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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
3.5
Special Products
3.5
OBJECTIVES
1. Square a binomial
2. Find the product of two binomials that differ
only in their sign
Certain products occur frequently enough in algebra that it is worth learning special formulas for dealing with them. First, let’s look at the square of a binomial, which is the product of two equal binomial factors.
(x y)2 (x y) (x y)
x 2 2xy y 2
(x y)2 (x y) (x y)
x2 2xy y 2
The patterns above lead us to the following rule.
Step by Step:
Step 1
Step 2
Step 3
To Square a Binomial
Find the first term of the square by squaring the first term of the
binomial.
Find the middle term of the square as twice the product of the two
terms of the binomial.
Find the last term of the square by squaring the last term of the
binomial.
Example 1
Squaring a Binomial
(a) (x 3)2 x2 2 · x · 3 32
CA UTI O N
A very common mistake in
squaring binomials is to forget
the middle term.
Square of
first term
Twice the
product of
the two terms
Square of
the last term
x2 6x 9
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(b) (3a 4b)2 (3a)2 2(3a)(4b) (4b)2
9a2 24ab 16b2
(c) (y 5)2 y2 2 · y · (5) ( 5)2
y2 10y 25
(d) (5c 3d )2 (5c)2 2(5c)(3d) (3d)2
25c2 30cd 9d 2
Again we have shown all the steps. With practice you can write just the square.
293
294
CHAPTER 3
POLYNOMIALS
CHECK YOURSELF 1
Multiply.
(a) (2x 1)2
(b) (4x 3y)2
Example 2
Squaring a Binomial
Find ( y 4)2.
2
2
( y 4)2
is not equal to
y2 42 or y2 16
The correct square is
( y 4)2 y2 8y 16
The middle term is twice the product of y and 4.
CHECK YOURSELF 2
Multiply.
(a) (x 5)2
(b) (3a 2)2
(c) (y 7)2
(d) (5x 2y)2
A second special product will be very important in the next chapter, which deals with
factoring. Suppose the form of a product is
(x y)(x y)
The two terms differ
only in sign.
Let’s see what happens when we multiply.
(x y)(x y)
x2 xy xy y2
x2 y2
0
Because the middle term becomes 0, we have the following rule.
Rules and Properties: Special Product
The product of two binomials that differ only in the sign between the terms is
the square of the first term minus the square of the second term.
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(2 3) 2 3 because
52 4 9
2
NOTE You should see that
SPECIAL PRODUCTS
SECTION 3.5
295
Let’s look at the application of this rule in Example 3.
Example 3
Multiplying Polynomials
Multiply each pair of binomials.
(a) (x 5)(x 5) x2 52
Square of
the first term
Square of
the second term
x2 25
NOTE
(b) (x 2y)(x 2y) x2 (2y)2
(2y) (2y)(2y)
2
4y 2
Square of
the first term
Square of
the second term
x2 4y2
(c) (3m n)(3m n) 9m2 n2
(d) (4a 3b)(4a 3b) 16a2 9b2
CHECK YOURSELF 3
Find the products.
(a) (a 6)(a 6)
(c) (5n 2p)(5n 2p)
(b) (x 3y)(x 3y)
(d) (7b 3c)(7b 3c)
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When finding the product of three or more factors, it is useful to first look for the pattern
in which two binomials differ only in their sign. Finding this product first will make it
easier to find the product of all the factors.
Example 4
Multiplying Polynomials
(a) x (x 3)(x 3)
x(x2 9)
x3 9x
These binomials differ only in the sign.
CHAPTER 3
POLYNOMIALS
(b) (x 1) (x 5)(x 5)
(x 1)(x2 25)
These binomials differ only in the sign.
With two binomials, use the FOIL method.
x3 x2 25x 25
(c) (2x 1) (x 3) (2x 1)
(x 3)(2x 1)(2x 1)
These two binomials differ only in the sign of
the second term. We can use the commutative
property to rearrange the terms.
(x 3)(4x2 1)
4x3 12x2 x 3
CHECK YOURSELF 4
Multiply.
(a) 3x(x 5)(x 5)
(c) (x 7)(3x 1)(x 7)
(b) (x 4)(2x 3)(2x 3)
CHECK YOURSELF ANSWERS
1.
2.
3.
4.
