3.4
Multiplying Polynomials
3.4
OBJECTIVES
1. Find the product of a monomial and a polynomial
2. Find the product of two polynomials
You have already had some experience in multiplying polynomials. In Section 1.7 we stated
the first property of exponents and used that property to find the product of two monomial
terms. Let’s review briefly.
Step by Step:
NOTE The first property of
Step 1
Step 2
To Find the Product of Monomials
Multiply the coefficients.
Use the first property of exponents to combine the variables.
exponents:
xm xn xmn
Example 1
Multiplying Monomials
Multiply 3x2y and 2x3y5.
Write
(3x2y)(2x3y5)
(3 2)(x2 x3)(y y5)
NOTE Once again we have
used the commutative and
associative properties to rewrite
the problem.
Multiply
Add the exponents.
the coefficients.
6x5y6
CHECK YOURSELF 1
Multiply.
(a) (5a2b)(3a2b4)
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NOTE You might want to
review Section 1.2 before going
on.
(b) (3xy)(4x3y5)
Our next task is to find the product of a monomial and a polynomial. Here we use the
distributive property, which we introduced in Section 1.2. That property leads us to the
following rule for multiplication.
Rules and Properties: To Multiply a Polynomial by a Monomial
NOTE Distributive property:
a(b c) ab ac
Use the distributive property to multiply each term of the polynomial by the
monomial.
Example 2
Multiplying a Monomial and a Binomial
(a) Multiply 2x 3 by x.
281
282
CHAPTER 3
POLYNOMIALS
Write
NOTE With practice you will
do this step mentally.
x(2x 3)
x 2x x 3
2x2 3x
Multiply x by 2x and then by 3, the
terms of the polynomial. That is,
“distribute” the multiplication
over the sum.
(b) Multiply 2a3 4a by 3a2.
Write
3a2(2a3 4a)
3a2 2a3 3a2 4a 6a5 12a3
CHECK YOURSELF 2
Multiply.
(a) 2y(y2 3y)
(b) 3w2(2w3 5w)
The patterns of Example 2 extend to any number of terms.
Example 3
Multiplying a Monomial and a Polynomial
Multiply the following.
(a) 3x(4x3 5x2 2)
3x 4x3 3x 5x2 3x 2 12x4 15x3 6x
all the steps of the process.
With practice you can write the
product directly, and you
should try to do so.
(b) 5y2(2y3 4)
5y2 2y3 5y2 4 10y5 20y2
(c) 5c(4c2 8c)
(5c) (4c2) (5c) (8c) 20c3 40c2
(d) 3c2d 2(7cd 2 5c2d 3)
3c2d2 7cd 2 3c2d2 5c2d 3 21c3d 4 15c4d 5
CHECK YOURSELF 3
Multiply.
(a) 3(5a2 2a 7)
(c) 5m(8m2 5m)
Example 4
Multiplying Binomials
(a) Multiply x 2 by x 3.
(b) 4x2(8x3 6)
(d) 9a2b(3a3b 6a2b4)
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NOTE Again we have shown
MULTIPLYING POLYNOMIALS
that each term, x and 2, of the
first binomial is multiplied by
each term, x and 3, of the
second binomial.
283
We can think of x 2 as a single quantity and apply the distributive property.
NOTE Note that this ensures
SECTION 3.4
(x 2)(x 3)
Multiply x 2 by x and then by 3.
(x 2)x (x 2) 3
xx2xx323
x2 2x 3x 6
x2 5x 6
(b) Multiply a 3 by a 4. (Think of a 3 as a single quantity and distribute.)
(a 3)(a 4)
(a 3)a (a 3)(4)
a a 3 a [(a 4) (3 4)]
a2 3a (4a 12)
a 3a 4a 12
2
Note that the parentheses are needed
here because a minus sign precedes
the binomial.
a2 7a 12
CHECK YOURSELF 4
Multiply.
(a) (x 4)(x 5)
(b) (y 5)(y 6)
Fortunately, there is a pattern to this kind of multiplication that allows you to write
the product of the two binomials directly without going through all these steps. We call it
the FOIL method of multiplying. The reason for this name will be clear as we look at the
process in more detail.
