Download Multiplying of Polynomials and Special Products

SECT
ION
5.4
5.4
5.4
OBJECTIVES
1. Find the product
of a monomial
and a polynomial
2. Find the product of
two polynomials
3. Square a polynomial
4. Find the product
of two binomials
that differ only in
sign
Multiplying of Polynomials and
Special Products
You have already had some experience in multiplying polynomials. In Section 5.1 we
stated the product rule for exponents and used that rule to find the product of two monomials. Let’s review briefly.
To Find the Product of Monomials
Step 1 Multiply the coefficients.
Step 2 Use the product rule for exponents to combine the variables:
axm bxn abxmn
Let’s look at an example in which we multiply two monomials.
Example 1
Multiplying Monomials
Multiply 3x2y and 2x3y5.
Write
Once again we have used
the commutative and
associative properties to
rewrite the problem.
(3x2y)(2x3y5)
(3 2)(x2 x3)(y y5)
6x y
5 6
✓ CHECK YOURSELF 1
■
Multiply.
(a) (5a2b)(3a2b4)
362
(b) (3xy)(4x3y5)









Multiply the coefficients.
Add the exponents.
Section 5.4
You might want to review
Section 0.5 before going on.
■
Multiplying of Polynomials and Special Products
363
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 0.5. That property leads us
to the following rule for multiplication.
To Multiply a Polynomial by a Monomial
Distributive property:
a(b c) ab ac
Example 2
Use the distributive property to multiply each term of the polynomial by the monomial and simplify the result.
Multiplying a Monomial and a Binomial
(a) Multiply 2x 3 by x.
Write
x(2x 3)
Note: With practice you will
do this step mentally.
x 2x x 3
Multiply x by 2x and then by 3,
the terms of the polynomial. That is,
“distribute” the multiplication over the sum.
2x2 3x
(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
364
Chapter 5
■
Polynomials
Again we have shown all the
steps of the process. With
practice you can write the
product directly, and you
should try to do so.
(b) 5c(4c2 8c)
(5c)(4c2) (5c)(8c) 20c3 40c2
(c) 3c2d2(7cd2 5c2d3)
3c2d2 7cd2 (3c2d2)(5c2d3) 21c3d4 15c4d5
✓ CHECK YOURSELF 3
■
Multiply.
Example 4
(a) 3(5a2 2a 7)
(b) 4x2(8x3 6)
(c) 5m(8m2 5m)
(d) 9a2b(3a3b 6a2b4)
Multiplying Binomials
(a) Multiply x 2 by x 3.
We can think of x 2 as a single quantity and apply the distributive property.



(x 2)(x 3)
Note that this ensures that
each term, x and 2, of the
first binomial is multiplied by
each term, x and 3, of the
second binomial.
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.
(a 3)(a 4)
(Think of a 3 as a single quantity and distribute.)
(a 3)a (a 3)(4)
a a 3 a [(a 4) (3 4)]
a2 3a (4a 12)
Note that the parentheses are needed
here because a minus sign precedes
the binomial.
✓ CHECK YOURSELF 4
■
Multiply.
(a) (x 4)(x 5)
(b) (y 5)(y 6)
Section 5.4
■
Multiplying of Polynomials and Special Products
365
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):
Remember this by F!
1.
(x 2)(x 3)
xx
Remember this by O!
2.
Find the product of the first terms of the factors.
(x 2)(x 3)
x3
Remember this by I!
3.
Find the product of the outer terms.
(x 2)(x 3)
2x
Remember this by L!
4.
Find the product of the inner terms.
(x 2)(x 3)
23
Find the product of the last terms.
Combining the four steps, we have
(x 2)(x 3)
It’s called FOIL to give you an
easy way of remembering the
steps: First, Outer, Inner, and
Last.
Example 5
x2 3x 2x 6
x2 5x 6
With practice, the FOIL method will let you write the products quickly and easily. Consider Example 5, which illustrates this approach.
Using the FOIL Method
Find the following products, using the FOIL method.
F
xx
L
45
(a) (x 4)(x 5)
4x
I
5x
O
When possible, you should
combine the outer and inner
products mentally and write
just the final product.
x 5x 4x 20
2
F
O
I
x 9x 20
2
L
366
Chapter 5
■
Polynomials
F
L
x x (7)(3)
b) (x 7)(x 3)
7x
I
Combine the outer and inner products as 4x.
3x
O
x 4x 21
2
✓ 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 leading
coefficients other than 1 or with more than one variable.
Example 6
Using the FOIL Method
Find the following products, using the FOIL method.
6
12x2
(a) (4x 3)(3x 2)
9x
Combine:
9x 9x x
8x
12x2 x 6
6x2
35y2
(b) (3x 5y)(2x 7y)
10xy
21xy
Combine:
10xy 21xy 31xy
6x2 31xy 35y2
Section 5.4
■
Multiplying of Polynomials and Special Products
367
The following rule summarizes our work in multiplying binomials.
To Multiply Two Binomials
Step 1 Find the first term of the product of the binomials by multiplying the first
terms of the binomials (F).
Step 2 Find the middle term of the product as the sum of the outer and inner
products (O I).
Step 3 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)
The FOIL method works well when multiplying any two binomials. But what if
one of the factors has three or more terms? The vertical format, shown in Example 7,
works for factors with any number of terms.
Example 7
Using the Vertical Method
Multiply x2 5x 8 by x 3.
Step 1
Step 2
Note: Using this vertical
method ensures that each
term of one factor multiples
each term of the other. That’s
why it works!
Step 3
x2 5x 8
x 3
———————
2
3x 15x 24
x2 5x 8
x 3
———————
2
3x 15x 24
x3 5x2 8x
——————————
x2 5x 8
x 3
———————
2
3x 15x 24
x3 5x2 8x
——————————
x3 2x2 7x 24
Multiply each term of x2 5x 8 by 3.
Now multiply each term by x.
Note that this line is shifted over so that like
terms are in the same columns.
Now add to combine like terms to write the
product.
368
Chapter 5
■
Polynomials
✓ CHECK YOURSELF 7
■
Multiply 2x2 5x 3 by 3x 4.
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)
x2 2xy y2
(x y)2 (x y)(x y)
x2 2xy y2
The patterns above lead us to the following rule.
To Square a Binomial
Step 1 Find the first term of the square by squaring the first term of the
binomial.
Step 2 Find the middle term of the square as twice the product of the two terms
of the binomial.
Step 3 Find the last term of the square by squaring the last term of the
binomial.
Squaring a Binomial
Example 8




