# Rational Expressions

## What does it mean?

### Definitions:

A rational expression is the ratio of two polynomials.

If f is a rational expression then f can be written in the form p/q where p and q are polynomials. Like polynomials or any other type of expression, the basic arithmetic operations, namely addition, subtraction, multiplication, and division, can be performed on rational expressions. A nice property of rational expressions is that when any of these operations are performed on two rational expressions, the result is always another rational expression. Contrary to polynomials, it is generally easy to multiply or divide but difficult to add or subtract two rational expressions.

## What does it look like?

To simplify a rational expression: $$\frac{x^2-16}{x^3+64}$$ a) Completely factor numerators and denominators. $$\frac{(x+4)(x-4)}{(x+4)(x^2-4x+16)}$$ b) Reduce common factors. $$\frac{(x-4)}{(x^2-4x+16)}$$

## You'll use it...

Rational equations can be used to solve a variety of problems that involve rates, times and work. Using rational expressions and equations can help us answer questions about how to combine workers or machines to complete a job on schedule.

## Video

### Simplifying Rational Expressions Introduction

Watch a Khan Academy Video »
Length: 15:23

## Video

Watch a Khan Academy Video »
Length: 16:30

## Video

### Subtracting Rational Expressions

Watch a Khan Academy Video »
Length: 5:35

## Video

### Multiplying and Simplifying Rational Expressions

Watch a Khan Academy Video »
Length: 5:05

## Video

### Example 1: Factoring difference of squares

Watch a Khan Academy Video »
Length: 1:49