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1: Addition and Subtraction of Polynomials

# 1: Addition and Subtraction of Polynomials

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5.1 • Addition and Subtraction of Polynomials

187

The commutative, associative, and distributive properties provide the basis for rearranging, regrouping, and combining similar terms.

Classroom Example

Add 9n ϩ 4, 2n Ϫ 5, and 3n Ϫ 11.

EXAMPLE 2

Add 5x Ϫ 1, 3x ϩ 4, and 9x Ϫ 7.

Solution

(5x Ϫ 1) ϩ (3x ϩ 4) ϩ (9x Ϫ 7) ϭ (5x ϩ 3x ϩ 9x) ϩ [Ϫ1 ϩ 4 ϩ (Ϫ7)]

ϭ 17x Ϫ 4

Classroom Example

Add Ϫy3 Ϫ 4y ϩ 1, 3y2 ϩ 6y ϩ 7,

and Ϫ10y ϩ 3.

EXAMPLE 3

Add Ϫx2 ϩ 2x Ϫ 1, 2x3 Ϫ x ϩ 4, and Ϫ5x ϩ 6.

Solution

(Ϫx2 ϩ 2x Ϫ 1) ϩ (2x3 Ϫ x ϩ 4) ϩ (Ϫ5x ϩ 6)

ϭ (2x3) ϩ (Ϫx2) ϩ (2x Ϫ x Ϫ 5x) ϩ (Ϫ1 ϩ 4 ϩ 6)

ϭ 2x3 Ϫ x2 Ϫ 4x ϩ 9

Subtracting Polynomials

Recall from Chapter 2 that a Ϫ b ϭ a ϩ (Ϫb). We define subtraction as adding the opposite.

This same idea extends to polynomials in general. The opposite of a polynomial is formed

by taking the opposite of each term. For example, the opposite of (2x2 Ϫ 7x ϩ 3) is

Ϫ2x2 ϩ 7x Ϫ 3. Symbolically, we express this as

Ϫ(2x2 Ϫ 7x ϩ 3) ϭ Ϫ2x2 ϩ 7x Ϫ 3

Now consider some subtraction problems.

Classroom Example

Subtract 5d 2 ϩ 2d Ϫ 6 from

9d 2 Ϫ 4d ϩ 1.

EXAMPLE 4

Subtract 2x2 ϩ 9x Ϫ 3 from 5x2 Ϫ 7x Ϫ 1.

Solution

Use the horizontal format.

(5x2 Ϫ 7x Ϫ 1) Ϫ (2x2 ϩ 9x Ϫ 3) ϭ (5x2 Ϫ 7x Ϫ 1) ϩ (Ϫ2x2 Ϫ 9x ϩ 3)

ϭ (5x2 Ϫ 2x2) ϩ (Ϫ7x Ϫ 9x) ϩ (Ϫ1 ϩ 3)

ϭ 3x2 Ϫ 16x ϩ 2

Classroom Example

Subtract Ϫ3x2 ϩ 2x Ϫ 4 from

6x2 Ϫ 3.

EXAMPLE 5

Subtract Ϫ8y2 Ϫ y ϩ 5 from 2y2 ϩ 9.

Solution

(2y2 ϩ 9) Ϫ (Ϫ8y2 Ϫ y ϩ 5) ϭ (2y2 ϩ 9) ϩ (8y2 ϩ y Ϫ 5)

ϭ (2y2 ϩ 8y2) ϩ (y) ϩ (9 Ϫ 5)

ϭ 10y2 ϩ y ϩ 4

Later when dividing polynomials, you will need to use a vertical format to subtract polynomials. Let’s consider two such examples.

188

Chapter 5 • Exponents and Polynomials

Classroom Example

Subtract 4n2 ϩ 9n Ϫ 7 from

12n2 Ϫ 5n ϩ 2.

Subtract 3x2 ϩ 5x Ϫ 2 from 9x2 Ϫ 7x Ϫ 1.

EXAMPLE 6

Solution

9x2 Ϫ 7x Ϫ 1

3x2 ϩ 5x Ϫ 2

Notice which polynomial goes on the bottom and the alignment of similar

terms in columns

Now we can mentally form the opposite of the bottom polynomial and add.

9x2 Ϫ 7x Ϫ 1

3x2 ϩ 5x Ϫ 2

6x2 Ϫ 12x ϩ 1

Classroom Example

Subtract 23x4 ϩ 11x3 ϩ 2x from

15x3 Ϫ 5x2 ϩ 3x.

The opposite of 3x 2 ϩ 5x Ϫ 2 is Ϫ3x 2 Ϫ 5x ϩ 2

EXAMPLE 7

Subtract 15y3 ϩ 5y2 ϩ 3 from 13y3 ϩ 7y Ϫ 1.

