee the Keystroke Guide:Scientific Notation forinstructions on entering anumber written in scientificnotation into a calculator.

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 213


1. � 2. 3. � 4.

5. 6. 7. � 8.

9. 10. 11. � 12.

13. � 14. 15. 16.

17. 18. 19. 20.

21. � 22. 23. � 24.

25. � 26. 27. � 28.

29. 30. 31. 32.

33. � 34. 35. � 36.

For Exercises 37 to 44, evaluate.

37. � 38. 39. � 40.

64 12127

125

1

12�1

1

8�23�35�2

54xy 2z 9

p2

2m 3

2b2c8

3a5

7xz8y 3

25x4y7z2

20x5y9z11

15mn9p3

30m4n9p

24a2b7c9

36a7b5c

14x4y6z2

16x3y9z

23a2

12y 3

1ab

1p3q

14a3b6

21a5b6

3x4y5

6x4y8

a4b5

a5b6

pq3

p4q4

56r 2

25m 3n8

ab

y 4

x 2

5r3t7

6r5t7

2m6n2

5m9n10

a3b2

a2b3

x6y5

x8y

23c 9

23x 5

34y

13x 3

�24c2

�36c11

�12x

�18x6

6y8

8y9

4x2

12x5

1m 6

1a 6

1z 2

1y 5

m

m7

a5

a11

z4

z6

y3

y8

�2b3

�2a3

2x 3

33r 2

2

�18b5

27b4

�16a7

24a6

8x9

12x6

6r4

4r2

y 4z 3m 5n 22m2k

y5z6

yz3

m9n7

m4n5

14m11

7m10

22k5

11k4

3z 42x 3w 8p 4

12z7

4z3

4x8

2x5

w9

w

p5

p

c 7a3z 7y 4

c12

c5

a8

a5

z9

z2

y7

y3

4.4 EXERCISES


OBJECTIVE A To divide monomials Suggested Assignment

Exercises 1–95, every other oddExercises 97–123, oddsMore challenging problems: Exercises 125, 126


�

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 214


41. 42. 43. � 44.

2 1


45. � 46. 47. � 48.

49. � 50. 51. � 52.

53. � 54. 55. � 56.

57. � 58. 59. � 60.

1 1

61. � 62. 63.

64. 65. � 66.

67. 68. � 69.

70. 71. � 72.

73. � 74. 75. 76.

77. � 78. 79. � 80.

1x12y12

1x6y

1x3

12x2y6

1x�3y�222

x6y8

1x�2y22

x2y3

1x�1y22

xy2

2x�1y�4

4xy2

2yx 3

3x 3

14x 3

12x 3

2x�1y4

x2y3

3x�2y

xy

2x�2y

8xy

3x�2y2

6xy2

1a5b6

10y 3

x 4

3a4

8b5

a�3b�4

a2b21�5x�2y2 1�2x�2y2213ab�22 12a�1b2�3

�a5

8b4

2a18

b3�5a 8

1�2ab�22 14a�2b2�212a�32 1a7b�1231�5a22 1a�522

�2x 22x 4

y 6

25x 2

1�2x�52x712x�12 1x�3215xy�32�2

9x 2y 4

9y 4

x 2�8x 3

y 6

13x�1y�2221�3x�1y2221�2xy�223

�1�1

�2

3xy0

�13p2q520132x3y4201ab520

2c6

5x 2

3�3v 35b 8

2

5c�6

1

3x�2

�3

v�3

5

b�8

45a 4

23z 2�

6y

4x 7

4

5a�42

3z�2�6y�14x�7

b4a61y10

1x 2

1

b�4

1

a�6y�10x�2

1625

127

32

32

2�2

2�3

5�3

5

3�2

3

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 215

81. � 82. 83. 84.

85. 86. 87. � 88.

89. 90. 91. � 92.

For Exercises 93 to 96, state whether the equation is true or false for all a � 0 and b � 0.

93. 94. 95. a�nan � 1 96.

False True True False

an

bm � a

bm�n

an�m �1

am�n

a4n

an � a4

4d 4

9a6b2c8

125p3

27m15n6

343x15z 9

27y15

s 8t 4

4r 12

18a4b�2c4

12ab�3d 2�215m3n�2p�1

25m�2n�4 �36x�4yz�1

14xy�4z2�313�1r4s�32�2

16r2s�1t�222

a6

b10�7b6

a2�8a3b4�a2b6c2

12a�2b32�2

14a2b�42�1

�114ab422

28a4b2

�18a2b423

64a3b8

22a2b4

�132b3c2

�2x 2y 2

11z 5�1

6x 3�4b9

a4

y10

�8x2y4

44y2z5

�16xy4

96x4y4

12a2b3

�27a2b2

1a�2y32�3

a2y


OBJECTIVE B To write a number in scientific notation

Quick Quiz

1. Evaluate: 5�2 2. Simplify: �5x�2 3. Simplify: 4. Simplify:z10

48x13y12

(3�1x 3y 2z�2)�3

(6�2x �2y �3z2)�2

1

z2

z4

z6�5

x2

1

25

For Exercises 97 to 105, write in scientific notation.

97. 0.00000000324 � 98. 0.00000012 99. 0.000000000000000003

100. 1,800,000,000 101. 32,000,000,000,000,000 102. 76,700,000,000,000

103. 0.000000000000000000122 � 104. 0.00137 105. 547,000,000

For Exercises 106 to 114, write in decimal notation.

