Partial Fraction Decomposition Calculator

A way to rewrite a rational function as a sum of simpler rational functions. For example, (3x + 1) / ((x − 1)(x − 2)) can be split into A / (x − 1) + B / (x − 2). The simpler pieces are easier to integrate or invert.

Multiply both sides by the common denominator, then evaluate at strategic x values (the roots of the original denominator). The calculator uses the closed-form A = (a·r₁ + b)/(r₁ − r₂) and B = (a·r₂ + b)/(r₂ − r₁).

When the denominator has a repeated factor (x − r)², the decomposition takes a different shape: A/(x − r) + B/(x − r)². That case is out of scope for this calculator.

If the denominator includes a factor like x² + 1 that has no real roots, the partial fraction piece uses (Ax + B)/(x² + 1). That is also out of scope for this calculator's distinct-linear-factor form.

Use polynomial long division first to write the rational function as a polynomial plus a proper rational remainder. Then decompose the remainder. The polynomial long division calculator handles the first step.

Mostly in integration (so you can integrate each simpler fraction with the log rule) and in inverse Laplace transforms (so you can invert each piece using a standard table).