Section 2.2: Multiply and Divide Rational Expressions
CHAPTER 2
Section 2.2: Multiply and Divide Rational Expressions
Section 2.2: Multiply and Divide
Rational Expressions
Objectives: Multiply rational expressions.
Divide rational expressions.
When, multiplying and dividing rational expressions, we will use the same process as we do
when multiplying and dividing fractions. Always be sure the answer is written in simplest form.
MULTIPLYING RATIONAL EXPRESSIONS
Example 1. Multiply, expressing the resulting fraction in its lowest terms.
15 14
?
49 45
1 2
? ?
7 3
?
2
21
First, reduce by dividing out the common factors from numerator and
denominator ( 15 and 7 )
Multiply the numerators together and the denominators together
Our Answer
When multiplying rational expressions, we first divide the numerators and denominators by
any common factors. Then we multiply the remaining factors straight across.
Example 2. Multiply.
25 x 2 24 y 4
?
9 y 8 55 x 7
Reduce coefficients by dividing out the common factors from the
numerator and the denominator ( 3 and 5 )
Reduce the variable terms by subtracting exponents
( x 2?7 ) ? x ?5 ; ( y 4?8 ) ? y ?4
Negative exponent ( x ?5 ) moves to the denominator as
Likewise, ( y ?4 ) to the denominator as
1
;
x5
1
y4
?
5
8
?
4
3 y 11x5
Multiply across (numerators together; denominators together)
?
40
33x5 y 4
Our Answer
Page 61
CHAPTER 2
Section 2.2: Multiply and Divide Rational Expressions
If the rational expression in either the numerator or the denominator is factorable, it must be
factored first. That way, any common factors can be divided out before multiplying.
Example 3. Multiply.
x2 ? 9
x 2 ? 8 x ? 16
?
x 2 ? x ? 20
3x ? 9
Factor each numerator and denominator
?
( x ? 3) ( x ? 3) ( x ? 4) ( x ? 4)
?
( x ? 4) ( x ? 5)
3 ( x ? 3)
Divide out the common factors ( x ? 3) and ( x ? 4)
?
x ?3 x ?4
?
x?5 3
Multiply across
?
( x ? 3) ( x ? 4)
3 ( x ? 5)
Our Answer
DIVIDING RATIONAL EXPRESSIONS
When dividing rational expressions, we change the division problem into an equivalent
multiplication problem. Multiply the first expression by the reciprocal of the divisor. In other
words, keep the first expression, change the operation from division to multiplication, and
¡°flip¡± the second expression. Then, multiply as shown in the examples above, writing the
answer in simplest form.
Example 4. Divide.
a 4b 2 b 4
?
a
4
Multiply the first expression by the reciprocal of the second
expression
?
a 4b 2 4
?
a b4
Subtract exponents; negative exponents move to denominator
?
a3 4
?
1 b2
Multiply across (numerators together; denominators together)
?
4a 3
b2
Our Answer
Page 62
CHAPTER 2
Section 2.2: Multiply and Divide Rational Expressions
Example 5. Divide.
x 2 ? x ? 12 5x 2 ? 15x
?
x2 ? 2 x ? 8 x2 ? x ? 2
Multiply by the reciprocal
?
x 2 ? x ? 12 x 2 ? x ? 2
?
x 2 ? 2 x ? 8 5 x 2 ? 15 x
Factor each numerator and denominator
?
( x ? 4) ( x ? 3) ( x ? 2) ( x ? 1)
?
( x ? 2) ( x ? 4) 5 x ( x ? 3)
Divide out the common factors ( x ? 4) , ( x ? 3) , and
( x ? 2)
1 x ?1
? ?
1 5x
?
x ?1
5x
Multiply across (numerators together; denominators
together)
Our Answer
The example below contains both multiplication and division. To perform these operations,
we change division to multiplication by the reciprocal of the divisor, factor wherever possible,
reduce if possible, and then multiply the remaining factors.
Example 6. Multiply and divide as indicated.
a 2 ? 7a ? 10 a ? 1 (a ? 1)
?
?
a 2 ? 6a ? 5 4a ? 8 (a ? 2)
Factor each numerator and denominator
?
(a ? 5) (a ? 2) (a ? 1) (a ? 1)
?
?
Multiply by the reciprocal of last fraction
(a ? 5) (a ? 1) 4(a ? 2) (a ? 2)
?
(a ? 5) (a ? 2) (a ? 1) (a ? 2)
?