106. 107. � 108.0.0000000000023 0.000167 2,000,000,000,000,000

109. 110. � 111.68,000,000 0.000000000000000000009 0.0000305

112. 113. � 114.

115. If n is a negative integer, how many zeros appear after the decimal point when 1.35 � 10n is written in decimal notation? �n � 1

0.00720.00000000102905,000,000,0007.2 � 10�31.02 � 10�99.05 � 1011

3.05 � 10�59 � 10�216.8 � 107

2 � 10151.67 � 10�42.3 � 10�12

5.47 � 1081.37 � 10�31.22 � 10�19

7.67 � 10133.2 � 10161.8 � 109

3 � 10�181.2 � 10�73.24 � 10�9�

�

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 216


116. If n is a positive integer greater than 1, how many zeros appear before the decimalpoint when 1.35 � 10n is written in decimal notation?n � 2

117. Technology See the news clipping at the right. Express in scientific notation thethickness, in meters, of the memristor.

118. Geology The approximate mass of the planet Earth is5,980,000,000,000,000,000,000,000 kg. Write the mass of Earth in scientific notation.

119. Physics The length of an infrared light wave is approximately 0.0000037 m. Writethis number in scientific notation.

120. Electricity The electric charge on an electron is 0.00000000000000000016coulomb. Write this number in scientific notation.

121. Physics Light travels approximately 16,000,000,000 mi in 1 day. Write this num-ber in scientific notation.

122. Astronomy One light-year is the distance traveled by light in 1 year. One light-year is 5,880,000,000,000 mi. Write this number in scientific notation.

123. Astronomy See the news clipping at the right. WASP-12b orbits a star that is 5.1156 � 1015 mi from Earth. (Source: news.yahoo.com) Write this number in decimal notation.5,115,600,000,000,000

124. Chemistry Approximately 35 teragrams of sulfur in the atmosphereare converted to sulfate each year. Write this number in decimal notation.35,000,000,000,000


125. Evaluate when 126. Evaluate when and 2. and 2.

4, 2, 1,12

, 14

14

, 12

, 1, 2, 4

x � �2, �1, 0, 1,2�xx � �2, �1, 0, 1,2x

13.5 � 1013 g2

5.88 � 1012

1.6 � 1010

1.6 � 10�19

3.7 � 10�6

5.98 � 1024 kg

1.5 � 10�8 m

HP Researchers View Imageof Memristor

AP Im

ages

Quick Quiz

1. Write in scientific notation.a. 41,300,000,000

4.13 � 1010

b. 0.000327 3.27 � 10�4

2. Write in decimal notation.a. 2.4 � 10�5 0.000024

b. 5.76 � 106 5,760,000

�

�

�

�

Hottest PlanetEver DiscoveredA planet called WASP-12b is the hottest planetever discovered, at about4000°F. It orbits its starfaster than any otherknown planet, completinga revolution once a day.

In the News

Source: news.yahoo.com

HP Introducesthe MemristorHewlett Packard hasannounced the design ofthe memristor, a newmemory technology withthe potential to be muchsmaller than the memorychips used in today’scomputers. HP has made a memristor with a thickness of0.000000015 m (15 nanometers).

In the News

Source: The New York Times

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 217

To divide a polynomial by a monomial, divide each term in the numerator by thedenominator and write the sum of the quotients.

Divide:

� 2x2 � x � 3

6x3 � 3x2 � 9x

3x�

6x3

3x�

3x2

3x�

9x

3x

6x3 � 3x2 � 9x

3x

S E C T I O N

Division of Polynomials


4.5OBJECTIVE A To divide a polynomial by a monomial

OBJECTIVE B To divide polynomials

12x2y � 6xy � 4x2

2xy�

12x2y

2xy�

6xy

2xy�

4x2

2xy� 6x � 3 �

2x

y


Divide:

Solution

Divide:

Your solution

4xy � 3 �1x

24x2y2 � 18xy � 6y

6xy

12x2y � 6xy � 4x2

2xy

Solution on p. S11

In-Class Examples

Divide.

1. 3x � 1 �

2. 3ab � 2 �1

2a

24a 2b 2 � 16ab � 4b8ab

2

x

18x 2y � 6xy � 12y

6xy

• Divide each term of the polynomialby the monomial.

• Simplify each term.

HOW TO • 1

HOW TO • 2

Instructor Note

It may help some students ifyou start with the divisionalgorithm for whole numbersand show them that a similarprocedure is used to dividepolynomials. You mayconsider using asan example.

676 21

Tips for SuccessAn important element ofsuccess is practice. Wecannot do anything well if wedo not practice it repeatedly.Practice is crucial to successin mathematics. In thisobjective you are learning anew skill, how to dividepolynomials. You will need topractice this skill over andover again in order to besuccessful at it.

The procedure for dividing two polynomials is similar to the one for dividing wholenumbers. The same equation used to check division of whole numbers is used to checkpolynomial division.

(Quotient � divisor) � remainder � dividend

Divide:

Step 1

Step 2

Check:

1x2 � 5x � 82 1x � 32 � x � 2 �2

x � 3

1x � 22 1x � 32 � 2 � x2 � 5x � 6 � 2 � x2 � 5x � 8

x � 2

�2x � 6

�2x � 8

x2 � 3x00 x � 3�x2 � 5x � 8

x � 2

�2x � 8

x2 � 3x

x � 3�x2 � 5x � 8 x

1x2 � 5x � 82 1x � 32

• Think: � xx 2

x x�x 2 �

• Multiply: x1x � 32 � x 2 � 3x

• Subtract:Bring down the 8.

