### 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

$\phantom{\rule{2em}{0ex}}\frac{4x+7}{{x}^{2}+3x+2}\phantom{\rule{1em}{0ex}}\text{isidenticallyequalto}\phantom{\rule{1em}{0ex}}\frac{1}{x+2}+\frac{3}{x+1}$

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

#### Contents

1 Proper and improper fractions2 Proper fractions with linear factors

3 Proper fractions with repeated linear factors

4 Proper fractions with quadratic factors

5 Engineering Example 3

5.1 Admittance

6 Improper fractions