(a) 4x 2 4x 1; (b) 16x 2 24xy 9y 2
(a) x 2 10x 25; (b) 9a 2 12a 4; (c) y 2 14y 49; (d) 25x 2 20xy 4y 2
(a) a 2 36; (b) x 2 9y 2; (c) 25n 2 4p 2; (d) 49b 2 9c 2
(a) 3x 3 75x; (b) 4x 3 16x 2 9x 36; (c) 3x 3 x 2 147x 49
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296
Name
3.5
Exercises
Section
Date
Find each of the following squares.
ANSWERS
1. (x 5)2
2. (y 9)2
1.
3. (w 6)2
2.
4. (a 8)2
3.
5. (z 12)2
6. ( p 20)2
7. (2a 1)
8. (3x 2)
4.
5.
2
2
9. (6m 1)2
6.
7.
10. (7b 2)2
8.
11. (3x y)2
9.
12. (5m n)2
10.
13. (2r 5s)2
14. (3a 4b)2
15. (8a 9b)
16. (7p 6q)
11.
12.
2
2
13.
14.
1
17. x 2
2
1
18. w 4
2
15.
16.
17.
Find each of the following products.
19. (x 6)(x 6)
20. ( y 8)( y 8)
18.
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19.
21. (m 12)(m 12)
22. (w 10)(w 10)
20.
21.
1
23. x 2
1
x
2
2
24. x 3
2
x
3
22.
23.
24.
297
ANSWERS
25.
25. (p 0.4)(p 0.4)
26. (m 0.6)(m 0.6)
27. (a 3b)(a 3b)
28. (p 4q)(p 4q)
29. (4r s)(4r s)
30. (7x y)(7x y)
31. (8w 5z)(8w 5z)
32. (7c 2d )(7c 2d)
33. (5x 9y)(5x 9y)
34. (6s 5t)(6s 5t)
35. x(x 2)(x 2)
36. a(a 5)(a 5)
37. 2s(s 3r)(s 3r)
38. 5w(2w z)(2w z)
39. 5r(r 3)2
40. 3x(x 2)2
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
For each of the following problems, let x represent the number, then write an expression
for the product.
39.
40.
41. The product of 6 more than a number and 6 less than that number
41.
42. The square of 5 more than a number
42.
43.
43. The square of 4 less than a number
44.
45.
44. The product of 5 less than a number and 5 more than that number
46.
48.
49.
45. (49)(51)
46. (27)(33)
47. (34)(26)
48. (98)(102)
49. (55)(65)
50. (64)(56)
50.
298
© 2001 McGraw-Hill Companies
Note that (28)(32) (30 2)(30 2) 900 4 896. Use this pattern to find each of
the following products.
47.
ANSWERS
51. Tree planting. Suppose an orchard is planted with trees in straight rows. If there are
5x 4 rows with 5x 4 trees in each row, how many trees are there in the orchard?
51.
52.
53.
54.
52. Area of a square. A square has sides of length 3x 2 centimeters (cm). Express the
55.
area of the square as a polynomial.
3x 2 cm
3x 2 cm
53. Complete the following statement: (a b)2 is not equal to a2 b2 because. . . . But,
wait! Isn’t (a b)2 sometimes equal to a2 b2 ? What do you think?
54. Is (a b)3 ever equal to a3 b3? Explain.
55. In the following figures, identify the length, width, and area of the square:
a
b
Length a
Width b
Area a
3
a
Width Area © 2001 McGraw-Hill Companies
3
x
x
Length x2
2x
Length 2x
Width Area 299
ANSWERS
56. The square below is x units on a side. The area is
56.
.
a.
Draw a picture of what happens when the sides are
doubled. The area is
.
b.
Continue the picture to show what happens when the sides are tripled.
The area is
.
c.
If the sides are quadrupled, the area is
d.
In general, if the sides are multiplied by n, the area is
.
e.
If each side is increased by 3, the area is increased by
.
f.
If each side is decreased by 2, the area is decreased by
g.
In general, if each side is increased by n, the area is increased by
each side is decreased by n, the area is decreased by
.
.
.
, and if
h.
x
x
Getting Ready for Section 3.6 [Section 1.7]
Divide.