1x 2 � 5x2 � 1x 2 � 3x2 � �2x

• Think: � �2�2x

x x��2x �

• Multiply: �21x � 32 � �2x � 6

• Subtract:• The remainder is 2.

1�2x � 82 � 1�2x � 62 � 2

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 218

SECTION 4.5 • Division of Polynomials 219

If a term is missing from the dividend, a zero can be inserted for that term. This helpskeep like terms in the same column.

Divide:

Check:

12x3 � 6x � 262 1x � 22 � 2x2 � 4x � 14 �2

x � 2

12x2 � 4x � 142 1x � 22 � 1�22 � 12x3 � 6x � 282 � 1�22 � 2x3 � 6x � 26

�2

14x � 28 14x � 26

� 4x2 � 8x � 4x2 � 6x

2x3 � 4x2 x � 2�2x3 � 0 � 6x � 26

2x2 � 4x � 14

2x3 � 6x � 26

x � 2

6x � 26 � 2x3

2 � x


Divide:

Solution

Divide:

Your solution

x 2 � 2x � 1 �6

2x � 3

12x3 � x2 � 8x � 32 12x � 32

� 2x2 � x � 1 �1

2x � 3

14x3 � 8x2 � x � 42 12x � 32

�1

� 2x � 3 � 2x � 4

2x2 � 3x 2x2 � 3x

4x3 � 6x2 � 6x 2x � 3�4x3 � 8x2 � 3x � 4

2x2 � 3x � 1

18x2 � 4x3 � x � 42 12x � 32

Solutions on p. S11


Divide:

Solution

Divide:

Your solution

x2 � x � 1

x3 � 2x � 1

x � 1

1x2 � 12 1x � 12 � x � 1

0

� x � 1 � x � 1

x2 � x � 1 x � 1�x2 � 0 � 1

x � 1

x2 � 1

x � 1

In-Class Examples

Divide.1.

2x2 � 3x � 2 �

2. x2 � 2x � 3x 3 � x � 6

x � 2

2

3x � 4

�6x 3 � x 2 � 18x � 10� �3x � 4�

Take NoteRecall that a fraction barmeans “divided by.”Therefore, can be

written , and can

be written ab

.

a b62

6 2

Instructor Note

Students are comfortablewriting the answer to

as 3 , which is .

Tell students that this is theform in which a remainder ofa quotient of polynomials iswritten.

34

3 �34

15 4

HOW TO • 3

• Arrange the terms of each polynomial indescending order.

• There is no term in .Insert a zero for the missing term.

2x 3 � 6x � 26x 2

• Insert a zero forthe missing term.

• Write the dividendin descendingpowers of x.

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 219

1. Every division problem has a related multiplication problem. What is the related

multiplication problem for the division problem � 5x � 4?15x2 � 12x � 3x(5x � 4)

For Exercises 2 to 22, divide.

2. � 3. � 4.

5. 6. � 7.

8. � 9. 10.

11. 12. � 13.

14. � 15. 16.

17. 18. � 19.

20. � 21. � 22.

For Exercises 23 to 49, divide.

23. 24. 25.

26. � 27. 28.2x � 42y � 72x � 114x2 � 162 12x � 4212y2 � 13y � 212 1y � 3212x2 � 5x � 22 1x � 22

y � 5x � 2b � 71y2 � 2y � 352 1y � 721x2 � x � 62 1x � 321b2 � 14b � 492 1b � 72

a � 3 � 6b2a � 1 � 3b4a � 5 � 6b

5a2b � 15ab � 30ab2

5ab

22a2b � 11ab � 33ab2

11ab

16a2b � 20ab � 24ab2

4ab

8y � 2 �3y

3x � 2 �1x

�2x 2 � 3

8y2 � 2y � 3y

3x2 � 2x � 1x

4x4 � 6x2

�2x2

�3y 3 � 5xy � 3xy � 2

9y6 � 15y3

�3y3

8x2y2 � 24xy

8xy

5x2y2 � 10xy

5xy

a6 � 5a3 � 3ax 4 � 3x 2 � 1a2 � 5a � 7

a8 � 5a5 � 3a3

a2

x6 � 3x4 � x2

x2

a3 � 5a2 � 7aa

x 2 � 3x � 5�y � 9�x � 2

x3 � 3x2 � 5xx

3y2 � 27y

�3y

5x2 � 10x

�5x

5y � 3x � 24b2 � 3

10y2 � 6y

2y

3x2 � 6x

3x

4b3 � 3b

b

6y � 42b � 52a � 5

6y2 � 4y

y16b � 40

8

10a � 25

5

15x2 � 12x

3x

4.5 EXERCISES


OBJECTIVE A To divide a polynomial by a monomial

Quick Quiz

Divide. 1. 4x3 � 5x2 � 2x � 312x 3 � 15x 2 � 6x � 9

3


Exercises 1–21, every other oddExercises 23–49, oddsMore challenging problems:

Exercises 51, 52

OBJECTIVE B To divide polynomials2. 2x2 � 3x � 1 �

2

x

10x 3y � 15x 2y � 5xy � 10y

5xy


�

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 220

SECTION 4.5 • Division of Polynomials 221

29. 30. � 31.

