Partial fraction decomposition

Partial fraction decomposition can be used to simplify a complex algebraic fraction, especially for the sake of integration, into a number of simpler partial fractions that are more easily dealt with.

Cases

Denominator with distinct linear factors

𝑓 (𝑥) = 𝑃 (𝑥) (𝑥 - 𝑎 1) \dots (𝑥 - 𝑎 𝑘) = 𝐴 1 𝑥 - 𝑎 1 + \dots + 𝐴 𝑘 𝑥 - 𝑎 𝑘

Repeated linear factors

𝑓 (𝑥) = 𝑃 (𝑥) (𝑥 - 𝑎) 𝑐 = 𝐴 1 𝑥 - 𝑎 + 𝐴 1 (𝑥 - 𝑎) 2 + \dots + 𝐴 1 (𝑥 - 𝑎) 𝑐 - 1 + 𝐴 1 (𝑥 - 𝑎) 𝑐

Irreducible factor

𝑓 (𝑥) = 𝑃 (𝑥) (𝑥 - 𝑎) (𝑥 2 + 𝑏 𝑥 + 𝑐) = 𝐴 𝑥 - 𝑎 + 𝐵 𝑥 + 𝐶 𝑥 2 + 𝑏 𝑥 + 𝑐

Note that the numerator is always one degree lower than the denominator.

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