(a)
2x2
2x
(b)
3a3
3a
(c)
6p3
2p2
(d)
10m4
5m2
(e)
20a3
5a3
(f)
6x2y
3xy
(g)
12r3s2
4rs
(h)
49c4d 6
7cd3
Answers
3. w2 12w 36
11. 9x2 6xy y2
15. 64a2 144ab 81b2
19. x2 36
21. m2 144
1
25. p2 0.16
29. 16r2 s2
4
64w2 25z2
33. 25x2 81y2
35. x3 4x
37. 2s3 18r2s
3
2
2
2
5r 30r 45r
41. x 36
43. x 8x 16
45. 2499
884
49. 3575
51. 25x2 40x 16
55.
a. x
b. a2
c. 3p
d. 2m2
f. 2x
300
1
4
2
27. a 9b2
17. x2 x 23. x2 31.
39.
47.
53.
5. z2 24z 144
7. 4a2 4a 1
2
13. 4r 20rs 25s2
g. 3r2s
h. 7c3d 3
e. 4
© 2001 McGraw-Hill Companies
1. x2 10x 25
9. 36m2 12m 1
3.6
Dividing Polynomials
3.6
OBJECTIVES
1. Find the quotient when a polynomial is divided
by a monomial
2. Find the quotient of two polynomials
In Section 1.7, we introduced the second property of exponents, which was used to divide
one monomial by another monomial. Let’s review that process.
Step by Step:
Step 1
Step 2
xm
xmn
xn
Divide the coefficients.
Use the second property of exponents to combine the variables.
Example 1
NOTE The second property
says: If x is not zero,
To Divide a Monomial by a Monomial
Dividing Monomials
Divide:
8
4
2
8x4
4x42
2x2
(a)
Subtract the exponents.
4x2
(b)
45a5b3
5a3b2
9a2b
CHECK YOURSELF 1
Divide.
(a)
16a5
8a3
(b)
28m4n3
7m3n
© 2001 McGraw-Hill Companies
Now let’s look at how this can be extended to divide any polynomial by a monomial. For
example, to divide 12a3 8a2 by 4a, proceed as follows:
NOTE Technically, this step
depends on the distributive
property and the definition
of division.
12a3
8a2
12a3 8a2
4a
4a
4a
Divide each term in
the numerator by the
denominator, 4a.
Now do each division.
3a2 2a
301
302
CHAPTER 3
POLYNOMIALS
The work above leads us to the following rule.
Step by Step:
To Divide a Polynomial by a Monomial
1. Divide each term of the polynomial by the monomial.
2. Simplify the results.
Example 2
Dividing by Monomials
Divide each term by 2.
(a)
4a2
8
4a2 8
2
2
2
2a2 4
Divide each term by 6y.
(b)
24y3 (18y2)
24y3
18y2
6y
6y
6y
4y2 3y
Remember the rules for
signs in division.
(c)
15x2 10x
15x2
10x
5x
5x
5x
3x 2
NOTE With practice you can
(d)
14x4 28x3 21x2
14x4
28x3
21x2
7x2
7x2
7x2
7x2
2x2 4x 3
(e)
9a3b4 6a2b3 12ab4
9a3b4
6a2b3
12ab4
3ab
3ab
3ab
3ab
3a2b3 2ab2 4b3
CHECK YOURSELF 2
Divide.
(a)
20y3 15y2
5y
(c)
16m4n3 12m3n2 8mn
4mn
(b)
8a3 12a2 4a
4a
© 2001 McGraw-Hill Companies
write just the quotient.
DIVIDING POLYNOMIALS
SECTION 3.6
303
We are now ready to look at dividing one polynomial by another polynomial (with more
than one term). The process is very much like long division in arithmetic, as Example 3
illustrates.
Example 3
Dividing by Binomials
Divide x2 7x 10 by x 2.
NOTE The first term in the
Step 1
dividend, x2, is divided by the
first term in the divisor, x.
Step 2
x
x 2B x2 7x 10
Divide x2 by x to get x.
x
x 2Bx2 7x 10
x2 2x
Multiply the divisor,
x 2, by x.
REMEMBER To subtract
Step 3
x2 2x, mentally change each
sign to x2 2x, and add. Take
your time and be careful here.
It’s where most errors are
made.
x
x 2Bx2 7x 10
x2 2x
5x 10
Subtract and bring down 10.
Step 4
x 5
x 2Bx2 7x 10
x2 2x
5x 10
NOTE Notice that we repeat
the process until the degree of
the remainder is less than that
of the divisor or until there is
no remainder.
Step 5
Divide 5x by x to get 5.
x 5
x 2Bx2 7x 10
x2 2x
5x 10
5x 10
0
© 2001 McGraw-Hill Companies
Multiply x 2 by 5 and then
subtract.