32. � 33. 34.

35. 36. � 37.

38. 39.

40. � 41.

42. � 43.

44. 45.

46. 47.

48. � 49.

50. True or false? When a sixth-degree polynomial is divided by a third-degree polyno-mial, the quotient is a second-degree polynomial.False


51. The product of a monomial and 4b is . Find the monomial.3ab

52. In your own words, explain how to divide exponential expressions.

12ab2

x 2 � 5x 2 � 31x4 � 3x2 � 102 1x2 � 221x4 � x2 � 62 1x2 � 22

x 2 � 5x � 2x 2 � 2x � 3

7x � x3 � 6x2 � 2

x � 1

5x � 3x2 � x3 � 3

x � 1

2a � 9 �33

3a � 13x � 5

124 � 6a2 � 25a2 13a � 1215 � 23x � 12x22 14x � 12

4a � 15a � 6 �4

3a � 2

12a2 � 25a � 7

3a � 7

15a2 � 8a � 8

3a � 2

5y � 3 �1

2y � 32x � 5 �

82x � 1

110 � 21y � 10y22 12y � 3218x � 3 � 4x22 12x � 12

3x � 17 �64

x � 4y � 6 �

262y � 3

3x2 � 5x � 4

x � 4

2y2 � 9y � 8

2y � 3

b � 5 �24

b � 3a � 3 �

4a � 2

6x � 12 �19

x � 2

1b2 � 8b � 92 1b � 321a2 � 5a � 102 1a � 2216x2 � 52 1x � 22

5x � 12 �12

x � 13y � 5 �

202y � 4

2x � 1 �2

3x � 2

5x2 � 7x

x � 1

6y2 � 2y

2y � 4

6x2 � 7x

3x � 2

x � 2 �8

x � 2x � 1 �

2x � 1

2y � 6 �25

y � 3

x2 � 4

x � 2

x2 � 1

x � 1

2y2 � 7

y � 3

Quick Quiz

Divide.1.

3x2 � 2x � 4

2.

4x2 � x � 3 �2

4x � 1

16x 3 � 13x � 14x � 1

�3x 3 � 11x 2 � 10x � 12� �x � 3�


46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 221

3 mi

15,840 ft


FOCUS ON PROBLEM SOLVING

In solving application problems, it may be useful to include the units in order to organ-ize the problem so that the answer is in the proper units. Using units to organize andcheck the correctness of an application is called dimensional analysis. We use the oper-ations of multiplying units and dividing units in applying dimensional analysis to appli-cation problems.

The Rule for Multiplying Exponential Expressions states that we multiply two expres-sions with the same base by adding the exponents.

In calculations that involve quantities, the units are operated on algebraically.

A rectangle measures 3 m by 5 m. Find the area of the rectangle.

The area of the rectangle is 15 m2 (square meters).

A box measures 10 cm by 5 cm by 3 cm. Find the volume of the box.

The volume of the box is 150 cm3 (cubic centimeters).

Find the area of a square whose side measures in.

The area of the square is (square inches).

Dimensional analysis is used in the conversion of units.

The following example converts the unit miles to feet. The equivalent measures 1 mi 5280 ft are used to form the following rates, which are called conversion

factors: and . Because 1 mi 5280 ft, both of the conversion factors

and are equal to 1.

To convert 3 mi to feet, multiply 3 mi by the conversion factor .

There are two important points in the above illustration. First, you can think of dividingthe numerator and denominator by the common unit “mile” just as you would divide thenumerator and denominator of a fraction by a common factor.

Second, the conversion factor is equal to 1, and multiplying an expression by

1 does not change the value of the expression.

5280 ft

1 mi

� 3 � 5280 ft � 15,840 ft3 mi � 5280 ft

1 mi�

5280 ft

1 mi �

3 mi

13 mi � 3 mi � 1 �

5280 ft

1 mi

5280 ft

1 mi

1 mi

5280 ft

�5280 ft

1 mi

1 mi

5280 ft

�

19x2 � 30x � 252 in2

A � s2 � 3 13x � 52 in. 4 2 � 13x � 522 in2 � 19x2 � 30x � 252 in2

13x � 52

V � LWH � 110 cm2 15 cm2 13 cm2 � 110 � 5 � 32 1cm � cm � cm2 � 150 cm3

A � LW � 13 m2 15 m2 � 13 � 52 1m � m2 � 15 m2

x4 � x6 � x4�6 � x10

Dimensional Analysis

3 m

5 m

(3x + 5) in.

10 cm5 cm

3 cm

1 mi

5280 ft

HOW TO • 1

HOW TO • 2

HOW TO • 3

For answers to the Focus on Problem Solving exercises, please see the Appendix in the Instructor’s Resource Binder that accompaniesthis textbook.

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 222

Focus on Problem Solving 223

In the application problem that follows, the units are kept in the problem while the prob-lem is worked.

In 2008, a horse named Big Brown ran a 1.25-mile race in 2.02 min. Find Big Brown’saverage speed for that race in miles per hour. Round to the nearest tenth.

Strategy To find the average speed, use the formula , where is the speed,

is the distance, and is the time. Use the conversion factor .

Solution

Big Brown’s average speed was 37.1 mph.

