Partial Fraction Decomposition Calculator
Decompose a rational function into partial fractions. Supports proper and improper fractions, real linear factors, and irreducible quadratics.
How the calculation works
Partial fraction decomposition splits a rational expression into simpler terms so you can integrate, invert a Laplace transform, or expand a generating function. Type polynomials in x, such as x^2-5x+6 or (x-1)(x+2). Parentheses and implicit multiplication (2x, (x+1)(x-3)) are allowed.
The solver factors the denominator, performs polynomial division when the fraction is improper, then solves for the undetermined coefficients. Results are printed in a form you can copy into homework or a CAS. Degree 4 and lower is the practical range for this calculator.
Formula
Worked example
(x+3)/(x²−x−6) = (x+3)/((x−3)(x+2)) = (6/5)/(x−3) + (−1/5)/(x+2).
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Frequently asked questions
How do I decompose a partial fraction?
Factor the denominator, write a template (A/(x−r) for each linear factor, Bx+C for each irreducible quadratic), then solve for the unknowns. This calculator does those steps for you.
What if the fraction is improper?
If the numerator degree is at least the denominator degree, divide polynomials first. The calculator shows the polynomial part plus the proper remainder decomposed.
Can I enter (x-1)(x+2)?
Yes. Factored denominators are expanded automatically before solving.
Does this work with repeated roots?
Yes for repeated linear factors (terms 1/(x−r)^k). Repeated irreducible quadratics are limited.
Why is this used in Laplace transforms?
Inverse Laplace of a rational F(s) is easiest after partial fractions, because each term has a known inverse.
Results are estimates for informational purposes and are not financial or professional advice.