This document contains examples and steps for simplifying complex rational expressions: 1) Complex rational expressions with fractions in the numerator and denominator must be simplified. The first step is to find the least common denominator (LCD) of the numerator and denominator. Then the expression is multiplied by the LCD over LCD, simplifying any common factors. 2) Example problems demonstrate multiplying complex rational expressions by the LCD and simplifying, including expressions with multiple terms in the numerator and denominator. 3) A shortcut is provided - if there is only one term in the numerator and denominator, the fractions can be divided by multiplying by the reciprocal, without needing the LCD.

1. Warm Up #13
1 (6x3 + 12x2 – 18x)  3x
2 (4x2 + 9x +2)  (x+2)
3 (x3 + 216)  (x + 6)

2. Warm Up #13:
1 (6x3 + 12x2 – 18x)  3x
2 (4x2 + 9x +2)  (x+2)
3 (x3 + 216)  (x + 6)
x2 – 6x +36
4x + 1
2x2 + 4x - 6

3. Complex Rational Expressions
3
2
1
x

1
3
2
4
3


x
x
x
y
x
x
x



2
3
2
8
1
Fractions inside of fractions are a
NO - NO
And must be simplified.

4. Steps
1. Find the LCD of the numerator and
denominator
2. Multiply the complex rational expression
by the LCD/LCD (so its equivalent to
multiplying by 1)
3. Simplify (factor and cross out common
factors)

5. Complex Rational Expressions
Example 1:
2
1
1
1
1
x
x



 2
2
x
x
1
2
2


x
x
x  
  
1
1
1



x
x
x
x
 
1

x
x

6. Example 2:
8
2
4
3
x
x



8
8
x
x
4
6 

7. x
x
 
1
5
x
x

Example 3:

8. y
y
  2
1
1
y
y


  
1
1 1
y
y y

 
 
1
1
y 
Example 4:

9. 2
1
3
2
x
x

3
5x
Example 5: You try

10. 8/5
Example 6: You try

11. Short Cut
• If there are 2 terms or more on either the
numerator or denominator you must
multiply by the LCD/LCD
• However, if there is only 1 term on the
numerator and 1 term on the denominator
then you can divide fractions (which means
multiply by the reciprocal)

12.   
2
1
2
2
3



x
x
x
x
 
1
2
3

x
x
   2
1
2
2
3



 x
x
x
x
   x
x
x
x
2
1
2
2
3 



Example 7:

13. Assignment:
Page 470
2-24 even