Solution

ϩ 7y Ϫ 1

13y3

15y ϩ 5y

ϩ3

3

2

Ϫ2y Ϫ 5y ϩ 7y Ϫ 4

3

Similar terms are arranged in columns

2

We mentally formed the opposite of the bottom

We can use the distributive property along with the properties a ϭ 1(a) and Ϫa ϭ Ϫ1(a)

when adding and subtracting polynomials. The next examples illustrate this approach.

Classroom Example

Perform the indicated operations.

(6a ϩ 5) ϩ (2a Ϫ 1) Ϫ (3a Ϫ 9)

EXAMPLE 8

Perform the indicated operations.

(3x Ϫ 4) ϩ (2x Ϫ 5) Ϫ (7x Ϫ 1)

Solution

(3x Ϫ 4) ϩ (2x Ϫ 5) Ϫ (7x Ϫ 1)

ϭ 1(3x Ϫ 4) ϩ 1(2x Ϫ 5) Ϫ 1(7x Ϫ 1)

ϭ 1(3x) Ϫ 1(4) ϩ 1(2x) Ϫ 1(5) Ϫ 1(7x) Ϫ 1(Ϫ1)

ϭ 3x Ϫ 4 ϩ 2x Ϫ 5 Ϫ 7x ϩ 1

ϭ 3x ϩ 2x Ϫ 7x Ϫ 4 Ϫ 5 ϩ 1

ϭ Ϫ2x Ϫ 8

Certainly we can do some of the steps mentally; Example 9 gives a possible format.

Classroom Example

Perform the indicated operations.

(Ϫ3x2 ϩ 2x Ϫ 1) Ϫ

(7x2 Ϫ 4x ϩ 9) ϩ (4x2 ϩ 9x Ϫ 10)

EXAMPLE 9

Perform the indicated operations.

(Ϫy2 ϩ 5y Ϫ 2) Ϫ (Ϫ2y2 ϩ 8y ϩ 6) ϩ (4y2 Ϫ 2y Ϫ 5)

Solution

(Ϫy2 ϩ 5y Ϫ 2) Ϫ (Ϫ2y2 ϩ 8y ϩ 6) ϩ (4y2 Ϫ 2y Ϫ 5)

ϭ Ϫy2 ϩ 5y Ϫ 2 ϩ 2y2 Ϫ 8y Ϫ 6 ϩ 4y2 Ϫ 2y Ϫ 5

ϭ Ϫy2 ϩ 2y2 ϩ 4y2 ϩ 5y Ϫ 8y Ϫ 2y Ϫ 2 Ϫ 6 Ϫ 5

ϭ 5y2 Ϫ 5y Ϫ 13

When we use the horizontal format, as in Examples 8 and 9, we use parentheses to indicate a quantity. In Example 8 the quantities (3x Ϫ 4) and (2x Ϫ 5) are to be added; from this

5.1 • Addition and Subtraction of Polynomials

189

result we are to subtract the quantity (7x Ϫ 1). Brackets, [ ], are also sometimes used as

grouping symbols, especially if there is a need to indicate quantities within quantities. To

remove the grouping symbols, perform the indicated operations, starting with the innermost

set of symbols. Let’s consider two examples of this type.

Classroom Example

Perform the indicated operations.

8q Ϫ [3q ϩ (q Ϫ 5)]

EXAMPLE 10

Perform the indicated operations.

3x Ϫ [2x ϩ (3x Ϫ 1)]

Solution

First we need to add the quantities 2x and (3x Ϫ 1).

3x Ϫ [2x ϩ (3x Ϫ 1)] ϭ 3x Ϫ (2x ϩ 3x Ϫ 1)

ϭ 3x Ϫ (5x Ϫ 1)

Now we need to subtract the quantity (5x Ϫ 1) from 3x.

3x Ϫ (5x Ϫ 1) ϭ 3x Ϫ 5x ϩ 1

ϭ Ϫ2x ϩ 1

Classroom Example

Perform the indicated operations.

17 Ϫ {6m Ϫ [3 Ϫ (m ϩ 2)]Ϫ9m}

EXAMPLE 11

Perform the indicated operations.

8 Ϫ {7x Ϫ [2 ϩ (x Ϫ 1)] ϩ 4x}

Solution

Start with the innermost set of grouping symbols (the parentheses) and proceed as follows:

8 Ϫ {7x Ϫ [2 ϩ (x Ϫ 1)] ϩ 4x} ϭ 8 Ϫ [7x Ϫ (x ϩ 1) ϩ 4x]

ϭ 8 Ϫ (7x Ϫ x Ϫ 1 ϩ 4x)

ϭ 8 Ϫ (10x Ϫ 1)

ϭ 8 Ϫ 10x ϩ 1

ϭ Ϫ10x ϩ 9

For a final example in this section, we look at polynomials in a geometric setting.

