Partial Fraction Decomposition (2 distinct linear)

P(x) / [(x−a)(x−b)] = A/(x−a) + B/(x−b). Distinct linear factors only.

Inputs

Result

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How to use this calculator

  • Enter linear numerator + 2 distinct roots.

About this calculator

Partial fraction decomposition splits a rational function into simpler fractions. For (px+q) / [(x−a)(x−b)] with distinct a, b: A = P(a)/(a−b), B = P(b)/(b−a). Used in integration (each simpler fraction has known antiderivative ln(x−c)) and Laplace transforms. For repeated factors, irreducible quadratics, or higher-order: more complex algorithms (cover-up method extended). Source: Wolfram MathWorld - Partial Fraction Decomposition.

Frequently asked

Why decompose?+
To integrate or transform. ∫1/(x−a) dx = ln|x−a|. Each piece becomes elementary.
Repeated root?+
(x − a)² in denominator: A/(x−a) + B/(x−a)². Different formula.
Irreducible quadratic?+
(x²+1) in denominator: needs (Ax+B)/(x²+1). Trig substitution after.
Cover-up method?+
Multiply both sides by (x−a), set x=a — A is what's left. Quick for simple cases.
Use in Laplace transforms?+
Inverse Laplace of complicated F(s) often requires partial fraction first.

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