A fraction like (5x + 1) / ((x + 2)(x − 3)) looks like a single tangled knot, but partial fraction decomposition unties it into a sum of simple pieces — something like 1/(x + 2) + 4/(x − 3). Each piece has a plain linear denominator, and that simplicity is the whole point: simple fractions are easy to integrate, easy to invert with Laplace transforms, and easy to reason about. Electrical engineers lean on this constantly when they convert a circuit's transfer function back into a time-domain signal.

The method runs in three moves. Factor the denominator completely, then write one unknown-topped fraction per factor: A/(x + 2) + B/(x − 3). Next, solve for the constants A and B by clearing denominators and matching coefficients (or by plugging in the handy x-values −2 and 3). Finally, recombine to verify. This calculator assists the verification step: type your solved constant A in the Variable A box and constant B in the Variable B box, and the Result box shows their product — cross-check it against the cross-terms of your recombined expansion to confirm the decomposition is correct.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Partial Fraction Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.