Download Multiplication of Polynomials and Special Products

6.4
Multiplication of Polynomials
and Special Products
6.4
OBJECTIVES
1.
2.
3.
4.
Evaluate f(x) g(x) for a given x
Multiply two polynomial functions
Square a binomial
Find the product of two binomials as a difference
of squares
In Section 1.4, you saw the first exponent property and used that property to multiply
monomials. Let’s review.
Example 1
Multiplying Monomials
Multiply.
Add exponents.
(8x2y)(4x3y4) (8 4)(x23)(y14)
NOTE
a a a
m n
mn
Multiply.
Notice the use of the associative
and commutative properties to
“regroup” and “reorder” the
factors.
32x y
5 5
CHECK YOURSELF 1
Multiply.
(a) (4a3b)(9a3b2)
(b) (5m3n)(7mn5)
We now want to extend the process to multiplying polynomial functions.
Example 2
Multiplying a Monomial and a Binomial Function
Given f(x) 5x2 and g(x) 3x2 5x, and letting h(x) f(x) g(x), find h(x).
h(x) f(x) g(x)
5x2 (3x2 5x)
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Apply the distributive property.
5x 3x 5x 5x
2
2
2
15x4 25x3
CHECK YOURSELF 2
Given f(x) 3x2 and g(x) 4x2 x, and letting h(x) f(x) g(x), find h(x).
We can check this result by comparing the values of h(x) and of f(x) g(x) for a specific
value of x. This is illustrated in Example 3.
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CHAPTER 6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
Example 3
Multiplying a Monomial and a Binomial Function
Given f(x) 5x2 and g(x) 3x25x, and letting h(x) f(x) g(x), compare f(1) g(1)
with h(1).
f(1) g(1) 5(1)2 (3(1)2 5(1))
5(3 5)
5(2)
10
From Example 2, we know that
h(x) 15x4 25x3
So
h(1) 15(1)4 25(1)3
15 25
10
Therefore, h(1) f(1) g(1).
CHECK YOURSELF 3
Given f(x) 3x2 and g(x) 4x2 x, and letting h(x) f(x) g(x), compare f (1) g(1)
with h(1).
The distributive property is also used to multiply two polynomial functions. To consider
the pattern, let’s start with the product of two binomial functions.
Example 4
Multiplying Binomial Functions
Given f(x) x 3 and g(x) 2x 5, and letting h(x) f(x) g(x), find the following.
(a) h(x)
h(x) f(x) g(x)
(x 3)(2x 5)
Apply the distributive property.
(x 3)(2x) (x 3)(5)
Apply the distributive property again.
(x)(2x) (3)(2x) (x)(5) (3)(5)
2x2 6x 5x 15
2x2 11x 15
Notice that this ensures that each term in the first polynomial is multiplied by each term
in the second polynomial.
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404
MULTIPLICATION OF POLYNOMIALS AND SPECIAL PRODUCTS
SECTION 6.4
405
(b) f(1) g(1)
f(1) g(1) (1 3)(2(1) 5)
4(7)
28
(c) h(1)
From part (a), we have h(x) 2x2 11x 15, so
h(1) 2(1)2 11(1) 15
2 11 15
28
Again, we see that h(1) f(1) g(1).
CHECK YOURSELF 4
Given f (x) 3x 2 and g(x) x 3, and letting h(x) f(x) g(x), find the following.
(b) f(1) g(1)
(a) h(x)
(c) h(1)
Certain products occur frequently enough in algebra that it is worth learning special
formulas for dealing with them. Consider these products of two equal binomial factors.
(a b)2 (a b)(a b)
NOTE
a 2ab b
2
2
and
a2 2ab b2
a2 2ab b2
(a b)2 (a b)(a b)
a2 2ab b2
are called perfect-square
trinomials.
We can summarize these statements as follows.
Rules and Properties: Squaring a Binomial
The square of a binomial has three terms, (1) the square of the first term, (2)
twice the product of the two terms, and (3) the square of the last term.