Classroom Example

Suppose that a triangle and a square

have the dimensions as shown below:

x

x

EXAMPLE 12

Suppose that a parallelogram and a rectangle have dimensions as indicated in Figure 5.1. Find

a polynomial that represents the sum of the areas of the two figures.

x

10

Find a polynomial that represents the

sum of the areas of the two figures.

x

x

20

Figure 5.1

Solution

Using the area formulas A ϭ bh and A ϭ lw for parallelograms and rectangles, respectively,

we can represent the sum of the areas of the two figures as follows:

Area of the parallelogram

Area of the rectangle

x(x) ϭ x2

20(x) ϭ 20x

We can represent the total area by x2 ϩ 20x.

190

Chapter 5 • Exponents and Polynomials

Concept Quiz 5.1

For Problems 1– 5, answer true or false.

1.

2.

3.

4.

5.

The degree of the monomial 4x2y is 3.

The degree of the polynomial 2x4 ϩ 5x3 ϩ 7x2 Ϫ 4x ϩ 6 is 10.

A three-term polynomial is called a binomial.

A polynomial is a monomial or a finite sum or difference of monomials.

Monomial terms must have whole number exponents for each variable.

For Problems 6 –10, match the polynomial with its description.

6.

7.

8.

9.

10.

5xy2

5xy2 ϩ 3x2

5x2y ϩ 3xy4

3x5 ϩ 2x3 ϩ 5x Ϫ 1

3x2y3

A.

B.

C.

D.

E.

Monomial of degree 5

Binomial of degree 5

Monomial of degree 3

Binomial of degree 3

Polynomial of degree 5

Problem Set 5.1

For Problems 1– 8, determine the degree of each polynomial.

(Objective 2)

24. 10x ϩ 3 from 14x ϩ 13

25. 5x Ϫ 2 from 3x Ϫ 7

1. 7x2y ϩ 6xy

2. 4xy Ϫ 7x

3. 5x Ϫ 9

4. 8x y Ϫ 2xy Ϫ x

26. 7x Ϫ 2 from 2x ϩ 3

5. 5x3 Ϫ x2 Ϫ x ϩ 3

6. 8x4 Ϫ 2x2 ϩ 6

27. Ϫx Ϫ 1 from Ϫ4x ϩ 6

7. 5xy

8. Ϫ7x ϩ 4

28. Ϫ3x ϩ 2 from Ϫx Ϫ 9

2

2 2

2

For Problems 9–22, add the polynomials. (Objective 3)

9. 3x ϩ 4 and 5x ϩ 7

10. 3x Ϫ 5 and 2x Ϫ 9

11. Ϫ5y Ϫ 3 and 9y ϩ 13

29. x2 Ϫ 7x ϩ 2 from 3x2 ϩ 8x Ϫ 4

30. 2x2 ϩ 6x Ϫ 1 from 8x2 Ϫ 2x ϩ 6

31. Ϫ2n2 Ϫ 3n ϩ 4 from 3n2 Ϫ n ϩ 7

32. 3n2 Ϫ 7n Ϫ 9 from Ϫ4n2 ϩ 6n ϩ 10

12. x2 Ϫ 2x Ϫ 1 and Ϫ2x2 ϩ x ϩ 4

33. Ϫ4x3 Ϫ x2 ϩ 6x Ϫ 1 from Ϫ7x3 ϩ x2 ϩ 6x Ϫ 12

13. Ϫ2x2 ϩ 7x Ϫ 9 and 4x2 Ϫ 9x Ϫ 14

34. Ϫ4x2 ϩ 6x Ϫ 2 from Ϫ3x3 ϩ 2x2 ϩ 7x Ϫ 1

14. 3a2 ϩ 4a Ϫ 7 and Ϫ3a2 Ϫ 7a ϩ 10

For Problems 35– 44, subtract the polynomials using a

vertical format. (Objective 4)

