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
polynomial and added
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?
Defend your answer.
Answers to the Concept Quiz
1. True
2. False
3. False
4. True
5. True
75. Is the sum of two binomials ever a trinomial? Defend
your answer.
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 )
Answers to the Concept Quiz
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