Mathematics » Rational Expressions and Equations » Simplify Rational Expressions

# Defining Rational Expressions

## Defining Rational Expressions

In a previous tutorial, we reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers, and the denominator is not zero.

In this tutorial, we will work with fractions whose numerators and denominators are polynomials. We call these rational expressions.

### Rational Expression

A rational expression is an expression of the form $$\cfrac{p\left(x\right)}{q\left(x\right)},$$ where p and q are polynomials and $$q\ne 0.$$

Remember, division by 0 is undefined.

Here are some examples of rational expressions:

$$-\cfrac{13}{42}\phantom{\rule{4em}{0ex}}\cfrac{7y}{8z}\phantom{\rule{4em}{0ex}}\cfrac{5x+2}{{x}^{2}-7}\phantom{\rule{4em}{0ex}}\cfrac{4{x}^{2}+3x-1}{2x-8}$$

Notice that the first rational expression listed above, $$-\cfrac{13}{42},$$ is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.

We will perform same operations with rational expressions that we do with fractions. We will simplify, add, subtract, multiply, divide, and use them in applications.