Partial Fraction Decomposition Calculator
Almost nobody gets the arithmetic wrong here. What goes wrong is the line before it — deciding what shape the answer is supposed to take.
(x−1)(x+2) gives A/(x−1) + B/(x+2). But (x−1)² needs both A/(x−1) and B/(x−1)², and x²+1 needs Ax+B on top rather than a single letter. Choose the wrong shape and the system you build has no solution, however carefully you solve it. This partial fraction decomposition calculator names the shape each factor produces and says why, before it works anything out — so you can see how to find the partial fraction decomposition rather than only what it turned out to be.
How It Finds the Decomposition, with Steps
Decomposition into partial fractions starts at the bottom: it factors the denominator first, because nothing can be decided until that is a product. Then it reads the shape off those factors and tells you what each one contributes: a plain linear factor takes one unknown, a factor repeated three times takes a term at every power from one to three, and an irreducible quadratic takes Ax + B.
If the top is not of lower degree than the bottom, it divides before anything else. That step is skipped more often than any other, and skipping it is why the algebra sometimes simply refuses to come out. The quotient sits outside the fractions and only the remainder gets split — x³/(2x−1), for instance, leaves a quotient with a half in it, which the calculator keeps exact rather than rounding.
Then it solves for the unknowns and shows every value as an exact fraction. Nothing is ever a decimal approximation.
How to Use This Partial Fraction Decomposition Calculator
Put the numerator in the top box and the denominator in the bottom, then press =. The bottom can be written factored, as (x−1)(x+2), or expanded as x²+x−2 — it is factored for you either way. The line under the boxes shows how your input was read.
The answer appears as the finished sum of pieces. One tap opens the shape table, where each factor of the denominator is listed with the reason it produces what it does, and below that the full step by step: divide if needed, factor, choose the shape, solve. It is a free online partial fraction decomposition calculator — no account, no download, and no limit on how many you work through.
If you are meeting this in calculus rather than algebra, the reason is integration: each piece integrates on its own while the original fraction does not, so an integral partial fraction decomposition is usually the first line of the working rather than the last.
Frequently Asked Questions
How do you find the partial fraction decomposition?
What if the numerator has a higher degree?
Why does a repeated factor need more than one term?
Why is partial fraction decomposition used in integration?
Factor the bottom, decide the shape from the factors, divide first if the top is too big, and only then solve. Get those four in the right order and the rest is arithmetic.