(a b)2 a2 2ab b2
and
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(a b)2 a2 2ab b2
Example 5
Squaring a Binomial
Find each of the following binomial squares.
NOTE Be sure to write out the
expansion in detail.
(1)
(a) (x 5)2 x2 2(x)(5) 52
Square of first term Twice the product Square of last term
of the two terms
x2 10x 25
(2)
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CHAPTER 6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CAUTION
(b) (2a 7)2 (2a)2 2(2a)(7) (7)2
4a2 28a 49
Be Careful! A very common
mistake in squaring binomials is
to forget the middle term!
(y 7)2
is not equal to
CHECK YOURSELF 5
y2 (7)2
Find each of the following binomial squares.
The correct square is
(a) (x 8)2
y2 14y 49
(b) (3x 5)2
The square of a binomial is
always a trinomial.
Another special product involves binomials that differ only in sign. It will be extremely
important in your work later in this chapter on factoring. Consider the following:
(a b)(a b) a2 ab ab b2
a2 b2
Rules and Properties: Product of Binomials Differing in Sign
(a b)(a b) a2 b2
In words, the product of two binomials that differ only in the signs of their
second terms is the difference of the squares of the two terms of the binomials.
Example 6
Finding a Special Product
Multiply.
(a) (x 3)(x 3) x2 (3)2
x2 9
CAUTION
The entire term 2x is squared,
not just the x.
(b) (2x 3y)(2x 3y) (2x)2 (3y)2
4x2 9y2
(c) (5a 4b2)(5a 4b2) (5a)2 (4b2)2
CHECK YOURSELF 6
Find each of the following products.
(a) (y 5)(y 5)
NOTE This format ensures that
each term of one polynomial
multiplies each term of the
other.
(b) (2x 3)(2x 3)
(c) (4r 5s2)(4r 5s2)
When multiplying two polynomials that don’t fit one of the special product patterns,
there are two different ways to set up the multiplication. Example 7 will illustrate the vertical approach.
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25a2 16b4
MULTIPLICATION OF POLYNOMIALS AND SPECIAL PRODUCTS
SECTION 6.4
407
Example 7
Multiplying Polynomials
Multiply 3x3 2x2 5 and 3x 2.
Step 1
3x3 2x2
6x 2
3
Step 2
Step 3
5
3x 2
10
4x
5
3x 2
6x3 4x2
10
4
9x 6x3
15x
Multiply by 2.
3x3 2x2
5
3x 2
6x3 4x2
10
4
9x 6x3
15x
9x4
4x2 15x 10
Multiply by 3x. Note that we align
the terms in the partial product.
3x3 2x2
Add the partial products.
CHECK YOURSELF 7
Find the following product, using the vertical method.
(4x3 6x 7)(3x 2)
A horizontal approach to the multiplication in Example 7 is also possible by the distributive property. As we see in Example 8, we first distribute 3x over the trinomial and then we
distribute 2 over the trinomial.
Example 8
Multiplying Polynomials
Multiply (3x 2)(3x3 2x2 5), using a horizontal format.
(3x 2)(3x3 2x2 5)
Step 2
9x4 6x3 15x 6x3 4x2 10
NOTE Again, this ensures that
each term of one polynomial
multiplies each term of the
other.
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Step 1
Step 1
Step 2
9x4 4x2 15x 10
Combine like terms.
Write the product in descending
form.
CHAPTER 6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CHECK YOURSELF 8
Find the product of Check Yourself 7, using a horizontal format.
Multiplication sometimes involves the product of more than two polynomials. In such
cases, the associative property of multiplication allows us to regroup the factors to make
the multiplication easier. Generally, we choose to start with the product of binomials. Example 9 illustrates this approach.
Example 9
Multiplying Polynomials
Find the products.
(a) x(x 3)(x 3) x(x2 9)
x3 9x
(b) 2x(x 3)(2x 1) 2x(2x2 5x 3)
4x3 10x2 6x
Find the product (x 3)(x 3).
Then distribute x as the last step.
Find the product of the binomials.
Then distribute 2x.
