Partial fraction decomposition takes a single rational expression and splits it into a sum of simpler fractions — the reverse of finding a common denominator. It shows up constantly in calculus (integrating rational functions), differential equations (inverse Laplace transforms), and control theory. This calculator handles the three cases every intro course covers: distinct linear factors, a repeated linear factor, and a linear factor paired with an irreducible quadratic.
Why decompose a fraction at all
Some operations are easy on a sum of simple fractions but hard on one complicated fraction. Integrating 1/(x − 1) is a one-line log rule; integrating (3x + 5)/[(x + 1)(x + 2)] directly is not obvious until it is rewritten as 2/(x + 1) + 1/(x + 2), at which point it is two easy log integrals added together. The same trick simplifies inverse Laplace transforms and partial-derivative work in engineering courses.
The cover-up method for distinct roots
When every denominator factor is a different linear term (x − r), each one gets exactly one unknown constant. The fastest way to find it is the cover-up (Heaviside) method: multiply the whole equation by that single factor, which cancels it on the left and turns every other term on the right into zero the moment you substitute x equal to that factor's root. What's left is the constant, computed directly — no system of equations required.
Repeated and irreducible-quadratic factors need more terms
A repeated factor (x − r)^k does not get one term — it needs a separate term for every power from 1 through k, because a single constant over (x − r)^k cannot reproduce every possible numerator degree below k. Similarly, an irreducible quadratic factor (one with no real roots) needs a linear numerator, Bx + C, instead of a plain constant, since a constant alone cannot match a numerator that varies with x. Get the term shapes wrong and no set of constants will make the decomposition equal the original expression.
Checking your work
The surest way to catch an arithmetic slip is to reverse the process: multiply the decomposition back out over the common denominator and confirm the numerator matches the one you started with. This calculator's Verify tab does exactly that — both symbolically (recombining the terms into a single polynomial) and numerically (evaluating the original expression and the decomposition at the same x-value and comparing the results).