1.3PureStretch

Partial Fractions

Partial fractions reverse the process of adding fractions: a single fraction with a factorised denominator is split back into a sum of simpler ones. This is essential for integration and binomial expansion later.

30 min Video by Zeeshan Zamurred Algebraic Methods

Edexcel A level Maths: 1.3 Partial FractionsWatch the full walkthrough before the notes below.

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What you'll be able to do

Split a fraction with distinct linear factors
Find the unknown constants (substitution method)
Use the cover-up method as a shortcut
Check the result by recombining

The form

A proper fraction whose denominator factorises into $distinct linear factors$ splits into a sum of fractions, one per factor, each with an unknown constant on top.

\frac{p x + q}{( x + a ) ( x + b )} = \frac{A}{x + a} + \frac{B}{x + b}

One term per linear factor, with constants

A

B

to find.

Finding the constants

Multiply both sides by the denominator to clear fractions, then substitute clever values of $x$ (each root) to isolate one constant at a time.

1Write

5 x + 1 = A (x - 2) + B (x + 1)

2Let

x = 2

11 = 3 B \Rightarrow B = \frac{11}{3}

3Let

x = - 1

- 4 = - 3 A \Rightarrow A = \frac{4}{3}

Answer

\frac{4/3}{x + 1} + \frac{11/3}{x - 2}

Tip — Substitute each root of the denominator — it kills all but one constant, solving it instantly.

The cover-up method

For distinct linear factors, you can find each constant directly: to get $A$ (over $x + a$ ), cover up $(x + a)$ and evaluate the rest at $x = - a$ . It is the substitution method done mentally.

Formula recap

\frac{p x + q}{( x + a ) ( x + b )} = \frac{A}{x + a} + \frac{B}{x + b}

Distinct linear factors.

substitute each root to find A, B

Solving the constants.

cover-up: evaluate at x = - a

Shortcut for each constant.

Common mistakes to avoid

Applying partial fractions to an improper fraction directly.

Divide first if the numerator degree ≥ denominator degree (see Algebraic Division).

Forgetting a term for each distinct factor.

There is exactly one partial fraction per linear factor.

Key takeaways

A proper fraction over distinct linear factors splits as A/(x+a) + B/(x+b).
Clear the denominator, then substitute each root to find the constants.
The cover-up method finds each constant quickly.

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