To multiply (x 2)(x 3):
1. (x 2)(x 3)
NOTE Remember this by F!
xx
Find the product of
the first terms of the
factors.
2. (x 2)(x 3)
NOTE Remember this by O!
Find the product of
the outer terms.
x3
3. (x 2)(x 3)
NOTE Remember this by I!
2x
Find the product of
the inner terms.
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4. (x 2)(x 3)
NOTE Remember this by L!
23
Find the product of
the last terms.
Combining the four steps, we have
NOTE Of course these are the
same four terms found in
Example 4a.
(x 2)(x 3)
x2 3x 2x 6
NOTE It’s called FOIL to give
x2 5x 6
you an easy way of
remembering the steps: First,
Outer, Inner, and Last.
With practice, the FOIL method will let you write the products quickly and easily. Consider Example 5, which illustrates this approach.
284
CHAPTER 3
POLYNOMIALS
Example 5
Using the FOIL Method
Find the following products, using the FOIL method.
F
x x
L
4 5
(a) (x 4)(x 5)
4x
I
5x
O
should combine the outer and
inner products mentally and
write just the final product.
x2 5x 4x 20
F
O
I
L
x 9x 20
2
F
x x
L
(7)(3)
(b) (x 7)(x 3)
7x
I
Combine the outer and inner products as 4x.
3x
O
x2 4x 21
CHECK YOURSELF 5
Multiply.
(a) (x 6)(x 7)
(b) (x 3)(x 5)
(c) (x 2)(x 8)
Using the FOIL method, you can also find the product of binomials with coefficients
other than 1 or with more than one variable.
Example 6
Using the FOIL Method
Find the following products, using the FOIL method.
F
12x2
L
6
(a) (4x 3)(3x 2)
9x
I
8x
O
12x2 x 6
Combine:
9x 8x x
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NOTE When possible, you
MULTIPLYING POLYNOMIALS
6x2
SECTION 3.4
285
35y2
(b) (3x 5y)(2x 7y)
10xy
Combine:
10xy 21xy 31xy
21xy
6x 31xy 35y2
2
The following rule summarizes our work in multiplying binomials.
Step by Step:
Step 1
Step 2
Step 3
To Multiply Two Binomials
Find the first term of the product of the binomials by multiplying the
first terms of the binomials (F).
Find the middle term of the product as the sum of the outer and inner
products (O I).
Find the last term of the product by multiplying the last terms of the
binomials (L).
CHECK YOURSELF 6
Multiply.
(a) (5x 2)(3x 7)
(b) (4a 3b)(5a 4b)
(c) (3m 5n)(2m 3n)
Sometimes, especially with larger polynomials, it is easier to use the vertical method to find
their product. This is the same method you originally learned when multiplying two large
integers.
Example 7
Multiplying Using the Vertical Method
Use the vertical method to find the product of (3x 2)(4x 1).
First, we rewrite the multiplication in vertical form.
3x 2
4x (1)
Multiplying the quantity 3x 2 by 1 yields
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3x 2
4x (1)
3x (2)
Note that we maintained the columns of the original binomial when we found the product.
We will continue with those columns as we multiply by the 4x term.
3x 2
4x (1)
3x (2)
12x2 8x
12x2 5x (2)
286
CHAPTER 3
POLYNOMIALS
We could write the product as (3x 2)(4x 1) 12x2 5x 2.
CHECK YOURSELF 7
Use the vertical method to find the product of (5x 3)(2x 1).
We’ll use the vertical method again in our next example. This time, we will multiply a
binomial and a trinomial. Note that the FOIL method can never work for anything but the
product of two binomials.
Example 8
Using the Vertical Method
Multiply x2 5x 8 by x 3.
Step 1
x2 5x 8
x 3
3x2 15x 24
x2 5x 8
x 3
Step 2
3x2 15x 24
x 5x2 8x
3
Note that this line is shifted
over so that like terms are in
the same columns.
x2 5x 8
x 3
Step 3
method ensures that each term
of one factor multiplies each
term of the other. That’s why it
works!
Now multiply each term by x.
3x2 15x 24
x 5x2 8x
3
x 2x 7x 24
3
2
Now add to combine like
terms to write the product.
CHECK YOURSELF 8
Multiply 2x2 5x 3 by 3x 4.