15. 5x Ϫ 2, 3x Ϫ 7, and 9x Ϫ 10

16. Ϫx Ϫ 4, 8x ϩ 9, and Ϫ7x Ϫ 6

35. 3x Ϫ 2 from 12x Ϫ 4

17. 2x Ϫ x ϩ 4, Ϫ5x Ϫ 7x Ϫ 2, and 9x ϩ 3x Ϫ 6

36. Ϫ4x ϩ 6 from 7x Ϫ 3

18. Ϫ3x2 ϩ 2x Ϫ 6, 6x2 ϩ 7x ϩ 3, and Ϫ4x2 Ϫ 9

37. Ϫ5a Ϫ 6 from Ϫ3a ϩ 9

19. Ϫ4n Ϫ n Ϫ 1 and 4n ϩ 6n Ϫ 5

38. 7a Ϫ 11 from Ϫ2a Ϫ 1

20. Ϫ5n2 ϩ 7n Ϫ 9 and Ϫ5n Ϫ 4

39. 8x2 Ϫ x ϩ 6 from 6x2 Ϫ x ϩ 11

21. 2x2 Ϫ 7x Ϫ 10, Ϫ6x Ϫ 2, and Ϫ9x2 ϩ 5

40. 3x2 Ϫ 2 from Ϫ2x2 ϩ 6x Ϫ 4

22. 7x Ϫ 11, Ϫx Ϫ 5x ϩ 9, and Ϫ4x ϩ 5

41. Ϫ2x3 Ϫ 6x2 ϩ 7x Ϫ 9 from 4x3 ϩ 6x2 ϩ 7x Ϫ 14

2

2

2

2

2

2

For Problems 23–34, subtract the polynomials using a

horizontal format. (Objective 4)

23. 7x ϩ 1 from 12x ϩ 6

42. 4x3 ϩ x Ϫ 10 from 3x2 Ϫ 6

43. 2x2 Ϫ 6x Ϫ 14 from 4x3 Ϫ 6x2 ϩ 7x Ϫ 2

44. 3x Ϫ 7 from 7x3 ϩ 6x2 Ϫ 5x Ϫ 4

5.1 • Addition and Subtraction of Polynomials

For Problems 45–64, perform the indicated operations.

(Objectives 3 and 4)

69. Find a polynomial that represents the perimeter of the

rectangle in Figure 5.2.

45. (5x ϩ 3) Ϫ (7x Ϫ 2) ϩ (3x ϩ 6)

1

3

46. (3x Ϫ 4) ϩ (9x Ϫ 1) Ϫ (14x Ϫ 7)

48. (Ϫ3x ϩ 6) ϩ (Ϫx Ϫ 8) Ϫ (Ϫ7x ϩ 10)

49. (x Ϫ 7x Ϫ 4) ϩ (2x Ϫ 8x Ϫ 9) Ϫ (4x Ϫ 2x Ϫ 1)

2

1

(3x + 2)(2x 5) = 6x 2

Figure 5.2

47. (Ϫx Ϫ 1) Ϫ (Ϫ2x ϩ 6) ϩ (Ϫ4x Ϫ 7)

2

191

2

50. (3x2 ϩ x Ϫ 6) Ϫ (8x2 Ϫ 9x ϩ 1) Ϫ (7x2 ϩ 2x Ϫ 6)

51. (Ϫx2 Ϫ 3x ϩ 4) ϩ(Ϫ2x2 Ϫ x Ϫ 2) Ϫ (Ϫ4x2 ϩ 7x ϩ 10)

52. (Ϫ3x2 Ϫ 2) ϩ (7x2 Ϫ 8) Ϫ (9x2 Ϫ 2x Ϫ 4)

53. (3a Ϫ 2b) Ϫ (7a ϩ 4b) Ϫ (6a Ϫ 3b)

2

3

11x

10

70. Find a polynomial that represents the area of the shaded

region in Figure 5.3. The length

of a radius of the larger circle is

r units, and the length of a

radius of the smaller circle is

4 units.

71. Find a polynomial that repre- Figure 5.3

sents the sum of the areas of the rectangles and squares

in Figure 5.4.

54. (5a ϩ 7b) ϩ (Ϫ8a Ϫ 2b) Ϫ (5a ϩ 6b)

55. (n Ϫ 6) Ϫ (2n2 Ϫ n ϩ 4) ϩ (n2 Ϫ 7)

56. 13n ϩ 42 Ϫ 1n2 Ϫ 9n ϩ 102 Ϫ 1Ϫ2n ϩ 42

2x

57. 7x ϩ [3x Ϫ (2x Ϫ 1)]

58. Ϫ6x ϩ [Ϫ2x Ϫ (5x ϩ 2)]

3x

4x

2x

59. Ϫ7n Ϫ [4n Ϫ (6n Ϫ 1)]

x

60. 9n Ϫ [3n Ϫ (5n ϩ 4)]

3x

x

3x

61. (5a Ϫ 1) Ϫ [3a ϩ (4a Ϫ 7)]

Figure 5.4

62. (Ϫ3a ϩ 4) Ϫ [Ϫ7a ϩ (9a Ϫ 1)]

72. Find a polynomial that represents the total surface area

of the rectangular solid in Figure 5.5.