CHECK YOURSELF 9
Find each of the following products.
(a) m(2m 3)(2m 3)
(b) 3a(2a 5)(a 3)
CHECK YOURSELF ANSWERS
1.
4.
5.
6.
8.
(a) 36a6b3; (b) 35m4n6
2. h(x) 12x4 3x3
3. f(1) g(1) 15 h(1)
2
(a) h(x) 3x 7x 6; (b) f(1) g(1) 4; (c) h(1) 4
(a) x2 16x 64; (b) 9x2 30x 25
(a) y2 25; (b) 4x2 9; (c) 16r2 25s4
7. 12x4 8x3 18x2 9x 14
4
3
2
3
12x 8x 18x 9x 14
9. (a) 4m 9m; (b) 6a3 3a2 45a
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408
Name
6.4 Exercises
Section
Date
In exercises 1 to 6, find each product.
1. (4x)(5y)
ANSWERS
2. (3m)(5n)
3. (6x2)(3x3)
4. (5y4)(3y2)
1.
2.
3.
4.
5.
5. (5r2s)(6r3s4)
6. (8a2b5)(3a3b2)
6.
7.
In exercises 7 to 14, f(x) and g(x) are given. Let h(x) f(x) g(x). Find (a) h(x),
(b) f(1) g(1), and (c) use the result of (a) to find h(1).
7. f(x) 3x and g(x) 2x2 3x
8. f(x) 4x and g(x) 2x2 7x
8.
9.
10.
9. f(x) 5x and g(x) 3x 5x 8
2
10. f(x) 2x and g(x) 7x 2x
2
2
11.
11. f(x) 4x3 and g(x) 9x2 3x 5
12. f(x) 2x3 and g(x) 2x3 4x
12.
13.
13. f(x) 3x3 and g(x) 5x2 4x
14. f(x) x2 and g(x) 7x3 5x2
14.
15.
In exercises 15 to 24, find each product.
15. (x y)(x 3y)
16.
16. (x 3y)(x 5y)
17.
18.
17. (x 2y)(x 7y)
18. (x 7y)(x 3y)
19.
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20.
19. (5x 7y)(5x 9y)
20. (3x 5y)(7x 2y)
21.
22.
21. (7x 5y)(7x 4y)
22. (9x 7y)(3x 2y)
23. (5x2 2y)(3x 2y2)
24. (6x2 5y2)(3x2 2y)
23.
24.
409
ANSWERS
25.
In exercises 25 to 38, multiply polynomial expressions using the special product formulas.
26.
25. (x 5)2
26. (x 7)2
27. (2x 3)2
28. (5x 3)2
29. (4x 3y)2
30. (7x 5y)2
31. (4x 3y2)2
32. (3x3 7y)2
33. (x 3y)(x 3y)
34. (x 5y)(x 5y)
35. (2x 3y)(2x 3y)
36. (5x 3y)(5x 3y)
37. (4x2 3y)(4x2 3y)
38. (7x 6y2)(7x 6y2)
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
In exercises 39 to 42, multiply using the vertical format.
40.
39. (3x y)(x2 3xy y2)
40. (5x y)(x2 3xy y2)
41. (x 2y)(x2 2xy 4y2)
42. (x 3y)(x2 3xy 9y2)
41.
42.
43.
In exercises 43 to 46, simplify each function.
44.
43. f(x) x(x 3)(x 1)
44. f(x) x(x 4)(x 2)
45. f(x) 2x(x 5)(x 4)
46. f(x) x2(x 4)(x2 5)
45.
46.
Multiply the following.
48.
49.
47.
50.
2 3 3 5
x
2
2x
2
49. [x (y 2)][x (y 2)]
410
48.
3 4 4 5
x
3
3x
3
50. [x (3 y)][x (3 y)]
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47.
ANSWERS
If the polynomial p(x) represents the selling price of an object, then the polynomial R(x),
in which R(x) x p(x), is the revenue produced by selling x objects. Use this information
to solve exercises 51 and 52.