Try each of the following problems. Round to the nearest tenth or nearest cent.

1. Convert 88 ft�s to miles per hour.

2. Convert 8 m�s to kilometers per hour (1 km 1000 m).

3. A carpet is to be placed in a meeting hall that is 36 ft wide and 80 ft long. At $21.50per square yard, how much will it cost to carpet the meeting hall?

4. A carpet is to be placed in a room that is 20 ft wide and 30 ft long. At $22.25 persquare yard, how much will it cost to carpet the area?

5. Find the number of gallons of water in a fish tank that is 36 in. long and 24 in. wide and is filled to a depth of 16 in. (1 gal 231 in3).

6. Find the number of gallons of water in a fish tank that is 24 in. long and 18 in. wide and is filled to a depth of 12 in. (1 gal 231 in3).

7. A -acre commercial lot is on sale for $2.15 per square foot. Find the sale price of

the commercial lot (1 acre 43,560 ft2).

8. A 0.75-acre industrial parcel was sold for $98,010. Find the parcel’s price per squarefoot (1 acre 43,560 ft2).

9. A new driveway will require 800 ft3 of concrete. Concrete is ordered by the cubicyard. How much concrete should be ordered?

10. A piston-engined dragster traveled 440 yd in 4.936 s at Ennis, Texas, on October 9, 1988. Find the average speed of the dragster in miles per hour.

11. The Marianas Trench in the Pacific Ocean is the deepest part of the ocean. Itsdepth is 6.85 mi. The speed of sound under water is 4700 ft�s. Find the timeit takes sound to travel from the surface of the ocean to the bottom of theMarianas Trench and back.

�

�

1

4

�

�

�

�75 mi

2.02 h� 37.1 mph

�1.25 mi

2.02 min�

1.25 mi

2.02 min�

60 min

1 h

d

t�r

60 min

1 htd

rd

tr �

“Big Brown”

AP Im

ages

© D

uom

o/Co

rbis

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 223


PROJECTS AND GROUP ACTIVITIES

1. Explain why the diagram at the right represents .

2. Draw similar diagrams representing each of the following.

Simplifying the power of a binomial is called expanding the binomial. The expansions ofthe first three powers of a binomial are shown below.

Find . [Hint: ]

Find . [Hint: ]

If we continue in this way, the results for are

Now expand . Before you begin, see whether you can find a pattern that will helpyou write the expansion of without having to multiply it out. Here are somehints.

1. Write out the variable terms of each binomial expansion from through. Observe how the exponents on the variables change.

2. Write out the coefficients of all the terms without the variable parts. It will behelpful if you make a triangular arrangement as shown at the left. Note that each rowbegins and ends with a 1. Also note (in the two shaded regions, for example) thatany number in a row is the sum of the two closest numbers above it. For instance,

and .

The triangle of numbers shown at the left is called Pascal’s Triangle. To find the expan-sion of , you need to find the eighth row of Pascal’s Triangle. First find rowseven. Then find row eight and use the patterns you have observed to write the expansion

.

Pascal’s Triangle has been the subject of extensive analysis, and many patterns have beenfound. See whether you can find some of them.

1a � b28

1a � b28

6 � 4 � 101 � 5 � 6

1a � b261a � b21

1a � b281a � b28

1a � b26 � a6 � 6a5b � 15a4b2 � 20a3b3 � 15a2b4 � 6ab5 � b6

1a � b26

1a � b25 � 1a � b241a � b21a � b25

1a � b24 � 1a � b231a � b21a � b24

1a � b23 � 1a � b221a � b2 � 1a2 � 2ab � b22 1a � b2 � a3 � 3a2b � 3ab2 � b3

1a � b22 � 1a � b2 1a � b2 � a2 � 2ab � b2

1a � b21 � a � b

Pascal’s Triangle

1x � 422

1x � 222

1a � b22 � a2 � 2ab � b2Diagramming the

Square of a Binomial

Point of InterestPascal did not invent thetriangle of numbers known asPascal’s Triangle. It wasknown to mathematicians inChina probably as early as1050 A.D. But Pascal’s Traitedu triangle arithmetique(Treatise Concerning theArithmetical Triangle)brought together all thedifferent aspects of thetriangle of numbers for thefirst time.

a2

ab

a

b b2

ab

a b

1

1

1 1

1

1

1

1

1

1

2

3

4

15 20 15 6

31

5101 105

6

6 4

For answers to the Projects and Group Activities exercises, please see the Appendix in the Instructor’s Resource Binder thataccompanies this textbook.

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 224

Chapter 4 Summary 225

SUMMARY

CHAPTER 4

KEY WORDS EXAMPLES

A monomial is a number, a variable, or a product of numbersand variables. [4.1A, p. 192]

A polynomial is a variable expression in which the terms are monomials. [4.1A, p. 192]

A polynomial of two terms is a binomial. [4.1A, p. 192]

A polynomial of three terms is a trinomial. [4.1A, p. 192]

The degree of a polynomial in one variable is the greatest exponent on a variable. [4.1A, p. 192]

A polynomial in one variable is usually written in descending order, where the exponents on the variable terms decrease from left to right. [4.1A, p. 192]

The opposite of a polynomial is the polynomial with the sign of every term changed to its opposite. [4.1B, p. 193]

ESSENTIAL RULES AND PROCEDURES EXAMPLES

Addition of Polynomials [4.1A, p. 192]To add polynomials, add the coefficients of the like terms.

