Partial Fractions

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 $\frac{4 x + 7}{x^{2} + 3 x + 2}$ has the same value as $\frac{1}{x + 2} + \frac{3}{x + 1}$ for any value of $x$ . We say that

$\frac{4 x + 7}{x^{2} + 3 x + 2} is identically equal to \frac{1}{x + 2} + \frac{3}{x + 1}$

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

1 Proper and improper fractions

2 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