Download Multiplication and Division of Rational Expressions and Functions

7.2
Multiplication and Division of
Rational Expressions and Functions
7.2
OBJECTIVES
1.
2.
3.
4.
Multiply two rational expressions
Divide two rational expressions
Multiply two rational functions
Divide two rational functions
Once again, let’s turn to an example from arithmetic to begin our discussion of multiplying
rational expressions. Recall that to multiply two fractions, we multiply the numerators and
multiply the denominators. For instance,
2 3
2 3
6
5 7
5 7
35
In algebra, the pattern is exactly the same.
Rules and Properties:
Multiplying Rational Expressions
For polynomials P, Q, R, and S,
NOTE For all problems with
rational expressions, assume
denominators are not 0.
P R
PR
Q S
QS
when Q 0 and
S0
Example 1
Multiplying Rational Expressions
Multiply.
2x3 10y
20x3y
5y2 3x2
15x2y2
5x2y 4x
5x2y 3y
4x
3y
Divide by the common factor
5x2y to simplify.
© 2001 McGraw-Hill Companies
CHECK YOURSELF 1
Multiply.
9a2b3 20ab2
5ab4 27ab3
NOTE The factoring methods
in Chapter 6 are used to
simplify rational expressions.
Generally, you will find it best to divide by any common factors before you multiply, as
Example 2 illustrates.
499
500
CHAPTER 7
RATIONAL EXPRESSIONS AND FUNCTIONS
Example 2
Multiplying Rational Expressions
Multiply as indicated.
x
6x 18
x 3x
9x
(a)
Factor.
2
1
2
x
6(x 3)
x(x 3)
9x
1
(b)
1
Divide by the common factors
of 3, x, and x 3.
3
2
3x
x2 y2
10xy
2
2
5x 5xy x 2xy y2
1
1
Factor and divide by the common
factors of 5, x, x y, and x y.
2
(x y)(x y)
10xy
5x(x y)
(x y)(x y)
1
1
1
2y
xy
4
10x 5x2
x2 2x 8x 24
(c)
that
2x
1
x2
1
1
4
5x(2 x)
x(x 2) 8(x 3)
2
5
2(x 3)
CHECK YOURSELF 2
Multiply as indicated.
(a)
x2 5x 14 8x 56
2
4x2
x 49
(b)
3x x2
x
2x 6
2
The following algorithm summarizes our work in multiplying rational expressions.
Step by Step: Multiplying Rational Expressions
Step 1 Write each numerator and denominator in completely factored form.
Step 2 Divide by any common factors appearing in both the numerator and
denominator.
Step 3 Multiply as needed to form the product.
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NOTE From Section 7.1, recall
MULTIPLICATION AND DIVISION OF RATIONAL EXPRESSIONS AND FUNCTIONS
SECTION 7.2
501
In dividing rational expressions, you can again use your experience from arithmetic.
Recall that
NOTE We invert the divisor
(the second fraction) and
multiply.
3
2
3 3
9
5
3
5 2
10
Once more, the pattern in algebra is identical.
Rules and Properties:
Dividing Rational Expressions
For polynomials P, Q, R, and S,
P
R
P S
PS
Q
S
Q R
QR
when Q 0 R 0 and
S0
To divide rational expressions, invert the divisor and multiply as before, as Example 3
illustrates.
Example 3
Dividing Rational Expressions
Divide as indicated.
NOTE Invert the divisor and
(a)
3x2
9x2y2
3x2
4y4
y
3
3 4 3 8x y
4y
8x y 9x2y2
6x
(b)
2x2 4xy
4x 8y
2x2 4xy 3x 6y
9x 18y
3x 6y
9x 18y 4x 8y
multiply.
CAUTION
Be Careful! Invert the divisor,
then factor.
1
1
x
2x(x 2y) 3(x 2y)
9(x 2y) 4(x 2y)
6
3
(c)
1
2
1
2x2 x 6
x2 4
2x2 x 6
4x
2
2
2
4x 6x
4x
4x 6x
x 4
1
1
2
(2x 3) (x 2)
4x
2
2x (2x 3)
(x 2)(x 2)
x2
© 2001 McGraw-Hill Companies
1
1
1
CHECK YOURSELF 3
Divide and simplify.
(a)
5xy
10y2
3 7x
14x3
(c)
x2 2x 15
x2 9
3
x 27
2x2 10x
(b)
3x 9y
x2 3xy
2
2x 10y
4x 20xy
CHAPTER 7
RATIONAL EXPRESSIONS AND FUNCTIONS
We summarize our work in dividing fractions with the following algorithm.
Step by Step: Dividing Rational Expressions
Step 1 Invert the divisor (the second rational expression) to write the problem
as one of multiplication.
Step 2 Proceed as in the algorithm for the multiplication of rational
expressions.
The product of two rational functions is always a rational function. Given two rational functions, f(x) and g(x), we can rename the product, so
h(x) f(x) g(x)
This will always be true for values of x for which both f and g are defined. So, for
example, h(1) f(1) g(1) as long as both f(1) and g(1) exist.
