Reducing a Rational Expression to Lowest Terms
In each rational expression that follows, we will assume that no value of a
variable will make a denominator 0.
Like a fraction in arithmetic, a rational expression is usually easier to work
with when written in lowest terms.
The procedure for reducing a rational expression to lowest terms is like the
procedure for reducing a fraction to lowest terms.
Procedure â€”
To Reduce a Rational Expression to Lowest Terms
Step 1 Factor the numerator and denominator.
Step 2 Cancel all pairs of factors common to the numerator and
denominator.
Note:
When we have reduced a rational
expression to lowest terms, we also say we
have simplified the rational expression.
Hereâ€™s an example from arithmetic.
Reduce to lowest terms:
Step 1 Factor the numerator and
denominator.


Step 2 Cancel all pairs of factors common
to the numerator and denominator. 

Thus,
Remember, that w ≠ 0, x
≠ 0, and y ≠ 0.
We can also use the properties of
exponents to simplify this fraction:
Example 1
Reduce to lowest terms:
Solution
Step 1

Factor the numerator
and denominator. 


Step 2 
Cancel all pairs of
factors common to
the numerator and
denominator. 


Thus,
Example 2
Reduce to lowest terms:
Note:
Remember, it is understood that x ≠ 3 and x
≠ 3.
Solution
Thus,
Note:
Like a fraction in arithmetic, a rational expression should normally be
written in lowest terms.