Subtraction of Polynomials [4.1B, p. 193] To subtract polynomials, add the opposite of the second polynomial to the first.

Rule for Multiplying Exponential Expressions [4.2A, p. 196]If m and n are integers, then . xm � xn � xm�n

5 is a number; y is a variable. is aproduct of numbers and variables. 5, y,and are monomials.2a3b2

2a3b2

is a polynomial. Eachterm of this expression is a monomial.5x2y � 3xy2 � 2

, , and arebinomials.

6a � 5by2 � 3x � 2

is a trinomial.x2 � 6x � 7

The degree of is 3.3x � 4x3 � 17x2 � 25

The polynomial iswritten in descending order.

2x4 � 3x2 � 4x � 7

The opposite of the polynomialis .�x2 � 3x � 4x2 � 3x � 4

� 3x3 � 2x2 � 5x � 9

� 1�4 � 52

� 3x3 � 12x2 � 4x22 � 13x � 2x2

12x2 � 3x � 42 � 13x3 � 4x2 � 2x � 52

� �2y2 � 2y � 12

� 1�9 � 32

� 13y2 � 5y22 � 1�8y � 10y2

� 13y2 � 8y � 92 � 1�5y2 � 10y � 32

13y2 � 8y � 92 � 15y2 � 10y � 32

a3 � a6 � a3�6 � a9

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 225


Rule for Simplifying the Power of an

Exponential Expression [4.2B, p. 197]

If m and n are integers, then .

Rule for Simplifying the Power of a Product [4.2B, p. 197]

If m, n, and p are integers, then .

To multiply a polynomial by a monomial, use the Distributive Property and the Rule for Multiplying ExponentialExpressions. [4.3A, p. 200]

To multiply two polynomials, multiply each term of one polynomial by each term of the other polynomial. [4.3B, p. 200]

FOIL Method [4.3C, p. 201]

To find the product of two binomials, add the products of the First terms, the Outer terms, the Inner terms, and the Last terms.

Product of the Sum and Difference of the Same Terms

[4.3D, p. 202]

Square of a Binomial [4.3D, p. 202]

Definition of Zero as an Exponent [4.4A, p. 208]

If , then .

Definition of a Negative Exponent [4.4A, p. 209]

If and n is a positive integer, then and .� xn1

x�n

1

xnx�n �x � 0

x0 � 1x � 0

�a � b�2 � a2 � 2ab � b2

1a � b22 � a2 � 2ab � b2

1a � b21a � b2 � a2 � b2

1xmyn2p � xmpynp

1xm2n � xmn 1c324 � c3 �4 � c12

1a3b224 � a3 �4b2 �4 � a12b8

� �20y3 � 12y2 � 32y

� 1�4y215y22 � 1�4y213y2 � 1�4y2182

1�4y215y2 � 3y � 82

x3 � x2 � 14x � 24

x3 � 5x2 � 6x

4x2 � 20x � 24

x � 4

x2 � 5x � 6

� 6x2 � 7x � 20

� 6x2 � 8x � 15x � 20

� 1�52142

� 12x213x2 � 12x2142 � 1�5213x2

12x � 5213x � 42

� 9x2 � 16

13x � 4213x � 42 � 13x22 � 42

� 9x2 � 24x � 16

13x � 422 � 13x22 � 213x2142 � 1�422 � 4x2 � 20x � 25

12x � 522 � 12x22 � 212x2152 � 52

; , c � 01�6c20 � 1170 � 1

and � x61

x�6

1

x6x�6 �

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 226

Chapter 4 Summary 227

Rule for Simplifying the Power of a Quotient [4.4A, p. 210]

If m, n, and p are integers and , then .

Rule for Negative Exponents on Fractional Expressions

[4.4A, p. 210]

If , , and n is a positive integer, then .

Rule for Dividing Exponential Expressions [4.4A, p. 210]

If m and n are integers and , then .

To Express a Number in Scientific Notation [4.4B, p. 213]

To express a number in scientific notation, write it in the form , where and n is an integer. If the number is

greater than 10, then n is a positive integer. If the number is between 0 and 1, then n is a negative integer.

To Change a Number in Scientific Notation

to Decimal Notation [4.4B, p. 213]

To change a number in scientific notation to decimal notation,move the decimal point to the right if n is positive and to the left if n is negative. Move the decimal point the same number of places as the absolute value of the exponent on 10.

To divide a polynomial by a monomial, divide each term in the numerator by the denominator and write the sum of the quotients.[4.5A, p. 218]

To check polynomial division, use the same equation used to check division of whole numbers:

(Quotient divisor) remainder dividend

[4.5B, p. 218]

��

1 a � 10a � 10n

� xm�nxm

xnx � 0

nb

a��na

bb � 0a � 0

xmp

ynp�pxm

yny � 0 c3

a52

�c3 �2

a5 �2�

c6

a10

xy�3

� y

x3

a7

a2� a7�2 � a5

0.0000078 � 7.8 � 10�6

367,000,000 � 3.67 � 108

9.06 � 10�5 � 0.0000906

2.418 � 107 � 24,180,000

� 2xy2 � y � 3

�8xy3

4y�

4y2

4y�

12y

4y

8xy3 � 4y2 � 12y

4y

Check:

1x2 � x � 102 1x � 32 � x � 4 �2

x � 3

� x2 � x � 10

1x � 421x � 32 � 2 � x2 � x � 12 � 2

2

�4x � 12

�4x � 10

x2 � 3x

x � 3�x2 � x � 10 x � 4

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 227


CONCEPT REVIEW

CHAPTER 4

Test your knowledge of the concepts presented in this chapter. Answer each question.Then check your answers against the ones provided in the Answer Section.