The quotient is x 5.
CHECK YOURSELF 3
Divide x2 9x 20 by x 4.
In Example 3, we showed all the steps separately to help you see the process. In practice,
the work can be shortened.
304
CHAPTER 3
POLYNOMIALS
Example 4
Dividing by Binomials
Divide x2 x 12 by x 3.
Step 1 Divide x2 by x to get x, the first
term of the quotient.
Step 2 Multiply x 3 by x.
Step 3 Subtract and bring down 12.
Remember to mentally change
the signs to x2 3x and add.
Step 4 Divide 4x by x to get 4, the
second term of the quotient.
Step 5 Multiply x 3 by 4 and
subtract.
4x 12
4x 12
0
The quotient is x 4.
CHECK YOURSELF 4
Divide.
(x2 2x 24) (x 4)
You may have a remainder in algebraic long division just as in arithmetic. Consider
Example 5.
Example 5
Dividing by Binomials
Divide 4x2 8x 11 by 2x 3.
2x 1
2x 3B4x2 8x 11
4x2 6x
Quotient
2x 11
2x 3
Divisor
8
Remainder
This result can be written as
4x2 8x 11
2x 3
2x 1 8
2x 3
Quotient
CHECK YOURSELF 5
Divide.
(6x2 7x 15) (3x 5)
Remainder
Divisor
© 2001 McGraw-Hill Companies
out a problem like 408 17, to
compare the steps.
x 4
x 3Bx x 12
x2 3x
2
NOTE You might want to write
DIVIDING POLYNOMIALS
SECTION 3.6
305
The division process shown in our previous examples can be extended to dividends of
a higher degree. The steps involved in the division process are exactly the same, as
Example 6 illustrates.
Example 6
Dividing by Binomials
Divide 6x3 x2 4x 5 by 3x 1.
2x2 x 1
3x 1B6x x2 4x 5
6x3 2x2
3
3x2 4x
3x2 x
3x 5
3x 1
6
The result can be written as
6
6x3 x2 4x 5
2x2 x 1 3x 1
3x 1
CHECK YOURSELF 6
Divide 4x3 2x2 2x 15 by 2x 3.
Suppose that the dividend is “missing” a term in some power of the variable. You can
use 0 as the coefficient for the missing term. Consider Example 7.
Example 7
Dividing by Binomials
Divide x3 2x2 5 by x 3.
x2 5x 15
x 3Bx 2x2 0x 5
x3 3x2
© 2001 McGraw-Hill Companies
3
5x2 0x
5x2 15x
Write 0x for the “missing”
term in x.
15x 5
15x 45
40
This result can be written as
40
x3 2x2 5
x2 5x 15 x3
x3
CHAPTER 3
POLYNOMIALS
CHECK YOURSELF 7
Divide.
(4x3 x 10) (2x 1)
You should always arrange the terms of the divisor and dividend in descending-exponent
form before starting the long division process, as illustrated in Example 8.
Example 8
Dividing by Binomials
Divide 5x2 x x3 5 by 1 x2.
Write the divisor as x2 1 and the dividend as x3 5x2 x 5.
x5
x2 1Bx3 5x2 x 5
x3
x
5x2
5x2
5
5
Write x3 x, the product
of x and x2 1, so that like
terms fall in the same
columns.
0
CHECK YOURSELF 8
Divide:
(5x2 10 2x3 4x) (2 x2)
CHECK YOURSELF ANSWERS
2. (a) 4y2 3y; (b) 2a2 3a 1; (c) 4m3n2 3m2n 2
20
6
3. x 5
4. x 6
5. 2x 1 6. 2x2 4x 7 3x 5
2x 3
11
7. 2x2 x 1 8. 2x 5
2x 1
1. (a) 2a2; (b) 4mn2
© 2001 McGraw-Hill Companies
306
Name
Exercises
3.6
Section
Date
Divide.
1.
18x6
9x2
2.
20a7
5a5
ANSWERS
1.
3.
35m3n2
7mn2
4.
42x5y2
6x3y
2.
3.
5.
3a 6
3
9b2 12
7.
3
16a3 24a2
9.
4a
4x 8
4
4.
10m2 5m
8.
5
6.
6.
5.
7.
8.
9x3 12x2
10.
3x
9.
10.
12m 6m
3m
2
11.
20b 25b
5b
3
12.
2
11.
12.
13.
18a4 12a3 6a2
6a
14.