(a) (x 3)2 x2 2 x 3 32
Caution
!
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
(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
Section 5.4
Again we have shown all the
steps. With practice you can
write just the square.
■
Multiplying of Polynomials and Special Products
369
(d) (5c 3d)2 (5c)2 2(5c)(3d) (3d)2
25c2 30cd 9d2
✓ CHECK YOURSELF 8
■
Multiply.
(a) (2x 1)2
Example 9
(b) (4x 3y)2
Squaring a Binomial
Find (y 4)2.
(y 4)2
is not equal to
y2 42 or
y2 16
The correct square is
(y 4)2 y2 8y 16
You should see that
(2 3)2 22 32 because
52 4 9
The middle term is twice the product of y and 4.
✓ CHECK YOURSELF 9
■
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 factors differ only in sign.
Let’s see what happens when we multiply.
(x y)(x y)





x2 xy xy y2
0
x y
2
2
370
Chapter 5
■
Polynomials
Since the middle term becomes 0, we have the following rule.
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.
Let’s look at the application of this rule in Example 10.
Example 10
Multiplying Polynomials
Multiply each pair of factors.
(a) (x 5)(x 5) x2 52
Square of
the first term
Square of
the second term
x2 25
Note: (2y)2 (2y)(2y)
4y2
(b) (x 2y)(x 2y) x2 (2y)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 10
■
Find the products.
(a) (a 6)(a 6)
(b) (x 3y)(x 3y)
(c) (5n 2p)(5n 2p)
(d) (7b 3c)(7b 3c)
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.
Section 5.4
Example 11
Multiplying of Polynomials and Special Products
■
371
Multiplying Polynomials