CHECK YOURSELF ANSWERS
1.
3.
4.
5.
6.
7.
(a) 15a4b5; (b) 12x4y6
2. (a) 2y3 6y2; (b) 6w5 15w3
(a) 15a2 6a 21; (b) 32x5 24x2; (c) 40m3 25m2; (d) 27a5b2 54a4b5
(a) x2 9x 20; (b) y2 y 30
(a) x2 13x 42; (b) x2 2x 15; (c) x2 10x 16
(a) 15x2 29x 14; (b) 20a2 31ab 12b2; (c) 6m2 19mn 15n2
10x2 x 3
8. 6x3 7x2 11x 12
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NOTE Using this vertical
Multiply each term
of x2 5x 8 by 3.
Name
3.4
Exercises
Section
Date
Multiply.
1. (5x2)(3x3)
ANSWERS
2. (7a5)(4a6)
1.
3. (2b2)(14b8)
4. (14y4)(4y6)
2.
3.
4.
6
7
5. (10p )(4p )
8
7
6. (6m )(9m )
5.
6.
7. (4m5)(3m)
8. (5r7)(3r)
7.
8.
9. (4x3y2)(8x2y)
10. (3r4s2)(7r2s5)
9.
10.
11. (3m5n2)(2m4n)
12. (7a3b5)(6a4b)
11.
12.
13.
13. 5(2x 6)
14. 4(7b 5)
14.
15.
15. 3a(4a 5)
16. 5x(2x 7)
16.
17.
17. 3s2(4s2 7s)
18. 9a2(3a3 5a)
18.
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19.
19. 2x(4x2 2x 1)
20. 5m(4m3 3m2 2)
20.
21.
22.
21. 3xy(2x y xy 5xy)
2
2
22. 5ab (ab 3a 5b)
2
23.
23. 6m2n(3m2n 2mn mn2)
24. 8pq2(2pq 3p 5q)
24.
287
ANSWERS
25.
Multiply.
26.
25. (x 3)(x 2)
26. (a 3)(a 7)
27. (m 5)(m 9)
28. (b 7)(b 5)
29. (p 8)(p 7)
30. (x 10)(x 9)
31. (w 10)(w 20)
32. (s 12)(s 8)
33. (3x 5)(x 8)
34. (w 5)(4w 7)
35. (2x 3)(3x 4)
36. (5a 1)(3a 7)
37. (3a b)(4a 9b)
38. (7s 3t)(3s 8t)
39. (3p 4q)(7p 5q)
40. (5x 4y)(2x y)
41. (2x 5y)(3x 4y)
42. (4x 5y)(4x 3y)
43. (x 5)2
44. (y 8)2
45. (y 9)2
46. (2a 3)2
47. (6m n)2
48. (7b c)2
49. (a 5)(a 5)
50. (x 7)(x 7)
51. (x 2y)(x 2y)
52. (7x y)(7x y)
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
48.
49.
50.
51.
52.
288
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47.
ANSWERS
53. (5s 3t)(5s 3t)
54. (9c 4d)(9c 4d)
53.
54.
55.
Multiply, using the vertical method.
56.
55. (x 2)(3x 5)
56. (a 3)(2a 7)
57.
58.
57. (2m 5)(3m 7)
58. (5p 3)(4p 1)
59.
60.
59. (3x 4y)(5x 2y)
60. (7a 2b)(2a 4b)
61.
62.
61. (a2 3ab b2)(a2 5ab b2)
62. (m2 5mn 3n2)(m2 4mn 2n2)
63.
64.
63. (x 2y)(x2 2xy 4y2)
64. (m 3n)(m2 3mn 9n2)
65.
66.
65. (3a 4b)(9a2 12ab 16b2)
66. (2r 3s)(4r2 6rs 9s2)
67.
68.
69.
Multiply.
70.
67. 2x(3x 2)(4x 1)
68. 3x(2x 1)(2x 1)
71.
72.
69. 5a(4a 3)(4a 3)
70. 6m(3m 2)(3m 7)
73.
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74.
71. 3s(5s 2)(4s 1)
72. 7w(2w 3)(2w 3)
75.
76.
