Partial Fractions

Partial fractions is the reverse process of adding algebraic fractions: you decompose a single fraction into a sum of simpler fractions. This technique is essential for integration and for working with series at A-Level, and is explicitly required by the Edexcel 9MA0 specification.

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Why Partial Fractions?

A fraction like (3x + 5)/((x + 1)(x + 2)) is difficult to integrate directly. But if we write it as A/(x + 1) + B/(x + 2), each part can be integrated easily using the natural logarithm.

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Type 1: Distinct Linear Factors

If the denominator has distinct (different) linear factors:

f(x)/((x − a)(x − b)) = A/(x − a) + B/(x − b)

Example: Express (5x + 3)/((x + 1)(x − 2)) in partial fractions.

Let (5x + 3)/((x + 1)(x − 2)) = A/(x + 1) + B/(x − 2).

Multiply both sides by (x + 1)(x − 2):

• 5x + 3 = A(x − 2) + B(x + 1)