Simplifying Rational Expressions

The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.

$\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}$

We can factor the numerator and denominator to rewrite the expression.

$\frac{{\left(x+4\right)}^{2}}{\left(x+4\right)\left(x+7\right)}$

Then we can simplify that expression by canceling the common factor $\left(x+4\right)$.

$\frac{x+4}{x+7}$

How To: Given a rational expression, simplify it.

1. Factor the numerator and denominator.
2. Cancel any common factors.

Example 1: Simplifying Rational Expressions

Simplify $\frac{{x}^{2}-9}{{x}^{2}+4x+3}\\$.

Solution

$\begin{array}\frac{\left(x+3\right)\left(x - 3\right)}{\left(x+3\right)\left(x+1\right)}\hfill & \hfill & \hfill & \hfill & \text{Factor the numerator and the denominator}.\hfill \\ \frac{x - 3}{x+1}\hfill & \hfill & \hfill & \hfill & \text{Cancel common factor }\left(x+3\right).\hfill \end{array}$

Analysis of the Solution

We can cancel the common factor because any expression divided by itself is equal to 1.

Can the ${x}^{2}$ term be cancelled in Example 1?

No. A factor is an expression that is multiplied by another expression. The ${x}^{2}$ term is not a factor of the numerator or the denominator.

Try It 1

Simplify $\frac{x - 6}{{x}^{2}-36}$.

Solution