63. 13x Ϫ {5x Ϫ [4x Ϫ (x Ϫ 6)]}

64. Ϫ10x Ϫ {7x Ϫ [3x Ϫ (2x Ϫ 3)]}

65. Subtract 5x Ϫ 3 from the sum of 4x Ϫ 2 and 7x ϩ 6.

66. Subtract 7x ϩ 5 from the sum of 9x Ϫ 4 and Ϫ3x Ϫ 2.

2

67. Subtract the sum of Ϫ2n Ϫ 5 and Ϫn ϩ 7 from Ϫ8n ϩ 9.

68. Subtract the sum of 7n Ϫ 11 and Ϫ4n Ϫ 3 from 13n Ϫ 4.

9

x

Figure 5.5

Thoughts Into Words

73. Explain how to subtract the polynomial

3x2 ϩ 6x Ϫ 2 from 4x2 ϩ 7.

74. Is the sum of two binomials always another binomial?

1. True

2. False

3. False

4. True

5. True

75. Is the sum of two binomials ever a trinomial? Defend

6. C

7. D

8. B

9. E

10. A

192

Chapter 5 • Exponents and Polynomials

5.2

Multiplying Monomials

OBJECTIVES

1

Apply the properties of exponents to multiply monomials

2

Multiply a polynomial by a monomial

3

Use products of monomials to represent the area or volume of geometric

ﬁgures

In Section 2.4, we used exponents and some of the basic properties of real numbers to

simplify algebraic expressions into a more compact form; for example,

(3x)(4xy) ϭ 3 и 4

и x и x и y ϭ 12x2y

Actually we were multiplying monomials, and it is this topic that we will pursue now. We

can make multiplying monomials easier by using some basic properties of exponents. These

properties are the direct result of the definition of an exponent. The following examples lead

to the first property:

и x3 ϭ (x и x)(x и x и x) ϭ x5

a3 и a4 ϭ (a и a и a)(a и a и a и a) ϭ a7

b и b2 ϭ (b)(b и b) ϭ b3

x2

In general,

bn

и bm ϭ (b и b и b и

. . . и b)(b и b и b и . . . и b)

14442443 14442443

n factors of b

ϭbиb

иbи

...

m factors of b

иb

144424443

(n ϩ m) factors of b

ϭ bnϩm

Property 5.1

If b is any real number, and n and m are positive integers, then

bn

# bm ϭ bnϩm

Property 5.1 states that when multiplying powers with the same base, add exponents.

Classroom Example

Multiply:

(a) m6 и m3    (b) d 5 и d 11

EXAMPLE 1

Multiply:

(a) x4

# x3

(b) a8

# a7

Solution

(a) x4

# x3 ϭ x4ϩ3 ϭ x7

(b) a8

# a7 ϭ a8ϩ7 ϭ a15

Another property of exponents is demonstrated by these examples.

(x2)3 ϭ x2 и x2 и x2 ϭ x2ϩ2ϩ2 ϭ x6

(a3)2 ϭ a3 и a3 ϭ a3ϩ3 ϭ a6

(b3)4 ϭ b3 и b3 и b3 и b3 ϭ b3ϩ3ϩ3ϩ3 ϭ b12

5.2 • Multiplying Monomials

193

In general,

(bn)m ϭ bn и bn и bn и . . . и bn

144424443

m factors of bn

m of these ns

64748

p

ϭ bnϩnϩnϩ ϩn

ϭ bmn

Property 5.2

If b is any real number, and m and n are positive integers, then

1bn 2 m ϭ bmn

Property 5.2 states that when raising a power to a power, multiply exponents.

Classroom Example

Raise each to the indicated power:

(a) (m2 ) 6    (b) (n7 ) 9

EXAMPLE 2

Raise each to the indicated power:

(a) (x4)3

(b) (a5)6

Solution

(a) (x4)3 ϭ x3 и 4 ϭ x12

(b) (a5)6 ϭ a6 и 5 ϭ a30

The third property of exponents we will use in this section raises a monomial to a power.

(2x)3 ϭ (2x)(2x)(2x) ϭ 2 и 2 и 2 и x и x и x ϭ 23 и x3

(3a4)2 ϭ (3a4)(3a4) ϭ 3 и 3 и a4 и a4 ϭ (3)2(a4)2

(Ϫ2xy5)2 ϭ (Ϫ2xy5)(Ϫ2xy5) ϭ (Ϫ2)(Ϫ2)(x)(x)(y5)(y5) ϭ (Ϫ2)2(x)2(y5)2

In general,

(ab)n ϭ ab и ab

и ab и

...