21x5 28x4 14x3
7x
13.
14.
15.
20x4y2 15x2y3 10x3y
5x2y
16.
16m3n3 24m2n2 40mn3
8mn2
15.
16.
17.
Perform the indicated divisions.
17.
x2 5x 6
x2
18.
x2 8x 15
x3
18.
© 2001 McGraw-Hill Companies
19.
19.
x2 x 20
x4
2x2 5x 3
21.
2x 1
23.
2x2 3x 5
x3
20.
x2 2x 35
x5
20.
21.
3x2 20x 32
22.
3x 4
22.
3x2 17x 12
x6
24.
24.
23.
307
ANSWERS
25.
25.
4x2 18x 15
x5
26.
3x2 18x 32
x8
27.
6x2 x 10
3x 5
28.
4x2 6x 25
2x 7
29.
x3 x2 4x 4
x2
30.
x3 2x2 4x 21
x3
31.
4x3 7x2 10x 5
4x 1
32.
2x3 3x2 4x 4
2x 1
33.
x3 x2 5
x2
34.
x3 4x 3
x3
35.
25x3 x
5x 2
36.
8x3 6x2 2x
4x 1
37.
2x2 8 3x x3
x2
38.
x2 18x 2x3 32
x4
39.
x4 1
x1
40.
x4 x2 16
x2
41.
x3 3x2 x 3
x2 1
42.
x3 2x2 3x 6
x2 3
43.
x4 2x2 2
x2 3
44.
x4 x2 5
x2 2
45.
y3 1
y1
46.
y3 8
y2
47.
x4 1
x2 1
48.
x6 1
x3 1
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
43.
44.
45.
46.
47.
48.
308
© 2001 McGraw-Hill Companies
42.
ANSWERS
49.
y2 y c
y2
49. Find the value of c so that
y1
50. Find the value of c so that
50.
x3 x2 x c
x1
x2 1
51.
51. Write a summary of your work with polynomials. Explain how a polynomial is
recognized, and explain the rules for the arithmetic of polynomials—how to add,
subtract, multiply, and divide. What parts of this chapter do you feel you understand
very well, and what part(s) do you still have questions about, or feel unsure of?
Exchange papers with another student and compare your questions.
52.
53.
(a)
(b)
(c)
52. A funny (and useful) thing about division of polynomials: To find out about this
(d)
funny thing, do this division. Compare your answer with another
student’s.
54.
(x 2)B2x2 3x 5
Is there a remainder?
(a)
(b)
Now, evaluate the polynomial 2x2 3x 5 when x 2. Is this value the same as the
remainder?
(c)
Try (x 3)B5x2 2x 1
Is there a remainder?
Evaluate the polynomial 5x2 2x 1 when x 3. Is this value the same as the
remainder?
What happens when there is no remainder?
Try (x 6)B3x3 14x2 23x 6
(d)
Is the remainder zero?
Evaluate the polynomial 3x3 14x 23x 6 when x 6. Is this value zero? Write
a description of the patterns you see. When does the pattern hold? Make up several
more examples, and test your conjecture.
53. (a) Divide
x2 1
x1
(b) Divide
x3 1
x1
© 2001 McGraw-Hill Companies
(d) Based on your results to (a), (b), and (c), predict
54. (a) Divide
(c) Divide
x2 x 1
x1
(c) Divide
x50 1
x1
(b) Divide
x4 x3 x2 x 1
x1
(d) Based on your results to (a), (b), and (c), predict
x4 1
x1
x3 x2 x 1
x1
x10 x9 x8 x 1
x1
309
Answers
1. 2x4
3. 5m2
3
13. 3a 2a2 a
21. x 3
5. a 2
7. 3b2 4
9. 4a2 6a
11. 4m 2
2
2
15. 4x y 3y 2x
17. x 3
19. x 5
23. 2x 3 4
x3
25. 4x 2 5
x5
5
8
31. x2 2x 3 29. x2 x 2
3x 5
4x 1
9
2
33. x2 x 2 35. 5x2 2x 1 x2
5x 2
2
2
41. x 3
37. x 4x 5 39. x3 x2 x 1
x2
1
43. x2 1 2
45. y2 y 1
47. x2 1
49. c 2
x 3
2
3
2
51.
53. (a) x 1; (b) x x 1; (c) x x x 1;
27. 2x 3 © 2001 McGraw-Hill Companies
(d) x49 x48 x 1
310