(a) x(x 3)(x 3)
TThese binomials differ only in the sign.
x(x 9)
2
x3 9x





(b) (x 1)(x 5)(x 5)
These binomials differ only in the sign.
(x 1)(x2 25)
With two binomials, use the FOIL method.
x x 25x 25
3
2
(c) (2x 1)(x 3)(2x 1)
(2x 1) (x 3) (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)(2x 1)(2x 1)
(x 3)(4x2 1)
4x3 12x2 x 3
✓ CHECK YOURSELF 11
■
Multiply.
(a) 3x(x 5)(x 5)
(b) (x 4)(2x 3)(2x 3)
(c) (x 7)(3x 1)(x 7)
✓ CHECK YOURSELF ANSWERS
■
1. (a) 15a4b5; (b) 12x4y6.
2. (a) 2y3 6y2; (b) 6w5 15w3.
3. (a) 15a2 6a 21; (b) 32x5 24x2; (c) 40m3 25m2; (d) 27a5b2 54a4b5.
4. (a) x2 9x 20; (b) y2 y 30.
(c) x2 10x 16.
5. (a) x2 13x 42; (b) x2 2x 15;
6. (a) 15x2 29x 14; (b) 20a2 31ab 12b2;
(c) 6m2 19mn 15n2.
(b) 16x2 24xy 9y2.
7. 6x3 7x2 11x 12.
9. (a) x2 10x 25; (b) 9a2 12a 4;
(c) y2 14y 49; (d) 25x2 20xy 4y2.
(c) 25n2 4p2; (d) 49b2 9c2.
(c) 3x3 x2 147x 49.
8. (a) 4x2 4x 1;
10. (a) a2 36; (b) x2 9y2;
11. (a) 3x3 75x; (b) 4x3 16x2 9x 36;
Exercises
1. 15x5
2. 28a11
3. 28b10
4. 56y10
13
■
5.4
Multiply.
1. (5x2)(3x3)
2. (7a5)(4a6)
3. (2b2)(14b8)
4. (14y4)(4y6)
5. (10p6)(4p7)
6. (6m8)(9m7)
7. (4m5)(3m)
8. (5r7)(3r)
15
6. 54m
5. 40p
6
8
7. 12m
8. 15r
5 3
10. 21r6s7
9. 32x y
11. 6m9n3
12. 42a7b6
13. 10x 30
14. 28b 20
15. 12a2 15a
16. 10x2 35x
11. (3m5n2)(2m4n)
12. (7a3b5)(6a4b)
17. 12s4 21s3
18. 27a5 45a3
13. 5(2x 6)
14. 4(7b 5)
15. 3a(4a 5)
16. 5x(2x 7)
17. 3s2(4s2 7s)
18. 9a2(3a3 5a)
19. 2x(4x2 2x 1)
20. 5m(4m3 3m2 2)
21. 3xy(2x2y xy2 5xy)
22. 5ab2(ab 3a 5b)
23. 6m2n(3m2n 2mn mn2)
24. 8pq2(2pq 3p 5q)
19. 8x3 4x2 2x
4
3
20. 20m 15m 10m
21. 6x3y2 3x2y3 15x2y2
2 3
2 2
3
22. 5a b 15a b 25ab
4 2
3 2
3 3
23. 18m n 12m n 6m n
24. 16p2q3 24p2q2 40pq3
25. x2 5x 6
26. a2 10a 21
27. m2 14m 45 28. b2 12b 35
9. (4x3y2)(8x2y)
10. (3r4s2)(7r2s5)
Multiply.
25. (x 3)(x 2)
26. (a 3)(a 7)
31. w2 30w 200
27. (m 5)(m 9)
28. (b 7)(b 5)
32. s2 20s 96
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)
29. p2 p 56
30. x2 x 90
2
33. 3x 29x 40
2
34. 4w 13w 35
35. 6x2 x 12
36. 15a2 38a 7
37. 12a2 31ab 9b2
38. 21s2 47st 24t2
39. 21p2 13pq 20q2
40. 10x2 13xy 4y2
41. 6x2 23xy 20y2
42. 16x2 8xy 15y2
372
Section 5.4
■
Multiplying of Polynomials and Special Products
43. x2 10x 25
Find each of the following squares.
44. y2 18y 81
43. (x 5)2
44. (y 9)2
45. (w 6)2
46. (a 8)2
47. z2 24z 144
47. (z 12)2
48. (p 20)2
48. p2 40p 400
49. (2a 1)2
50. (3x 2)2
51. (6m 1)2
52. (7b 2)2
51. 36m2 12m 1
53. (3x y)2
54. (5m n)2
52. 49b2 28b 4
55. (2r 5s)2
56. (3a 4b)2
57. (8a 9b)2
58. (7p 6q)2
45. w2 12w 36
46. a2 16a 64
49. 4a2 4a 1
50. 9x2 12x 4
53. 9x2 6xy y2
54. 25m2 10mn n2
55. 4r2 20rs 25s2
56. 9a2 24ab 16b2
1
59. x 2
2
1
60. w 4
2
57. 64a2 144ab 81b2
58. 49p2 84pq 36q2
Find each of the following products.
1
59. x2 x 4
61. (x 6)(x 6)
62. (y 8)(y 8)
1
1
60. w2 w 2
16
63. (m 12)(m 12)
64. (w 10)(w 10)
61. x2 36
1
1
65. x x 2
2
2
2
66. x x 3
3
64. w2 100
1
4
65. x2 66. x2 4
9
67. (p 0.4)(p 0.4)
68. (m 0.6)(m 0.6)
69. (a 3b)(a 3b)
70. (p 4q)(p 4q)
67. p2 0.16
71. (4r s)(4r s)
72. (7x y)(7x y)
73. (8w 5z)(8w 5z)
74. (7c 2d)(7c 2d)
75. (5x 9y)(5x 9y)
76. (6s 5t)(6s 5t)
62. y2 64
63. m2 144
2
68. m 0.36
69. a2 9b2
70. p2 16q2
71. 16r2 s2
72. 49x2 y2
73. 64w2 25z2
74. 49c2 4d2
75. 25x2 81y2
76. 36s2 25t2
373
374
Chapter 5
■
Polynomials
77. 24x3 10x2 4x
Multiply.
78. 12x3 3x
77. 2x(3x 2)(4x 1)
78. 3x(2x 1)(2x 1)
79. 80a 45a
79. 5a(4a 3)(4a 3)
80. 6m(3m 2)(3m 7)
80. 54m3 162m2 84m
81. 3s(5s 2)(4s 1)
82. 7w(2w 3)(2w 3)
81. 60s 39s 6s
83. (x 2)(x 1)(x 3)
84. (y 3)(y 2)(y 4)
82. 28w3 63w
85. (a 1)
86. (x 1)3
3
3
2
3
83. x3 4x2 x 6
84. y3 3y2 10y 24
3
