In multiplying rational expressions, we use the following rule: Dividing by a rational expression is the same as multiplying by its reciprocal. 5.2 Multiplying.

In multiplying rational expressions, we use the following rule:

Dividing by a rational expression is the same as multiplying by its reciprocal.

5.2 Multiplying and Dividing Rational Expressions.

P R P R

Q S Q S

P R P S

Q S Q R

To multiply rational expressions we use the following steps:

1. Multiply the numerator and multiply the denominator.

2. Factor completely the numerator and the denominator.

3. Cancel common factors and simplify

Multiply:2 3

4 2

16 5

4

x x

y y

2 3

4 2

16 5

4

x x

y y

2 3

4 2

6 5

4

1 x x

y y

5

6

20x

y

Factor 16 and use product rule to multiply.

2 3

4 2

4 4 5

4

x

y

Cancel common

factors and simplify

Multiply:4 12 5

5 7 5

80 14

49 25

m x y

x y m

12 516 2x

7

2

32

35

x

y m

4 12 5

57580 14

2549

m x y

mx y

4 12 5

5 7 5

80 14

49 25

m x y

x y m

12

5

5

5 7

45 16 2 7

7 7 5 5

ym

m

x

x y

Factor numbers on numerator and denominator.

Cancel common factors and use quotient rule.

7 5 5 47 5y m

Multiply:2

2

( 2) 2

4 2

t t

t t

2

2

( 2) 2

4 2

t t

t t

2 ( 2)( 2)

2 2 ( 2)

t t t

t t t

( 2)

2

t

t

Multiply:2( ) 30

5 2( )

a b

a b

2( ) 30

5 2( )

a b

a b

2( )

5

3

2 )

0

(

a b

a b

2

1

( )

5 2 ( )

2 3 5 a b

a b

2 13( )

1

a b

= 3(a + b)

Factor numerator

Cancel common factors and use quotient rule

Divide:8 6

5 5

35 25

9 10

q q

q q

Dividing by a rational expression is the same as multiplying by its reciprocal.

Change the division to a multiplication and invert the divisor.

8 6

5 5

35 25

9 10

q q

q q

8 5

5 6

35 10

9 25

q q

q q

8 5

5 69

35 10

25

q q

q q

8 5

5 6

5 5

5

7 2

9 5

q

q

Factor numbers and use product rule

13

11

14

9

q

q

13 1114

9

q

214

9q

Cancel common factors

Use quotient rule

8 5

5 6

5 5

5

7 2

9 5

q

q

Divide:2

2

2

4 ( 4)

a a

a a

Change the division to a multiplication and invert the divisor.

2

2

2

4 ( 4)

a a

a a

2

2

2 ( 4)

4

a a

a a

2

2( 4)2

(

4)

aa

a a

2 1

2 1

( 4)2

a

a

2( 4)a

a

Use quotient rule

Divide: 2 2

( 1) 3 3

17 30 7 18

x x

x x x x

Change the division to a multiplication and invert the divisor.

2 2

( 1) 3 3

17 30 7 18

x x

x x x x

2

2

( 1) 7 18

17 30 3 3

x x x

x x x

Factor each expression

2

2

( 1) 7 18

17 30 3 3

x x x

x x x

( 1) ( 9)( 2)

( 15)( 2) 3( 1)

x x x

x x x

Cancel common factors

( 9)

3( 15)

x

x

x2 + 7x – 18

x x

+9 -2

= (x + 9)(x – 2)

x2 – 17x + 30 x x

-15 -2

= (x – 15)(x – 2)

3x + 3 = 3(x + 1)

Divide:3 2 3 2

2 2

6 2

2 1 1

x x x x x x

x x x

Change the division to a multiplication and invert the divisor.

Factor common factors.

Cancel common factors

3 1

1

x

x

3 2 2

2 3 2

6 1

2 1 2

x x x x

x x x x x

2 2

2 2

(6 1) 1

2 1 ( 2 1)

x x x x

x x x x x

Factor all expressions

(3 1)(2 1) ( 1)( 1)

(2 1)( 1) ( 1)( 1)

x x x x x

x x x x x

Divide:2 2 2

2 2 2

( )

2 ( )

x y x y

x xy y x y

Change the division to a multiplication and invert the divisor.

Factor all expressions.

x2 – 2xy + y2 =

2 2 2

2 2 2

( )

2 ( )

x y x y

x xy y x y

x2 – y2 = (x + y) (x – y)x x

- y- y

2

2

( )( ) ( )

( ( )) ( )

x y x y

x y xy y

x y

x

(x – y) (x – y) = (x – y)2

2

2

( )( ) ( )

( ( )) ( )

x y x y

x y xy y

x y

x

x y

x y

Cancel common factors

Multiply:3 2 2

2 2

2 7 3 3

2 3 ( 3)

x x x x x

x x x

Factor common factors.

Factor all expressions2

2 2

(2 7 3) ( 3)

2 3 ( 3)

x x x x x

x x x

2x2 – 7x + 3 =

2x x

- 1 - 3

(2x – 1) (x – 3)

x2 + 2x – 3 =

x x

+ 3 - 1 (x + 3) (x – 1)

2

2 2

(2 7 3) ( 3)

2 3 ( 3)

x x x x x

x x x

(2 1)( 3) ( 3)

( 3)( 1) ( 3)( 3)

x x x x x

x x x x

Cancel common factors

(2 1)( 3) ( 3)

( 3)( 1) ( 3)( 3)

x x x x x

x x x x

2 (2 1)

( 1)( 3)

x x

x x

Divide:2 2 2 2

2 2 2 2

3 17 10 6 2

6 13 5 6 5

r rs s r rs s

r rs s r rs s

Change division to multiplication.

Factor all expressions

3r r

+2s +5s

3r 2r

- s +5s

2 2 2 2

2 2 2 2

3 17 10 6 5

6 13 5 6 2

r rs s r rs s

r rs s r rs s

3r 2r

-s-s3r 2r

+2s -s

(3 2 )( 5 )r s r s

5

2 5

r s

r s

Cancel common factors

(3 )(2 5 )r s r s (3 )(2 )r s r s

(3 2 )(2 )r s r s

Divide:

Change the division to a multiplication.

Factor all expressions.

Cancel common factors

r2 + 4r – 12 =

r2 + r – 6 = (r + 3) (r – 2)r r

- 2+6(r + 6) (r – 2)

2

2

6 3

4 12 1

r r r

r r r

2

2

6 1

4 12 3

r r r

r r r

2

2

6 1

4 12 3

r r r

r r r

( 3)( 2)( 1)

( 6)( 2)( 3)

r r r

r r r

1

6

r

r