73. (x 2)(x 1)(x 3)
74. (y 3)(y 2)(y 4)
75. (a 1)3
76. (x 1)3
289
ANSWERS
Multiply the following.
77.
78.
79.
77.
2 3 3 5
78.
3 4 4 5
x
2
2x
2
80.
81.
x
3
3x
3
82.
79. [x (y 2)][x (y 2)]
83.
84.
80. [x (3 y)][x (3 y)]
85.
86.
Label the following as true or false.
87.
81. (x y)2 x2 y2
88.
82. (x y)2 x2 y2
83. (x y)2 x2 2xy y2
84. (x y)2 x2 2xy y2
85. Length. The length of a rectangle is given by 3x 5 centimeters (cm) and the width
is given by 2x 7 cm. Express the area of the rectangle in terms of x.
86. Area. The base of a triangle measures 3y 7 inches (in.) and the height is
87. Revenue. The price of an item is given by p 2x 10. If the revenue generated is
found by multiplying the number of items (x) sold by the price of an item, find the
polynomial which represents the revenue.
88. Revenue. The price of an item is given by p 2x2 100. Find the polynomial that
represents the revenue generated from the sale of x items.
290
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2y 3 in. Express the area of the triangle in terms of y.
ANSWERS
89. Work with another student to complete this table and write the polynomial. A paper
box is to be made from a piece of cardboard 20 inches (in.) wide and 30 in. long. The
box will be formed by cutting squares out of each of the four corners and folding up
the sides to make a box.
89.
90.
30 in.
x
20 in.
a.
If x is the dimension of the side of the square cut out of the corner, when the sides are
folded up, the box will be x inches tall. You should use a piece of paper to try this to
see how the box will be made. Complete the following chart.
b.
c.
d.
Length of Side
of Corner Square
1 in.
2 in.
3 in.
Length of
Box
Width of
Box
Depth of
Box
Volume of
Box
e.
f.
g.
h.
n in.
Write a general formula for the width, length, and height of the box and a general
formula for the volume of the box, and simplify it by multiplying. The variable will be the
height, the side of the square cut out of the corners. What is the highest power of the
variable in the polynomial you have written for the volume ______?
90. (a) Multiply (x 1) (x 1)
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(b) Multiply (x 1)(x2 x 1)
(c) Multiply (x 1)(x3 x2 x 1)
(d) Based on your results to (a), (b), and (c), find the product
(x 1) (x29 x28 x 1).
Getting Ready for Section 3.5 [Section 1.4]
Simplify.
(a)
(c)
(e)
(g)
(3a)(3a)
(5x)(5x)
(2w)(2w)
(4r)(4r)
(b)
(d)
(f)
(h)
(3a)2
(5x)2
(2w)2
(4r)2
291
Answers
1. 15x5
3. 28b10
5. 40p13
7. 12m6
9. 32x5y3
11. 6m9n3
2
4
3
3
13. 10x 30
15. 12a 15a
17. 12s 21s
19. 8x 4x2 2x
3 2
2 3
2 2
4 2
3 2
3 3
21. 6x y 3x y 15x y
23. 18m n 12m n 6m n
25. x2 5x 6
2
2
2
27. m 14m 45
29. p p 56
31. w 30w 200
33. 3x2 29x 40
35. 6x2 x 12
37. 12a2 31ab 9b2
2
2
2
39. 21p 13pq 20q
41. 6x 23xy 20y2
43. x2 10x 25
2
2
2
2
45. y 18y 81
47. 36m 12mn n
49. a 25
51. x2 4y2
2
2
2
2
53. 25s 9t
55. 3x 11x 10
57. 6m m 35
59. 15x2 14xy 8y2
61. a4 2a3b 15a2b2 8ab3 b4
63. x3 8y3
65. 27a3 64b3
67. 24x3 10x2 4x
3
3
2
69. 80a 45a
71. 60s 39s 6s
73. x3 4x2 x 6
75. a3 3a2 3a 1
81. False
89.
h. 16r2
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g. 16r2
83. True
a. 9a2
x2
11x
4
79. x2 y2 4y 4
3
45
15
85. 6x2 11x 35cm2
87. 2x2 10x
2
2
2
b. 9a
c. 25x
d. 25x
e. 4w2
f. 4w2
77.
292