Partial fractions

The idea

Split a fraction with a factorised denominator into simpler fractions: $\frac{5x + 1}{(x - 1)(x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2}$ . A repeated factor $(x + 1)^2$ needs both $\frac{B}{x + 1}$ and $\frac{C}{(x + 1)^2}$ .

Write the form with unknown constants.
Multiply through by the denominator.
Substitute values of $x$ that make a bracket zero, then compare coefficients for any constant left.

Watch out A repeated factor needs a term for each power.

See it worked

Worked example

Express $\frac{5x + 1}{(x - 1)(x + 2)}$ in partial fractions.

$5x + 1 = A(x + 2) + B(x - 1)$ .
$x = 1$ : $6 = 3A$ , so $A = 2$ . $x = -2$ : $-9 = -3B$ , so $B = 3$ .
$\frac{2}{x - 1} + \frac{3}{x + 2}$ .

Answer $\frac{2}{x - 1} + \frac{3}{x + 2}$

Your turn

Question 1Non-calculator3 marks

$\frac{3x + 7}{(x + 1)(x + 3)} \equiv \frac{A}{x + 1} + \frac{B}{x + 3}$ . Find $A$ and $B$ .

Question 2Non-calculator3 marks

Simplify $\frac{x^2 - 9}{x^2 + x - 6}$ .

Question 3Non-calculator3 marks

$\frac{x + 5}{(x - 1)(x + 1)^2} \equiv \frac{A}{x - 1} + \frac{B}{x + 1} + \frac{C}{(x + 1)^2}$ . Find $A$ , $B$ and $C$ .

In past papers

38 official questions in 37 papers since 2018. The latest:

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A-level Maths Pro, Pearson Edexcel 9MA0

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