Writing a Rational Expression in Lowest Terms
A rational expression is a fraction where the numerator and the
denominator are polynomials. Reducing a rational expression to lowest
terms is similar to reducing an arithmetic fraction to lowest terms.
Procedure
To Reduce a Rational Expression to Lowest Terms
Step 1 Factor the numerator and denominator.
Step 2 Cancel pairs of factors that are common to the numerator
and denominator.
Example 1
Reduce to lowest terms:
Solution 

Step 1 Factor the numerator and denominator.


Step 2 Cancel common factors.


Simplify.


Thus, the result is
Note:
To factor x^{2}  2x  15:
â€¢ Find two integers whose product is 15 and whose sum is 2.
They are 3 and 5.
â€¢ Use these integers to write the
factorization (x + 3)(x  5). To factor x^{2}  7x + 10:
â€¢ Find two integers whose product is 10
and whose sum is 7.
They are 2 and 5.
â€¢ Use these integers to write the
factorization (x  2)(x  5).
Example 2
Reduce to lowest terms:
Solution 

Step 1 Factor the numerator and denominator. 
Factor 1 out of the numerator. Notice that in the numerator, 8 + x,
can be written as x  8. 

Step 2 Cancel common factors. 
Cancel the common factor of x  8.
Simplify. 
= 1 
So, the fraction reduces to 1. Note:
Notice that in
, the numerator and
denominator are opposites.
Therefore,
reduces to 1.
Example 3
Reduce to lowest terms:
Solution 

Step 1 Factor the numerator and denominator.

Factor.
In the numerator, write 4  x
as 1(x  4). 

Step 2 Cancel common factors. 
Cancel the common factor of x 4.
Simplify.


Thus, the fraction reduces to
.
Notice that 4  x can be written as a
product where one factor is x  4: 4  x = 1(4 + x) = 1(x  4)
