Lecture complex fractions

Section 6.3
Complex
Rational
Expressions
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

Objective #1
Simplify complex rational expressions by
multiplying by 1.

Simplifying Complex Fractions
Complex rational expressions, also called complex
fractions, have numerators or denominators containing one
or more fractions.
5
x
x

5
1 1

5
x
Woe is me,
for I am
complex.

Complex Rational Expressions
Simplifying a Complex Rational
Expression by Multiplying by 1 in the
Form LCD
T
LCD
1) Find the LCD of all rational expressions within the
complex rational expression.
2) Multiply both the numerator and the denominator of the
complex rational expression by this LCD.
3) Use the distributive property and multiply each term in the
numerator and denominator by this LCD. Simplify each term.
No fractional expressions should remain within the numerator
and denominator of the main fraction.
4) If possible, factor and simplify.

x


x
EXAMPLE

Simplify: .
1
1
5
5
5
x
x

SOLUTION
The denominators in the complex rational expression are 5 and
x. The LCD is 5x. Multiply both the numerator and the
denominator of the complex rational expression by 5x.





 

x
5
 


x
x
x
x
x
x
1
1
5
5
5
5
1
1
5
5
5 Multiply the numerator and
denominator by 5x.

 Divide out common factors.
x
  
1
x x
x
x
x
x
1
5
5
5
5
5
5
5
  

Use the distributive
property.
CONTINUED
x
  
1
x x
x
x
x
x
1
5
5
5
5
5
5
5
  
2

25 
5

x
x
Simplify.
 x  5  x

5

1   
5

x
Factor and simplify.

Simplify.
CONTINUED
5 
1

x
Simplify. 5   x

1
x x


x x x x
EXAMPLE
Simplify: .
6
1
6


SOLUTION
The denominators in the complex rational expression are x + 6
and x. The LCD is (x + 6)x. Multiply both the numerator and
the denominator of the complex rational expression by (x + 6)x.
Multiply the numerator and
denominator by (x + 6)x.
 
  6
1
6
1
6
6
6
1
6
1









 x x
x x

Use the distributive
property.
CONTINUED
   
  

 6 6
1
6
  
6
1
6
 
 

x x
x
x x
x
x x
Divide out common factors.
   

 6 6
1
6
6
1
6
 
 

x x
x
x x
x
x x
Simplify.
 6

x x
 
  6 
6

x x
x x
6 36
6
 
x 2 
x Simplify.


CONTINUED
6

6x 2 
36x Subtract.

Factor and simplify.
 
6  
1
6  
6

x x

1

 6 Simplify.

x x

x x x 
y
 1  1  1

  

 1  1  1

xy  y y x 
y y
x y x y x y x y
  
   
1a. Simplify:
2
2
1
1
x
y
x
y


Multiply the numerator and denominator by the LCD of 2. y
2
2
2
2 2 2 2 2
2
2 2 2
y
y y y y
x y x x y
y
y y y
2
2 2
( )
( )( )
Objective #1: Example

x x x 
y
 1  1  1

  

 1  1  1

xy  y y x 
y y
x y x y x y x y
  
   
1a. Simplify:
2
2
1
1
x
y
x
y


Multiply the numerator and denominator by the LCD of 2. y
2
2
2
2 2 2 2 2
2
2 2 2
y
y y y y
x y x x y
y
y y y
2
2 2
( )
( )( )


1b. Simplify:
1 1
7
7
x 
x
Multiply the numerator and denominator
by the LCD of x(x  7).
1 1 1 1 ( 7) ( 7)
7 ( 7) 7 7
7 ( 7) 7 7 ( 7)
( 7) 7 7
7 ( 7) 7 ( 7) 7 ( 7)
1 1
( 7) ( 7)
x x x x
x x x x x x x x
x x x x
x x x x
x x x x x x
x x x x
 
  
      
 
    
  
  

  
 

Objective #2
Simplify complex rational expressions by dividing.