1. Why is it important to write the terms of a polynomial in descending order before adding in a vertical format?

2. What is the opposite of �7x3 � 3x2 � 4x � 2?

3. When multiplying the terms 4p3 and 7p6, what happens to the exponents?

4. Why is the simplification of the expression �4b(2b2 � 3b � 5) � �8b3 � 12b � 20 not true?

5. How do you multiply two binomials?

6. Simplify .

7. Simplify .

8. How do you write a very large number in scientific notation?

9. What is wrong with this simplification? � 7x2 � 8x2 � 6x

10. How do you check polynomial division?

14x3 � 8x2 � 6x

2x

a0

b�2�2

w2x4yz6

w3xy4z0

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 228

REVIEW EXERCISES

CHAPTER 4

Chapter 4 Review Exercises 229

1. Multiply:[4.3C]

3. Simplify:[4.2A]

5. Multiply:[4.3A]

7. Simplify:[4.2B]

9. Subtract:[4.1B]

11. Simplify:[4.2B]

13. Simplify:[4.2B]

15. Evaluate:

[4.4A]

17. Divide:

[4.5B]

19. Multiply:[4.3B]

2. Add:[4.1A]

4. Simplify:

[4.4A]

6. Simplify:

[4.4A]

8. Evaluate:[4.2B]

10. Simplify:

[4.4A]

12. Expand:[4.3D]

14. Divide:

[4.5A]

16. Subtract:[4.1B]

18. Multiply:[4.3C]

20. Divide:

[4.5B]b 2 � 5b � 2 �7

b � 7

1b3 � 2b2 � 33b � 72 1b � 72

2ax � 4ay � bx � 2by12a � b21x � 2y2

13y 3 � 12y 2 � 5y � 1113y3 � 7y � 22 � 112y2 � 2y � 12

4b 4 � 12b 2 � 1

12b7 � 36b5 � 3b3

3b3

25y 2 � 70y � 4915y � 722

b6

a4

a�1b3

a3b�3

6412322

�1

2a

3ab4

�6a2b4

2x 3

3

8x12

12x9

21y 2 � 4y � 1112y2 � 17y � 42 � 19y2 � 13y � 32

6y 3 � 17y 2 � 2y � 2113y2 � 4y � 72 12y � 32

�x � 2 �1

x � 3

7 � x � x2

x � 3

�1

16

�4�2

100a15b1315a7b62214ab2

�108x181�2x3221�3x423

2x 2 � 3x � 815x2 � 2x � 12 � 13x2 � 5x � 72

16u 12v 161�2u3v424

�8x 3 � 14x 2 � 18x�2x 14x2 � 7x � 92

x 4y 8z 41xy5z32 1x3y3z2

8b 2 � 2b � 1512b � 32 14b � 52

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 229


21. Multiply:[4.3A]

23. Multiply:[4.3B]

25. Divide:

[4.5A]

27. Write 37,560,000,000 in scientific notation.[4.4B]

29. Simplify:[4.2A]

31. Simplify:

[4.4A]

33. Write 0.000000127 in scientific notation.[4.4B]

22. Multiply:[4.3D]

24. Add:[4.1A]

26. Multiply:[4.3D]

28. Write in decimal notation.14,600,000 [4.4B]

30. Divide:[4.5B]

32. Multiply:[4.3C]

34. Write in decimal notation.[4.4B]0.0000000000032

3.2 � 10�12

10a2 � 31a � 6315a � 72 12a � 92

2y � 916y2 � 35y � 362 13y � 42

1.46 � 107

a 2 � 491a � 72 1a � 72

2x 3 � 9x 2 � 3x � 1212x3 � 7x2 � x2 � 12x2 � 4x � 122

4a2 � 25b212a � 5b2 12a � 5b2

1.27 � 10�7

x4y 6

9

1�3x�2y�32�2

�54a13b 5c 712a12b32 1�9b2c62 13ac2

3.756 � 1010

�4y � 8

16y2 � 32y

�4y

12b5 � 4b4 � 6b3 � 8b2 � 516b3 � 2b2 � 52 12b2 � 12

8a 3b 3 � 4a 2b 4 � 6ab 52ab314a2 � 2ab � 3b22

35. Geometry The length of a table-tennis table is 1 ft less than twice the width of thetable. Let represent the width of the table-tennis table. Express the area of thetable in terms of the variable .

ft2 [4.3E]

36. Geometry The side of a checkerboard is in. Express the area of thecheckerboard in terms of the variable .

in2 [4.3E]19x2 � 12x � 42x

13x � 22

12w2 � w2w

w

© D

uom

o/Co

rbis

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 230

Chapter 4 Test 231

TEST

CHAPTER 4

1. Multiply:[4.3A]

3. Simplify:

[4.4A]

5. Divide:

[4.5B]

7. Simplify:[4.2B]

9. Multiply:[4.3C]

11. Divide:[4.5B]

13. Multiply:[4.3B]

2. Divide:

[4.5A]

