# Simplifying Rational Expressions

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

$\frac{p}{q}$ ,

where $p$ and $q$ are integers (and $q\ne 0$ ).

Similarly, a
**
rational expression
**
(sometimes called an
**
algebraic fraction
**
) is one that can be expressed as a quotient of
polynomials
, i.e.
$\frac{p}{q}$
where
$p$
and
$q$
are polynomials (and
$q\ne 0$
).

**
Example 1:
**

$\frac{3}{{x}^{3}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}5{x}^{2}y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}7{y}^{3}}$

is a rational expression, since both the numerator and the denominator are polynomials . (" $3$ " counts as a polynomial... it's just a very simple one, with only one term.)

$\frac{5x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}3}{6x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{x}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{x}^{y}}$

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.

$\frac{3x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6}{9{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}9x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}54}$

First, factor out a constant from both numerator and denominator. Write the $9$ as $3\cdot 3$ .

$\frac{3x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6}{9{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}9x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}54}=\frac{3x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6}{3\cdot 3({x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}6)}$

Next, factor the quadratic in the denominator. (Look for two numbers with a product of $-6$ and a sum of $-1$ .)

$=\frac{3(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2)}{3\cdot 3(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2)(x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}3)}$

Finally, cancel common factors.

$=\frac{1}{3(x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}3)}$

## IMPORTANT NOTE: EXCLUDED VALUES

When we factored out $x+2$ in the above expression, we made an important change. The new expression

$\frac{1}{3(x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}3)}$

is defined for $x=-2$ ; it equals $-\frac{1}{5}$ . But the original expression we were trying to simplify,

$\frac{3x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6}{9{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}9x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}54}$

is
**
undefined
**
for
$x=-2$
, 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
**
.