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

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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

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