Multiply the following.
2
85. a 3a 3a 1
86. x3 3x2 3x 1
x2
11x
4
87. 3
45
15
x2
29
9
88. x 4
80
20
x
2 2x
2
87. 2
3 3
5
89. [x (y 2)][x (y 2)]
x
3 3x
3
88. 3
4 4
5
90. [x (3 y)][x (3 y)]
Label the following as true or false.
89. x2 y2 4y 4
90. x2 y2 6y 9
91. (x y)2 x2 y2
92. (x y)2 x2 y2
91. False
93. (x y)2 x2 2xy y2
94. (x y)2 x2 2xy y2
92. False
95. 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.
93. True
94. True
95. 6x2 11x 35 cm2
5
21
96. 3y y in2
2
2
2
97. 10x 3x2
3
98. 100x 2x
99. 25x2 40x 16
96. Area. The base of a triangle measures 3y 7 inches (in.) and the height is
2y 3 in. Express the area of the triangle in terms of y.
97. Revenue. The price of an item is given by p 10 3x. 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.
98. Revenue. The price of an item is given by p 100 2x2. Find the polynomial
that represents the revenue generated from the sale of x items.
99. 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?
Section 5.4
100. 9x2 12x 4 cm2
2
101. x(x 2) or x 2x
102. x(3x 6) or 3x2 6x
■
Multiplying of Polynomials and Special Products
375
100. Area of a square. A square has sides of length 3x 2 centimeters (cm). Express
the area of the square as a polynomial.
101. Area of a rectangle. The length and width of a rectangle are given by two consecutive odd integers. Write an expression for the area of the rectangle.
102. Area of a rectangle. The length of a rectangle is 6 less than three times the width.
Write an expression for the area of the rectangle.
103. 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.
30 in.
x
20 in.
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.
Length of Side of
Corner Square
Length of
Box
Width of
Box
Depth of
Box
Volume of
Box
1 in.
2 in.
3 in.
n in.
Write general formulas 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?
Extend the table to decide what the dimensions are for a box with maximum volume. Draw a sketch of this box and write in the dimensions.
104. 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?
105. Is (a b)3 ever equal to a3 b3? Explain.
376
Chapter 5
■
Polynomials
106. In the following figures, identify the length and the width of the square:
a
b
Length __________
a
Width __________
Area __________
b
x
2
Length __________
x
Width __________
Area __________
2
107. The square shown is x units on a side. The area is __________.
Draw a picture of what happens when the sides are doubled. The area is
__________.
Continue the picture to show what happens when the sides are tripled. The area
is __________.
x
If the sides are quadrupled, the area is __________.
In general, if the sides are multiplied by n, the area is __________.
x
If each side is increased by 3, the area is increased by __________.
If each side is decreased by 2, the area is decreased by __________.
In general, if each side is increased by n, the area is increased by __________,
and if each side is decreased by n, the area is decreased by __________.
108. x2 36
109. x2 10x 25
For each of the following problems, let x represent the number, then write an expression for the product.
110. x2 8x 16
108. The product of 6 more than a number and 6 less than that number
111. x2 25
109. The square of 5 more than a number
112. 2499
110. The square of 4 less than a number
113. 891
111. The product of 5 less than a number and 5 more than that number
114. 884
115. 9996
Note that (28)(32) (30 2)(30 2) 900 4 896. Use this pattern to find each
of the following products.
116. 3575
112. (49)(51)
113. (27)(33)
117. 3584
114. (34)(26)
115. (98)(102)
116. (55)(65)
117. (64)(56)