Section 6.3 Complex Rational Expressions
aka “Complex Polynomial Fractions”


Definition of a Complex Fraction:
An expression with more than one fraction bar
Simplifying Complex Fractions:


Method 1: Clearing all Denominators using the LCD
Method 2: Using the Division Rule for Fractions
2
1
x
x3
a b

b a
a
b
 2
2
b
a
x 2  3x
5
x
6.3
1
Arithmetic Example – Method 1

Clearing all Denominators using the LCD
1 1 2  3  5  7 1  1 



3 5 
3 5 
1 1
1 1


2  3  5  7  
2 7
2 7
2  5  7  2  3  7 70  42 112


3  5  7  2  3  5 105  30 135
6.3
2
Arithmetic Example – Method 2

Simplify top and bottom, use the division rule
1 1
1 5 1 3
8


3 5  3  5 5  3  15  8  14  112
1 1
1 7 1 2
9
15
9
135


2 7
2  7 7  2 14
6.3
3
Definition and Examples
6.3
4
Simplifying CRE’s –
Method 1: Multiplying by 1
2 
2
2
 1 x    1 x     x(1)
2 x
x2
x 
x


x




x  3 x  x  3
x  x)  x(3 x( x  3) x( x  3)
6.3
5
Simplifying CRE’s –
Method 2: Dividing 2 Rational Expr’s
2
2 x 2 x
1


2  x  1
2 x
x2
x
x
x
x






x  x  3 x( x  3) x( x  3)
x3 x3 x3
1
1
6.3
6
Which Method to Use?

When choosing a method, look at the problem carefully.
1
1
a 2  3a  2 
a 2  3a  2

3
2
3  (a  1)
2  (a  2)


a  2 a  1 (a  2)  (a  1) (a  1)  (a  2)
1
2
2
(
1
)
(
a
 3a  2)
1
a  3a  2 


2
3a  3  2a  4 (a  3a  2)
(a  1)
a 1
a 2  3a  2
6.3
7
Monomials - What’s the LCD?
1
3 2 1
1 1
ab  3  

3
3
b

a
b
a
b
b

 
ab b 

3
1
1
1  aa
3 2 1
 2 ab  2 2 2
2 2
ab b
a b b 
LCD  a 3b 2
b(1  a ) b(1  a )(1  a  a ) b(1  a  a )


2
a(1  a )
a (1  a)(1  a)
a (1  a)
3
2
6.3
2
8
Method 1
Practice – What’s the LCD? And the Lacks?
3
1

2 x  2 x  1  3( x  1)  1  2( x  1)  3 x  3  2 x  2  x  5
1
x
1  2( x  1)  2 x
2x  2  2x
4x  2
 2
x 1 x 1
2 x  2  2( x  1)
x 1 
( x  1)
( x  1)
x  1  ( x  1)
2( x  1)
2( x  1)
x  1  ( x  1)( x  1)
2
Big LCD  2( x  1)( x  1)
2
Lacks 
6.3
9
Method 2
Practice – What’re the LCDs? And Lacks?
3 2
3x  2
 2
2
2
3x  2
x ( x  2)
( x  2)
x x
x
 2



2
3
1
3
x

x

2
x
(
3
x

2
)(
x

1
)
(
x

1
)
 2
x2 x
x 2 ( x  2)
xx
x x
2
x  2  ( x  2)
x
2
Top LCD  x Lacks 
2
x x
1
2
2
x2
( x  2)
Bot LCD  x ( x  2) Lacks 
2
6.3
10
Practice – What’s the LCD?
2
1
x
4
1 2
x
6.3
11
Practice – What’s the LCD?
a 1  b 1
3
3
a b
1 1

 a b 
1 1
 3
3
a b
6.3
12
What Next?

Present Section 6.4
Rational Equations
6.3
13