4. Simplify:[4.2A]

6. Multiply:[4.3B]

8. Simplify:

[4.4A]

10. Divide:

[4.5A]

12. Multiply:[4.3A]

14. Multiply:[4.3D]16y 2 � 914y � 32 14y � 32

6y 4 � 9y 3 � 18y 2�3y21�2y2 � 3y � 62

4x 4 � 2x 2 � 5

16x5 � 8x3 � 20x

4x

9y 10

x 10

(3x�2y3)3

3x4y�1

x3 � 7x 2 � 17x � 151x � 32 1x2 � 4x � 52

�6x 3y 61�2xy22 13x2y42

4x � 1 �3x 2

12x3 � 3x2 � 9

3x2

�4x4 � 8x3 � 3x2 � 14x � 211�2x3 � x2 � 72 12x � 32

x � 71x2 � 6x � 72 1x � 12

a 2 � 3ab � 10b 21a � 2b2 1a � 5b2

�8a6b31�2a2b23

x � 1 �2

x � 1

1x2 � 12 1x � 12

�4x 6

12x2

�3x8

4x 3 � 6x 22x12x2 � 3x2

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 231


15. Simplify:

[4.2A]

17. Divide:

[4.5A]

19. Expand:[4.3D]

21. Simplify:

[4.4A]

23. Add:[4.1A]

16. Simplify:

[4.4A]

18. Subtract:[4.1B]

20. Divide:

[4.5B]

22. Multiply:[4.3C]

24. Write 0.00000000302 in scientific notation.[4.4B]3.02 � 10�9

10x2 � 43xy � 28y 212x � 7y2 15x � 4y2

2x � 3 �2

2x � 3

14x2 � 72 12x � 32

�5a3 � 3a2 � 4a � 313a2 � 2a � 72 � 15a3 � 2a � 102

8ab 4

2a�1b

2�2a�2b�3

3x 3 � 6x 2 � 8x � 313x3 � 2x2 � 42 � 18x2 � 8x � 72

�2x 3

�(2x2y)3

4x3y3

4x 2 � 20x � 2512x � 522

4a � 7

20a � 35

5

a 4b 7

1ab22 1a3b52

25. Geometry The radius of a circle is m. Use the equation , whereis the radius, to find the area of the circle in terms of the variable x. Leave the

answer in terms of .[4.3E]1�x2 � 10�x � 25�2 m2

�r

A � �r 21x � 52x − 5

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 232

Cumulative Review Exercises 233

1. Simplify:

[1.6C]

3. Simplify:

[1.7B]

5. Simplify:[2.2A]

7. Simplify:[2.2D]

9. Solve:[3.3A]

11. 35.2 is what percent of 160?22% [3.1D]

13. Subtract:[4.1B]

15. Simplify:[4.2A]

2. Evaluate .

[1.7A]

4. Evaluate when

and .

[2.1A]

6. Simplify:

[2.2B]

8. Solve:

[3.1C]

10. Solve:[3.3B]

12. Add:[4.1A]

14. Simplify:[4.2B]

16. Multiply:[4.3A]6y 4 � 8y 3 � 16y 2

�2y21�3y2 � 4y � 82

a9b151a3b523

4b3 � 4b2 � 8b � 414b3 � 7b2 � 72 � 13b2 � 8b � 32

152 � 314 � x2 � 2x � 5

�16

x�3

412 �

�9x

�3

4112x2

�229

b � 3

a � �2b � 1a � b22

b2

53

�5

83

�2

3�32 �

�8x 3y 614xy32 1�2x2y32

3y 3 � 2y 2 � 10y13y3 � 5y � 82 � 1�2y2 � 5y � 82

�162x � 9 � 3x � 7

�18x � 12�2 33x � 2 14 � 3x2 � 2 4

5x � 3xy�2x � 1�xy2 � 7x � 4xy

2511

� 23

8�

5

63

�1

2

5144

3

16� �

5

8 �7

9

CUMULATIVE REVIEW EXERCISES

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 233


17. Multiply:[4.3B]

19. Simplify:

[4.4A]

18. Multiply:[4.3C]

20. Divide:[4.5B]a � 71a2 � 4a � 212 1a � 32

15b2 � 31b � 1413b � 22 15b � 722

12b 2

(�2a2b322

8a4b8

10a 3 � 39a 2 � 20a � 2112a � 72 15a2 � 2a � 32

21. Write in decimal notation.[4.4B]

22. Translate “the difference between eight times a number and twice the number iseighteen” into an equation and solve.

[3.4B]

23. Mixtures Fifty ounces of orange juice are added to 200 oz of a fruit punch that is10% orange juice. What is the percent concentration of orange juice in the result-ing mixture?28% [3.6B]

24. Transportation A car traveling at 50 mph overtakes a cyclist who, riding at 10 mph, has had a 2-hour head start. How far from the starting point does the carovertake the cyclist?25 mi [3.6C]

25. Geometry The width of a rectangle is 40% of the length. The perimeter of the rec-tangle is 42 m. Find the length and width of the rectangle.Length: 15 m; width: 6 m [3.1D]

8x � 2x � 18; 3

0.00006096.09 � 10�5

© B

rian

Tolb

ert/

Corb

is

46043_04_Ch04_0191-0234.qxd 10/27/09 12:49 PM Page 234

polynomials-02.pdf

Documents