Simplifying a Complex Rational
Expression by Dividing
1) If necessary, add or subtract to get a single rational
expression in the numerator.
2) If necessary, add or subtract to get a single rational
expression in the denominator.
3) Perform the division indicated by the main fraction
bar: Invert the denominator of the complex rational
expression and multiply.
4) If possible, simplify.

m
2 2
m m m
3
2
4 4
9
2 2
m
2
m
m
m m m
9 
2
m m m m
   
2 2 3 3
m m
2  3 
3
EXAMPLE
Simplify: .
5 6  
6

 
 


m m
m m
SOLUTION
1) Subtract to get a single rational expression in the
numerator.
2
2 2 4 4  3  3
  2
2 
 

 

 m m m
 
2

m m
   
  
    
    
   2
2 2
2
3 3 2
2 3 3
3 3 2
  

  

  

m m m
m m m
m m m

3 2
CONTINUED
m m m m
   
4 4 2 18
2
3 2 2
m m m
  
6 4 18

  3   3  
2
   3   3  
2
2
2) Add to get a single rational expression in the denominator.
3
m
m
2 2  
3
m m m
  

m m
9
3 9 3 2 2

m   m 
m
m

m m m
m m m
 3 2  2 3
3
5 6 6

 

 

  m m m m
m m
m m
 m

3 
3
   
 
   
3  3   3

 3 2 3
2 3 3
3 2 3
  

  

  

m m m
m m m
m m m
 3  2  3
   3  2 
3

  

m m m
m m m

3) & 4) Perform the division indicated by the main fraction
bar: Invert and multiply. If possible, simplify.
m m m
  
6 4 18
 m  3  m  3  m

2

9

m
 3 2 3
2
m
2 2
m m m
m
5 6 6
m m m
  
3 2 3
m m m
  
6 4 18
m m m
  
3 2 3
m m m
  
6 4 18
m m m
  
6 4 18
3
4 4
9
2
2
3 2
2 2
  

 

 
 


m m m
m m
m m
   
   
9
3 3 2
2 2
3 2


  

m
m m m
   
   
9
3 3 2
2 2
3 2


  

m
m m m
 2 2
9
3 2
 

m m
CONTINUED

x x
x x
x x
x x
 
2a. Simplify:
1 1

1 1
1 1
1 1
 
 

 
The LCD of the numerator is (x 1)(x 1).
The LCD of the denominator is (x 1)(x 1).

x x x x x x x x x x
x x x x x x x x x x
x x x x x x x x x x
x x x x x x x x x x
 1  1 (  1)(  1) (  1)(  1)  2  1  2  1
  
 1  1 (  1)(  1) (  1)(  1) (  1)(  1) (  1)(  1)
 
 1  1 (  1)(  1) (  1)(  1)    2  1  2  1
 1  1 (  1)(  1) (  1)(  1)  (  1)(  1) (  1)( 
1)
x x x x
    
x x x x
2 1 2 1
x x x
1)( 1) ( 1)( 1)
x x x x x x x x
x x x x
x
x x x x x x
2 2
2 2
2 2
2 1 ( 2 1)
(
x

2 2
2 2 2 2
2 1 2 1 2 1 2 1
( 1)( 1) ( 1)( 1)
4
( 1)( 1) 4 ( 1)( 1) 4
2 x 2 2 ( x 1)( x 1) 2 x 2 2 2( x
2 1)
( x 1)( x
1)
2
x
x
2 1
    
   

         
   
   
   
    
 


CONTINUED

x

1 
4
 
x x
2b. Simplify:
2
1 2
1 7 10
 
Rewrite the expression without negative exponents.
Then multiply the numerator and denominator
by the LCD of 2. x

x x
2 2
1 2
2
2
2
2 2 2
2 2 2
2
2 2
2
2
4
1
1 4
1 7 10 7 10 1
4 4
1 1
7 10 7 10 1 1
4 ( 2)( 2) 2
7 10 ( 5)( 2) 5
x x
x x
x
x
x x x
x x x
x x x x x
x x x x
x x x x x

 



   

  
  
      
   
  
    
CONTINUED

Lecture complex fractions

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Lecture complex fractions