и ab

144424443

n factors of ab

ϭ (a и a и a и . . . и a)(b и b и b и . . . и b)

14442443 14442443

n factors of a

n factors of b

ϭab

n n

Property 5.3

If a and b are real numbers, and n is a positive integer, then

(ab)n ϭ anbn

Property 5.3 states that when raising a monomial to a power, raise each factor to that power.

Classroom Example

Raise each to the indicated power:

(a) (3m2n) 2    (b) (Ϫ5c2 d 6 ) 3

EXAMPLE 3

Raise each to the indicated power:

(a) (2x2y3)4

Solution

(a) (2x2y3)4 ϭ (2)4(x2)4(y3)4 ϭ 16x8y12

(b) (Ϫ3ab5)3 ϭ (Ϫ3)3(a1)3(b5)3 ϭ Ϫ27a3b15

(b) (Ϫ3ab5)3

194

Chapter 5 • Exponents and Polynomials

Consider the following examples in which we use the properties of exponents to help

simplify the process of multiplying monomials.

1. (3x3)(5x4) ϭ 3 и 5

ϭ 15x7

и x3 и x4

x3

2. (Ϫ4a2b3)(6ab2) ϭ Ϫ4 и 6 и a2

ϭ Ϫ24a3b5

3. (xy)(7xy5) ϭ 1 и 7

ϭ 7x2y6

и a и b3 и b2

и x и x и y и y5

3

1

3 1

4. a x2y3 b a x3y5 b ϭ и

4

2

4 2

3

ϭ x5y8

8

и x 4 ϭ x 3ϩ4 ϭ x 7

The numerical coefficient of xy is 1

и x2 и x3 и y3 и y5

It is a simple process to raise a monomial to a power when using the properties of exponents. Study the next examples.

5.

(2x3)4 ϭ (2)4(x3)4

ϭ (2)4(x12)

by using (ab)n ϭ anbn

by using (bn)m ϭ bmn

ϭ 16x12

6. (Ϫ2a4)5 ϭ (Ϫ2)5(a4)5

ϭ Ϫ32a20

3

2

2 3

7. a x2y3 b ϭ a b (x2)3(y3)3

5

5

8 6 9

ϭ

xy

125

8. (0.2a6b7)2 ϭ (0.2)2(a6)2(b7)2

ϭ 0.04a12b14

Sometimes problems involve first raising monomials to a power and then multiplying the

resulting monomials, as in the following examples.

9. (3x2)3(2x3)2 ϭ (3)3(x2)3(2)2(x3)2

ϭ (27)(x6)(4)(x6)

ϭ 108x12

10. (Ϫx2y3)5(Ϫ2x2y)2 ϭ (Ϫ1)5(x2)5(y3)5(Ϫ2)2(x2)2(y)2

ϭ (Ϫ1)(x10)(y15)(4)(x4)(y2)

ϭ Ϫ4x14y17

The distributive property along with the properties of exponents form a basis for finding

the product of a monomial and a polynomial. The next examples illustrate these ideas.

11. (3x)(2x2 ϩ 6x ϩ 1) ϭ (3x)(2x2) ϩ (3x)(6x) ϩ (3x)(1)

ϭ 6x3 ϩ 18x2 ϩ 3x

12. (5a2)(a3 Ϫ 2a2 Ϫ 1) ϭ (5a2)(a3) Ϫ (5a2)(2a2) Ϫ (5a2)(1)

ϭ 5a5 Ϫ 10a4 Ϫ 5a2

13. (Ϫ2xy)(6x2y Ϫ 3xy2 Ϫ 4y3)

ϭ (Ϫ2xy)(6x2y) Ϫ (Ϫ2xy)(3xy2) Ϫ (Ϫ2xy)(4y3)

ϭ Ϫ12x3y2 ϩ 6x2y3 ϩ 8xy4

Once you feel comfortable with this process, you may want to perform most of the work

mentally and then simply write down the final result. See whether you understand the following examples.

5.2 • Multiplying Monomials

195

14. 3x(2x ϩ 3) ϭ 6x2 ϩ 9x

15. Ϫ4x(2x2 Ϫ 3x Ϫ 1) ϭ Ϫ8x3 ϩ 12x2 ϩ 4x

16. ab(3a2b Ϫ 2ab2 Ϫ b3) ϭ 3a3b2 Ϫ 2a2b3 Ϫ ab4

We conclude this section by making a connection between algebra and geometry.

EXAMPLE 4

Classroom Example

Suppose that the dimensions of a

right circular cylinder are represented

by radius x and height 4x. Express

the volume and total surface area of

the cylinder.

