### Introduction

It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For example it can be shown that $\frac{4x+7}{{x}^{2}+3x+2}$ has the same value as $\frac{1}{x+2}+\frac{3}{x+1}$ for any value of $x$ . We say that

and that the partial fractions of $\frac{4x+7}{{x}^{2}+3x+2}$ are $\frac{1}{x+2}$ and $\frac{3}{x+1}$ .

The ability to express a fraction as its partial fractions is particularly useful in the study of Laplace transforms, $Z$ -transforms, Control Theory and Integration. In this Section we explain how partial fractions are found.

#### Prerequisites

• be familiar with addition, subtraction, multiplication and division of algebraic fractions

#### Learning Outcomes

• distinguish between proper and improper fractions
• express an algebraic fraction as the sum of its partial fractions