Example 4 illustrates this concept.
Example 4
Multiplying Rational Functions
Given the rational functions
f(x) x2 3x 10
x1
g(x) and
x2 4x 5
x5
find the following.
(a) f(0) g(0)
Because f(0) 10 and g(0) 1, then f(0) g(0) (10)(1) 10.
(b) f(5) g(5)
Although we can find f(5), g(5) is undefined. 5 is an excluded value for the domain of
the function g. Therefore, f(5) g(5) is undefined.
(c) h(x) f(x) g(x)
h(x) f(x) g(x)
x2 3x 10 x2 4x 5
x1
x5
1
1
(x 5)(x 2) (x 1)(x 5)
(x 1)
(x 5)
1
(x 5)(x 2)
1
x 1, x 5
(d) h(0)
h(0) (0 5)(0 2) 10
(e) h(5)
Although the temptation is to substitute 5 for x in part (c), notice that the function is undefined when x is 1 or 5. As was true in part (b), the function is undefined at that point.
© 2001 McGraw-Hill Companies
502
MULTIPLICATION AND DIVISION OF RATIONAL EXPRESSIONS AND FUNCTIONS
SECTION 7.2
503
CHECK YOURSELF 4
Given the rational functions
f(x) x2 2x 8
x2
g(x) and
x2 3x 10
x4
find the following.
(a) f(0) g(0)
(b) f(4) g(4)
(c) h(x) f(x) g(x)
(d) h(0)
(e) h(4)
When we divide two rational functions to create a third rational function, we must be certain
to exclude values for which the denominator is equal to zero, as Example 5 illustrates.
Example 5
Dividing Polynomial Functions
Given the rational functions
f(x) x3 2x2
x2
and
g(x) x2 3x 2
x4
complete the following.
(a) Find
f(0)
.
g(0)
1
Because f(0) 0 and g(0) , then
2
f(0)
0
0
g(0)
1
2
(b) Find
f(1)
.
g(1)
Although we can find both f(1) and g(1), g(1) 0, so division is undefined when x 1.
1 is an excluded value for the domain of the quotient.
(c) Find h(x) © 2001 McGraw-Hill Companies
h(x) f(x)
.
g(x)
f(x)
g(x)
x3 2x2
x2
2
x 3x 2
x4
Invert and multiply.
x3 2x2
x4
2
x2
x 3x 2
1
x4
x2(x 2)
x2
(x 1)(x 2)
1
x (x 4)
(x 2)(x 1)
2
x 2, 1, 2, 4
CHAPTER 7
RATIONAL EXPRESSIONS AND FUNCTIONS
(d) For which values of x is h(x) undefined?
h(x) will be undefined for any value of x for which f(x) is undefined, g(x) is undefined,
or g(x) 0.
h(x) is undefined for the values 2, 1, 2, and 4.
CHECK YOURSELF 5
Given the rational functions
f(x) x2 2x 1
x3
and
g(x) x2 5x 4
x2
complete the following.
f(0)
f(1)
(b) Find
.
g(0)
g(1)
(d) For which values of x is h(x) undefined?
(a) Find
(c) Find h(x) f(x)
.
g(x)
CHECK YOURSELF ANSWERS
4a
2(x 2)
x2
x
2x
2.
(a)
;
(b)
3. (a) ; (b) 6; (c) 2
3b2
x2
4
y
x 3x 9
4. (a) 10; (b) undefined; (c) h(x) (x 5)(x 2) x 4, x 2; (d) 10;
1
(x 1)(x 2)
and (e) undefined
5. (a) ; (b) undefined; (c) h(x) ; and
6
(x 3)(x 4)
(d) x 3, 1, 2, 4
1.
© 2001 McGraw-Hill Companies
504
Name
7.2
Exercises
Section
Date
In exercises 1 to 36, multiply or divide as indicated. Express your result in simplest form.
1.
x2 6x
3 x4
2.
ANSWERS
y3 15y
6
10
y
1.
2.
3.
a
a2
3 7a
21
4.
p5
p2
8
12p
3.
4.
5.
4xy2 25xy
15x3 16y3
6.
5xy2
3x3y
3 10xy 9xy3
5.
6.
7.
4x2y2
8y2
8.
3 9x
27xy
8b3
2ab2
7.
15ab
20ab3
8.
9.
9.
m3n 6mn2
3mn
3 2mn m n
5m2n
10.
4cd 2 3c3d
9cd
2 5cd 2c d
20cd 3
10.
11.
12.
5x 15
9x2
11.
3x
2x 6
a2 3a
20a2
12.
5a
3a 9
13.
14.
13.
3b 15
4b 20
6b
9b2
14.
7m2 28m
5m 20
4m
12m2
15.
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16.
17.
15.
x 3x 10
15x
5x
3x 15
16.
y 8y
12y
2
4y
y 64
17.
c2 2c 8
5c 20
6c
18c
18.
m2 49
3m 21
5m
20m2
2
2
2
2
18.