Suppose that the dimensions of a rectangular solid are represented by x, 2x, and 3x as shown

in Figure 5.6. Express the volume and total surface area of the figure.

x

2x

3x

Figure 5.6

Solution

Using the formula V ϭ lwh, we can express the volume of the rectangular solid as (2x)(3x)(x),

which equals 6x3. The total surface area can be described as follows:

Area of front and back rectangles: 2(x)(3x) ϭ 6x2

Area of left side and right side: 2(2x)(x) ϭ 4x2

Area of top and bottom:

2(2x)(3x) ϭ 12x2

We can represent the total surface area by 6x2 ϩ 4x2 ϩ 12x2 or 22x2.

Concept Quiz 5.2

For Problems 1–10, answer true or false.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

When multiplying factors with the same base, add the exponents.

32 и 32 ϭ 94

2x2 и 3x3 ϭ 6x6

(x2 ) 3 ϭ x5

(Ϫ4x3 ) 2 ϭ Ϫ4x6

To simplify (3x2y)(2x3y2 ) 4, use the order of operations to first raise 2x3y2 to the fourth

power, and then multiply the monomials.

(3x2y) 3 ϭ 27x6y3

(Ϫ2xy)(3x2y3 ) ϭ Ϫ6x3y4

(Ϫx2y)(xy3 )(xy) ϭ x4y5

(2x2y3 ) 3 (Ϫxy2 ) ϭ Ϫ8x7y11

Problem Set 5.2

For Problems 1– 30, multiply using the properties of exponents to help with the manipulation. (Objective 1)

1. (5x)(9x)

2

2. (7x)(8x)

3. (3x )(7x)

4. (9x)(4x3)

5. (Ϫ3xy)(2xy)

6. (6xy)(Ϫ3xy)

7. (Ϫ2x2y)(Ϫ7x)

8. (Ϫ5xy2)(Ϫ4y)

9. (4a2b2)(Ϫ12ab)

11. (Ϫxy)(Ϫ5x3)

10. (Ϫ3a3b)(13ab2)

12. (Ϫ7y2)(Ϫx2y)

13. (8ab2c)(13a2c)

14. (9abc3)(14bc2)

15. (5x2)(2x)(3x3)

16. (4x)(2x2)(6x4)

17. (4xy)(Ϫ2x)(7y2)

18. (5y2)(Ϫ3xy)(5x2)

19. (Ϫ2ab)(Ϫab)(Ϫ3b)

20. (Ϫ7ab)(Ϫ4a)(Ϫab)

21. (6cd)(Ϫ3c2d)(Ϫ4d)

22. (2c 3d)(Ϫ6d 3)(Ϫ5cd)

2

3

23. a xyb a x2y4 b

3

5

5

8

24. aϪ xb a x2yb

6

3

196

25. aϪ

Chapter 5 • Exponents and Polynomials

69. Ϫ4(3x ϩ 2) Ϫ 5[2x Ϫ (3x ϩ 4)]

7 2

8

a bb a b4b

12

21

70. Ϫ5(2x Ϫ 1) Ϫ 3[x Ϫ (4x Ϫ 3)]

9

15

26. aϪ a3b4 baϪ ab2 b

5

6

27. (0.4x5)(0.7x3)

3

29. (Ϫ4ab)(1.6a b)

For Problems 71– 80, perform the indicated operations and

simplify. (Objective 1)

28. (Ϫ1.2x4)(0.3x2)

2

2 4

30. (Ϫ6a b)(Ϫ1.4a b )

For Problems 31– 46, raise each monomial to the indicated

power. Use the properties of exponents to help with the manipulation. (Objective 1)

31. (2x4)2

32. (3x3)2

33. (Ϫ3a2b3)2

34. (Ϫ8a4b5)2

2 3

35. (3x )

4 3

36. (2x )

4 3

37. (Ϫ4x )

38. (Ϫ3x3)3

39. (9x4y5)2

40. (8x6y4)2

41. (2x2y)4

42. (2x2y3)5

43. (Ϫ3a3b2)4

44. (Ϫ2a4b2)4

45. (Ϫx2y)6

46. (Ϫx2y3)7

71. (3x)2(2x3)

72. (Ϫ2x)3(4x5)

73. (Ϫ3x)3(Ϫ4x)2

74. (3xy)2(2x2y)4

75. (5x2y)2(xy2)3

76. (Ϫx2y)3(6xy)2

77. (Ϫa2bc3)3(a3b)2

78. (ab2c3)4(Ϫa2b)3

79. (Ϫ2x2y2)4(Ϫxy3)3

80. (Ϫ3xy)3(Ϫx2y3)4

For Problems 81– 84, use the products of polynomials to

represent the area or volume of the geometric figure.

(Objective 3)

81. Express in simplified form the sum of the areas of the

two rectangles shown in Figure 5.7.

