Partial Fraction Calculator
Split a fraction of polynomials into partial fractions, with repeated linear factors, irreducible quadratic factors and improper fractions, step by step and in exact fractions.
About the Partial Fraction Calculator
A proper fraction N(x)/D(x) (the top of lower degree) is written as a sum of simpler fractions, one for each factor of D: A/(x − a) for a linear factor, A/(x − a) + B/(x − a)² for a repeated one, and (Bx + C)/(x² + px + q) for a quadratic with no real roots. If the fraction is improper, divide first to get a polynomial plus a proper fraction.
The constant over a non-repeated linear factor comes quickly by the cover-up rule: cover the factor and put x = a in what is left. The rest come from comparing coefficients. Partial fractions are used to integrate rational functions and for inverse Laplace transforms.
The formula
(px + q)/((x − a)(x − b)) = A/(x − a) + B/(x − b)
Worked example
(3x + 5)/((x − 1)(x + 2)²)
- = A/(x − 1) + B/(x + 2) + C/(x + 2)²
- Cover-up at x = 1: A = 8/9; at x = −2: C = 1/3
- Comparing x²: A + B = 0, so B = −8/9
Questions people ask
Why divide first?
Partial fractions only work for a proper fraction. Long division gives a polynomial plus a proper fraction, and only the proper part is split.
Does the denominator have to factorise?
Yes, into factors with rational coefficients: linear factors and quadratics with no real roots. A cubic with only irrational roots can't be split here.