Multiplying and dividing rational expressions

1. R.5 day2 Multiply and Divide
Rational Expressions
Learning Target: You will be able to multiply and divide rational
expressions, and simplify the product or quotient.

2. Multiply rational expressions.
• The product of two fractions is found by multiplying the
numerators and multiplying the denominators. Rational
expressions are multiplied in the same way.
• The product of the rational expressions and is
• That is, to multiply rational expressions, multiply the
numerators and multiply the denominators.
Copyright © 2008 Pearson Education, Inc. Publishing as
Pearson Addison-Wesley
Slide 7.2 - 3
P
Q
R
S
P R PR
Q S QS
 

3. EXAMPLE 1
• Multiply. Write each answer in lowest terms.
Copyright © 2008 Pearson Education, Inc. Publishing as
Pearson Addison-Wesley
Solution:
Multiplying Rational
Expressions
Slide 7.2 - 4
2 5
7 10

2
2
8 9
3
p q
pq

2 5
7 10



2
2
8 9
3
p q
p q
  

 
7
2 5
2 5


 
1
7

3
8
3
3
p q
p
p q q
    

  
24p
q

It is also possible to divide out common factors in the numerator
and denominator before multiplying the rational expressions.

4. Multiply the fractions
Reduce before multiply.

1
6x2
y
8x
21y3 
7y2
16x3

2 2 2 x 7 y y
7 3 y y y 2 2 2 2 x x x

5. Multiply the fractions
Reduce before multiply.

2ac3
3b2
5a4
c
12b

24bc2
15a3
b2

5 a a a a c 2 2 2 3 b c c
2 3 2 b 3 5 a a a b b

6. Example 2
Multiply rational expressions
Step 1: Factor and Multiply
2x2
 4x
x2
 4x 12

x2
9x 18
2x
2x(x 2)(x  3)(x 6)
(x 2)(x 6)2x
x  3

7. Checkpoint
Multiply the expression
6x² + 18x x² - x – 2
x² + x – 6 * x² - 7x – 8
6x(x + 3)(x-2)(x+1)
(x+3)(x-2)(x-8)(x+1)
6x
x-8

8. More Examples
• Multiply the expressions. Simplify the result.
3x5
y2
8xy

6xy2
9x3
y

2x2
10x
x2
 25

x  3
2x2

x2
y2
4

x  3
x2
 5x

9. Divide rational expressions.
Copyright © 2008 Pearson Education, Inc. Publishing as
Pearson Addison-Wesley
Division of rational expressions is defined as follows.
If and are any two rational expressions with
then
That is, to divide one rational expression by another rational
expression, multiply the first rational expression by the reciprocal
of the second rational expression.
Slide 7.2 - 12
.
P R P S PS
Q S Q R QR
   
,
0
R
S

R
S
P
Q

10. EXAMPLE 4
•Divide. Write each answer in lowest terms.
Copyright © 2008 Pearson Education, Inc. Publishing as
Pearson Addison-Wesley
Solution:
4
3 4
4
5
 


3 16
4 5
 
12
5

3
2
3
3
p p
p
p p


 
 
 
3 5
4 16

2
3
3 4
3 4
9
6
p p
p p
 


Dividing Rational Expressions
Slide 7.2 - 13
2 3
9 6
3 4 3 4
p p
p p

 
3
2 p


11. EXAMPLE 5
• Divide. Write the answer in lowest terms.
Copyright © 2008 Pearson Education, Inc. Publishing as
Pearson Addison-Wesley
Solution:
2 2
5 10
2 8
a b ab

Dividing Rational Expressions
Slide 7.2 - 14
2
2
5 8
2 10
a b
ab
 
5 2 2
2 2 5
2
a b
a
a
b
b
 
   

    
2a
b


12. EXAMPLE 6
• Divide. Write the answer in lowest terms.
Copyright © 2008 Pearson Education, Inc. Publishing as
Pearson Addison-Wesley
Dividing Rational Expressions
Slide 7.2 - 15
   
2
2
4 3 3
2 1 4 1
x x x x
x x
  

 
 
 
2
2
4 4
1
3
2 3
1
x
x x
x x x

 
 


  
 
2 1
1
4 2
2
1
1
x x
x
x
x
x
 

 
 

 
4 2 1
x
x

 
Solution:

13. EXAMPLE 7
•Divide. Write in the answer in lowest terms.
Copyright © 2008 Pearson Education, Inc. Publishing as
Pearson Addison-Wesley
2
2 2
1 2 1
ab a a b
a a a
 

  
Dividing Rational Expressions
(Factors Are Opposites)
Slide 7.2 - 16
Solution:
   
   
1 1
1
1
1
a a
a b a
a b
a a
 


 
 
2 2
2
2 1
1
ab a a a
a a b
  
 
 
 
 
1
1
a a
a

 

Remember to write −1 when dividing out factors that are opposite of each
other. It may be written in the numerator or denominator, but not both.

14. Divide the Rational Expressions
You can only Reduce when Multiplying
 4dx
10px2
3c2
d

5px
6c2
d2
2 5 p x x 2 3 c c d d
3 c c d 5 p x

10px2
3c2
d

6c2
d2
5px

15. Example 4
Divide rational expressions
3
x  7

8x2
8x
x2
6x  7
3
8x
Step 1: Multiply by reciprocal
Step 2: Factor and Multiply
Step 3: Simplify

16. More Examples
• Divide each expression. Simplify the result.
4x
5x  20

x2
 2x
x2
 6x  8

2x2
 3x  5
6x
 2x2
 5x
 

4
5

(x 1)(2x 5)
6x

1
x(2x 5)
x 1
6x2

17. Homework
•R.5 (pg 53) #33-49 odd