# Hotmath

# Simplifying Rational Expressions

As you know, a rational number is one that can be expressed as a fraction , that is,

,

where and are integers (and ).

Similarly, a
**
rational expression
**
(sometimes called an
**
algebraic fraction
**
) is one that can be expressed as a quotient of
polynomials
, i.e.
where
and
are polynomials (and
).

**
Example 1:
**

is a rational expression, since both the numerator and the denominator are polynomials . (" " counts as a polynomial... it's just a very simple one, with only one term.)

is
**
not
**
a rational expression. The denominator is
**
not
**
a polynomial.

A rational expression can be simplified if the numerator and denominator contain a common factor .

**
Example 2:
**

Simplify.

First, factor out a constant from both numerator and denominator. Write the as .

Next, factor the quadratic in the denominator. (Look for two numbers with a product of and a sum of .)

Finally, cancel common factors.

## IMPORTANT NOTE: EXCLUDED VALUES

When we factored out in the above expression, we made an important change. The new expression

is defined for ; it equals . But the original expression we were trying to simplify,

is
**
undefined
**
for
, because the denominator equals zero (and division by zero is a no-no).

So, our simplification is not really true for all points. When you simplify rational expressions, you should make note of these
**
excluded values
**
.