4

3

x−1

x+2

Figure 5.7

For Problems 47– 60, multiply by using the distributive property. (Objective 2)

47. 5x(3x ϩ 2)

48. 7x(2x ϩ 5)

49. 3x2(6x Ϫ 2)

50. 4x2(7x Ϫ 2)

51. Ϫ4x(7x2 Ϫ 4)

52. Ϫ6x(9x2 Ϫ 5)

53. 2x(x2 Ϫ 4x ϩ 6)

54. 3x(2x2 Ϫ x ϩ 5)

55. Ϫ6a(3a2 Ϫ 5a Ϫ 7)

56. Ϫ8a(4a2 Ϫ 9a Ϫ 6)

57. 7xy(4x2 Ϫ x ϩ 5)

58. 5x2y(3x2 ϩ 7x Ϫ 9)

59. Ϫxy(9x2 Ϫ 2x Ϫ 6)

60. xy2(6x2 Ϫ x Ϫ 1)

82. Express in simplified form the volume and the total surface area of the rectangular solid in Figure 5.8.

x

4

2x

Figure 5.8

83. Represent the area of the shaded region in Figure 5.9.

The length of a radius of the smaller circle is x, and the

length of a radius of the larger circle is 2x.

For Problems 61– 70, remove the parentheses by multiplying

and then simplify by combining similar terms; for example,

3(x Ϫ y) ϩ 2(x Ϫ 3y) ϭ 3x Ϫ 3y ϩ 2x Ϫ 6y ϭ 5x Ϫ 9y

(Objective 2)

61. 5(x ϩ 2y) ϩ 4(2x ϩ 3y)

62. 3(2x ϩ 5y) ϩ 2(4x ϩ y)

Figure 5.9

84. Represent the area of the shaded region in Figure 5.10.

63. 4(x Ϫ 3y) Ϫ 3(2x Ϫ y)

x−2

64. 2(5x Ϫ 3y) Ϫ 5(x ϩ 4y)

65. 2x(x2 Ϫ 3x Ϫ 4) ϩ x(2x2 ϩ 3x Ϫ 6)

x

4

66. 3x(2x2 Ϫ x ϩ 5) Ϫ 2x(x2 ϩ 4x ϩ 7)

67. 3[2x Ϫ (x Ϫ 2)] Ϫ 4(x Ϫ 2)

68. 2[3x Ϫ (2x ϩ 1)] Ϫ 2(3x Ϫ 4)

3x + 2

Figure 5.10

5.3 • Multiplying Polynomials

197

Thoughts Into Words

85. How would you explain to someone why the product of

x3 and x4 is x7 and not x12?

86. Suppose your friend was absent from class the day that

this section was discussed. How would you help her understand why the property (bn ) m ϭ bmn is true?

87. How can Figure 5.11 be used to geometrically demonstrate that x(x ϩ 2) ϭ x2 ϩ 2x?

x

x

2

Figure 5.11

Further Investigations

For Problems 88–97, find each of the indicated products.

Assume that the variables in the exponents represent positive integers; for example,

(x2n)(x4n) ϭ x2nϩ4n ϭ x6n

n

3n

2n

88. (x )(x )

91. (x5nϩ2)(xnϪ1)

92. (x3)(x4nϪ5)

93. (x6nϪ1)(x4)

94. (2xn)(3x2n)

95. (4x3n)(Ϫ5x7n)

96. (Ϫ6x2nϩ4)(5x3nϪ4)

97. (Ϫ3x5nϪ2)(Ϫ4x2nϩ2)

5n

89. (x )(x )

1. True

2. False

3. False

4. False

9. False

10. True

5.3

90. (x2nϪ1)(x3nϩ2)

5. False

6. True

7. True

8. True

Multiplying Polynomials

OBJECTIVES

1

Use the distributive property to ﬁnd the product of two binomials

2

Use the shortcut pattern to ﬁnd the product of two binomials

3

Use a pattern to ﬁnd the square of a binomial

4

Use a pattern to ﬁnd the product of (a ϩ b)(a Ϫ b)

In general, to go from multiplying a monomial times a polynomial to multiplying two polynomials requires the use of the distributive property. Consider some examples.

Classroom Example

Find the product of (a ϩ 6) and

(b ϩ 3).

EXAMPLE 1

Find the product of (x ϩ 3) and (y ϩ 4).

Solution

(x ϩ 3)(y ϩ 4) ϭ x(y ϩ 4) ϩ 3(y ϩ 4)

ϭ x(y) ϩ x(4) ϩ 3(y) ϩ 3(4)

ϭ xy ϩ 4x ϩ 3y ϩ 12

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