?
(a ? 5) (a ? 1) 4(a ? 2) (a ? 1)
Divide out the common factors (a ? 5) , (a ? 2) ,
and (a ? 1)
?
a?2
4(a ? 1)
Our Answer
Page 63
CHAPTER 2
Section 2.2: Multiply and Divide Rational Expressions
Practice Exercises
Section 2.2: Multiply and Divide Rational Expressions
Multiply.
1)
8x2 9
?
9 2
5)
6 x( x ? 4) ( x ? 3)( x ? 6)
?
x ?3
6 x( x ? 6)
2)
9n 7
?
2n 5n
6)
25n ? 25
4
?
5
30n ? 30
3)
5x2 6
?
4 5
7)
v ?1
4
? 2
4 v ? 11v ? 10
4)
7(m ? 6) 5m(7m ? 5)
?
m?6
7(7m ? 5)
8)
x2 ? 6 x ? 7 x ? 5
?
x?5
x?7
9)
8x 4
?
3x 7
13)
7r
r ?6
?
7r (r ? 10) (r ? 6) 2
10)
9m 7
?
5m 2 2
14)
9
b?5
? 2
b ? b ? 12 b ? b ? 12
11)
10 p 8
?
5 10
15)
x ? 10
7
?
35 x ? 21 35 x ? 21
12)
7
n?2
?
10(n ? 3) (n ? 3)(n ? 2)
16)
8k
1
?
24k ? 40k 15k ? 25
Divide.
2
2
Perform the indicated operation.
17)
1 8a ? 80
?
a?6
8
20)
x 2 ? 7 x ? 10
x ? 10
? 2
x?2
x ? x ? 20
18)
p ?8
1
?
p ? 12 p ? 32 p ? 10
21)
4m ? 36 m ? 5
?
m ? 9 5m2
6
10n ? 80
22)
2r
2r
?
r ? 6 7r ? 42
2
19) (n ? 8) ?
The Practice Exercises are continued on the next page.
Page 64
CHAPTER 2
Section 2.2: Multiply and Divide Rational Expressions
Practice Exercises: Section 2.2 (continued)
Perform the indicated operation.
23)
3x ? 6
? ( x ? 3)
12 x ? 24
33) (10m2 ? 100m) ?
24)
2n2 ? 12n ? 54
? (2n ? 6)
n?7
34)
n?7
9n ? 54
?
n ? 2n ? 35 10n ? 50
25)
b?2
(5b ? 3)
40b2 ? 24b
35)
7 p 2 ? 25 p ? 12
3p ?8
?
2
6 p ? 48
21 p ? 44 p ? 32
26)
21v 2 ? 16v ? 16 35v ? 20
?
3v ? 4
v ?9
7 x 2 ? 66 x ? 80 7 x 2 ? 39 x ? 70
36)
?
49 x 2 ? 7 x ? 72 49 x 2 ? 7 x ? 72
27)
n?7
12 ? 6n
? 2
6n ? 12 n ? 13n ? 42
37)
10b2
30b ? 20
? 2
30b ? 20 2b ? 10b
28)
x 2 ? 11x ? 24 6 x3 ? 6 x 2
?
6 x3 ? 18 x 2 x 2 ? 5 x ? 24
38)
35n2 ? 12n ? 32 7n2 ? 16n ? 15
?
49n2 ? 91n ? 40
5n ? 4
29)
27a ? 36 6a ? 8
?
9a ? 63
2
39)
7r 2 ? 53r ? 24 49r ? 21
?
7r ? 2
49r ? 14
30)
k ?7
7k 2 ? 28k
?
k 2 ? k ? 12 8k 2 ? 56k
40)
12 x ? 24
15 x ? 21
?
2
10 x ? 34 x ? 28
5
31)
x 2 ? 12 x ? 32 7 x 2 ? 14 x
?
x 2 ? 6 x ? 16 7 x 2 ? 21x
41)
x2 ?1 x2 ? 4
x2 ? x ? 2
? 2
?
2x ? 4 x ? x ? 2
3x ? 6
32)
9 x3 ? 54 x 2 x 2 ? 5 x ? 14
?
x 2 ? 5 x ? 14
10 x 2
a 3 ? b3
3a ? 6b
a 2 ? 4b2
42) 2
?
?
a ? 3ab ? 2b2 3a 2 ? 3ab ? 3b2
a ? 2b
18m3 ? 36m2
20m2 ? 40m
2
Page 65
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