505
ANSWERS
19.
19.
x2 2x 8
10x
2
4x 16
x 4
20.
y2 7y 10
2y
2
y2 5y
y 4
21.
d 2 3d 18
d2 9
16d 96
20d
22.
b2 6b 8
b2 4
2
b 4b
2b
23.
2x2 x 3 3x2 11x 20
3x2 7x 4
4x2 9
24.
3p2 13p 10
4p2 1
2p 9p 5
9p2 4
25.
a2 9
2a2 5a 3
2a2 6a
4a2 1
26.
2x2 5x 7
5x2 5x
4x2 9
2x2 3x
27.
2w 6
3w
2
w 2w 3 w
28.
3y 15
4y
2
y 3y 5 y
29.
a7
21 3a
2
2a 6
a 3a
30.
x4
16 4x
2
x 2x
3x 6
31.
x2 9y2
4x 10y
2
2 2
2x xy 15y
x 3xy
32.
2a2 7ab 15b2 2a2 3ab
2
2ab 10b2
4a 9b2
33.
3m2 5mn 2n2
m3 m2n
9m2 4n2
9m2 6mn
34.
4x2 20xy
2x2y 5xy2
4x2 25y2
2x2 15xy 25y2
35.
x3 8
5x 10
3
2
x 4 x 2x2 4x
36.
a3 27
a3 3a2 9a
a2 9
3a3 9a2
20.
21.
22.
23.
24.
2
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
(b)
(c)
(d)
(e)
x2 2x 8
x2 3x 4
and g(x) . Find (a) f(0) g(0),
x2
x4
(b) f(4) g(4), (c) h(x) f(x) g(x), (d) h(0), and (e) h(4).
37. Let f(x) 506
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37. (a)
ANSWERS
x2 4x 3
x2 7x 10
and g(x) . Find (a) f(1) g(1),
38. Let f(x) x5
x3
(b) f(3) g(3), (c) h(x) f(x) g(x), (d) h(1), and (e) h(3).
38. (a)
(b)
(c)
(d)
(e)
39. (a)
2x2 3x 5
3x2 5x 2
and g(x) . Find (a) f(1) g(1),
x2
x1
(b) f(2) g(2), (c) h(x) f(x) g(x), (d) h(1), and (e) h(2).
(b)
(c)
(d)
(e)
39. Let f(x) 40. (a)
(b)
(c)
(d)
(e)
x2 1
x2 9
and g(x) . Find (a) f(2) g(2), (b) f(3) g(3),
x3
x1
(c) h(x) f(x) g(x), (d) h(2), and (e) h(3).
40. Let f(x) 41. (a)
(b)
(c)
3x2 x 2
x2 4x 5
f(0)
f(1)
and g(x) . Find (a)
41. Let f(x) , (b)
,
x2
x4
g(0)
g(1)
f(x)
, and (d) the values of x for which h(x) is undefined.
(c) h(x) g(x)
(d)
42. (a)
(b)
(c)
(d)
x2 x
x2 x 6
f(0)
f(2)
and g(x) . Find (a)
, (b)
,
x5
x5
g(0)
g(2)
f(x)
, and (d) the values of x for which h(x) is undefined.
(c) h(x) g(x)
42. Let f(x) 43.
44.
45.
© 2001 McGraw-Hill Companies
The results from multiplying and dividing rational expressions can be checked by using a
graphing calculator. To do this, define one expression in Y1 and the other in Y2. Then
define the operation in Y3 as Y1 Y2 or Y1 Y2. Put your simplified result in Y4 (sorry,
you still must simplify algebraically). Deselect the graphs for Y1 and Y2. If you have
correctly simplified the expression, the graphs of Y3 and Y4 will be identical. Use this
technique in exercises 43 to 46.
43.
x3 3x2 2x 6 5x2 15x
x2 9
20x
44.
3a3 a2 9a 3 3a2 9
4
15a2 5a
a 9
45.
x4 16
(x3 4x)
x x6
46.
w3 27
(w3 3w2 9w)
w 2w 3
2
46.
2
507
Answers
1.
15.
25.
37.
39.
41.
3.
3
a4
5.
© 2001 McGraw-Hill Companies
43.
5
16b3
15x
9b
7.
9. 5mn
11.
13.
12x
3a
2
8
3(c
2)
5x
5d
x5
x2 2x
17.
19.
21.
23.
5
2(x 2)
4(d 3)
2x 3
2a 1
6
a
2
3
5
27.
29.
31.
33.
35.
2a
w2
6
x
m
x
(a) 4; (b) undefined; (c) (x 1)(x 4) x 2, 4; (d) 4; (e) undefined
(a) 6; (b) undefined; (c) (2x 5)(3x 1) x 2, 1; (d) 6; (e) undefined
4
5
(3x 2)(x 4)
(a) ; (b) ; (c)
x 4, 1, 2, 5; (d) 2, 4, 1, 5
5
4
(x 2)(x 5)
x2 2
x2
45.
4
x(x 3)
2
x
508