51. If p(x) 100 0.2x, find R(x).
52. If p(x) 250 0.5x, find R(x).
Find R(50).
Find R(20).
52.
53.
54.
55.
In exercises 53 to 56, label the statements as true or false.
53. (x y)2 x2 y2
51.
56.
54. (x y)2 x2 y2
57.
55. (x y) x 2xy y
2
2
2
56. (x y) x 2xy y
2
2
2
58.
57. Area. The length of a rectangle is given by 3x 5 centimeters (cm) and the width is
59.
given by 2x 7 cm. Express the area of the rectangle in terms of x.
60.
58. Area. The base of a triangle measures 3y 7 inches (in.) and the height is 2y 3 in.
61.
Express the area of the triangle in terms of y.
62.
2y 3
3y 7
59. 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 that represents the revenue.
60. 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.
61. Tree planting. Suppose an orchard is planted with trees in straight rows. If there are
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5x 4 rows with 5x 4 trees in each row, how many trees are there in the orchard?
62. Area of a square. A square has sides of length 3x 2 centimeters (cm). Express the
area of the square as a polynomial.
411
ANSWERS
63.
63. 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.
64.
64. Area of a rectangle. The length of a rectangle is 6 less than three times the width.
65.
Write an expression for the area of the rectangle.
65. Work with another student to complete this table and write the polynomial. A paper
66.
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.
67.
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.
66. 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?
67. Is (a b)3 ever equal to a3 b3? Explain.
412
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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.
ANSWERS
68. In the following figures, identify the length and the width of the square, and then find
the area.
a
68.
69.
b
Length ________
70.
a
Width ________
b
Area ________
2
x
71.
72.
73.
Length ________
x
Width ________
2
Area ________
69. The square shown is x units on a side. The area is _______.
x
x
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
_______.
If the sides are quadrupled, the area is _______.
In general, if the sides are multiplied by n, the area is _______.
If each side is increased by 3, the area is increased by _______.
If each side is decreased by 2, the area is decreased by _______.
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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 _______.
For each of the following problems, let x represent the number, then write an expression
for the product.
70. The product of 6 more than a number and 6 less than that number.
71. The square of 5 more than a number.
72. The square of 4 less than a number.
73. The product of 5 less than a number and 5 more than that number.
413
ANSWERS
74.
Note that (28)(32) (30 2)(30 2) 900 4 896. Use this pattern to find each of
the following products.
75.
76.
74. (49)(51)
75. (27)(33)
76. (34)(26)
77. (98)(102)
78. (55)(65)
79. (64)(56)
77.
78.
79.
Answers
1. 20xy
3. 18x5
5. 30r5s5
7. (a) 6x3 9x2; (b) 3; (c) 3
3
2
9. (a) 15x 25x 40x; (b) 0; (c) 0
11. (a) 36x5 12x4 20x3; (b) 28; (c) 28
5
4
2
13. (a) 15x 12x ; (b) 3; (c) 3
15. x 4xy 3y2
2
2
2
17. x 5xy 14y
19. 25x 80xy 63y2
21. 49x2 63xy 20y2
3
2 2
3
2
23. 15x 10x y 6xy 4y
25. x 10x 25
27. 4x2 12x 9
2
2
2
2
4
29. 16x 24xy 9y
31. 16x 24xy 9y
33. x2 9y2
2
2
4
2
3
2
35. 4x 9y
37. 16x 9y
39. 3x 8x y 6xy2 y3
3
3
3
2
41. x 8y
43. x 2x 3x
45. 2x3 2x2 40x
x2
11x
4
49. x2 y2 4y 4
51. 100x 0.2x2, 4500
3
45
15
53. False
55. True
57. 6x2 11x 35 cm2
59. 10x 3x2
2
2
61. 25x 40x 16
63. x(x 2) or x 2x
65.
47.
67.
73. x2 25
75. 891
77. 9996
79. 3584
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71. x2 10x 25
69. x2; 4x2; 9x2; 16x2; n2x2; 6x 9; 4x 4; 2xn n2; 2xn n2
414