Partial Fractions (Pure Mathematics 3 – 9709) — Mathematics

Introduction

Partial fractions is the process of splitting a single rational function into a sum of simpler fractions. In 9709 Paper 3, partial fractions appear as a standalone skill and as an essential tool for integration, binomial expansions of rational functions, and inverse Laplace transforms in further work. Mastering the correct form for each denominator type is the key examiner requirement — an incorrect form scores zero, however much algebra follows.

The syllabus restricts the cases to:

Distinct linear factors: denominator of the form $(ax+b)(cx+d)(ex+f)$
Repeated linear factor: denominator of the form $(ax+b)(cx+d)^2$
Irreducible quadratic factor: denominator of the form $(ax+b)(cx^2+d)$

In all cases the degree of the numerator must be strictly less than the degree of the denominator (proper fraction). If it is not, you must first perform polynomial division — but the syllabus explicitly excludes that situation from this objective, so every exam question here will already be a proper fraction.

Core Concept

A rational function is $\dfrac{f(x)}{g(x)}$ where $f$ and $g$ are polynomials. When $\deg(f) < \deg(g)$ the fraction is proper and can be decomposed into partial fractions whose denominators are the individual factors of $g(x)$ .

Why does this work? Every real polynomial factors into linear and irreducible quadratic factors over $\mathbb{R}$ . Each such factor contributes a term (or terms, for repeated factors) to the partial fraction decomposition. The constants in the numerators of those terms are uniquely determined — that uniqueness is what makes the method valid.

Choosing the correct form

Denominator type	Correct partial fraction form
$(ax+b)(cx+d)(ex+f)$ — three distinct linear factors	$\dfrac{A}{ax+b}+\dfrac{B}{cx+d}+\dfrac{C}{ex+f}$
$(ax+b)(cx+d)^2$ — one linear, one repeated linear	$\dfrac{A}{ax+b}+\dfrac{B}{cx+d}+\dfrac{C}{(cx+d)^2}$
$(ax+b)(cx^2+d)$ — one linear, one irreducible quadratic	$\dfrac{A}{ax+b}+\dfrac{Bx+C}{cx^2+d}$

Critical rule for the quadratic factor: the numerator over $cx^2+d$ must be linear ( $Bx+C$ ), not a single constant. This is the most common mark-losing error.

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Introduction

Core Concept

Choosing the correct form

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