Partial Fraction Decomposition Calculator
Free partial fraction decomposition calculator with steps. Break any rational function into simple fractions — distinct, repeated and irreducible quadratic factors all handled, with exact fractions and the full coefficient solve.
Enter a numerator and a denominator — expanded or already factored — and get the decomposition with exact fractions. The calculator factors the denominator, sets up the template, and shows the linear system it solves for the unknown coefficients.
e.g. 3x+11
Expanded or factored, e.g. (x-2)(x+3) or x^2+x-6
How to do partial fraction decomposition
- 1
Make the fraction proper
If the numerator's degree is at least the denominator's, do polynomial long division first.
- 2
Factor the denominator completely
Into linear factors and irreducible quadratics, noting which ones repeat.
- 3
Write the template
One constant per power of each linear factor, one linear numerator per power of each quadratic factor.
- 4
Solve for the coefficients
Multiply through by the denominator and either match coefficients of each power or substitute convenient values of x.
The four setups
| Denominator factor | Terms it contributes | Example |
|---|---|---|
| Distinct linear (x − a) | A/(x − a) | 1/((x−1)(x+1)) |
| Repeated linear (x − a)ᵏ | A₁/(x−a) + A₂/(x−a)² + … | x²/(x−1)³ |
| Irreducible quadratic | (Bx + C)/(x² + bx + c) | 1/(x³−1) |
| Repeated quadratic | (B₁x+C₁)/q + (B₂x+C₂)/q² + … | (x³+1)/(x²+4)² |
Worked examples
Distinct linear factors
Three linear factors
Non-monic denominator
Repeated linear factor
Cancelling factor
Irreducible quadratic
Repeated quadratic
Mixed factors
Where you will need it
Integrating rational functions. There is no general antiderivative for P(x)/Q(x), but every decomposed piece integrates to a logarithm, a power, or an arctangent.
Inverse Laplace transforms. The transform table only covers simple denominators, so decomposition is the step that makes the inverse transform possible.
Control theory and signal processing. Splitting a transfer function into partial fractions separates a system's response into one mode per pole.
Series and generating functions. Decomposition turns a rational generating function into a sum of geometric series, which is how closed forms for recurrences such as Fibonacci are found.
Frequently asked questions
What is partial fraction decomposition? ▾
It is the reverse of adding fractions: rewriting a single rational function as a sum of simpler fractions whose denominators are the factors of the original denominator. It is the standard preparation step for integrating rational functions and for inverse Laplace transforms.
What are the rules for setting up the decomposition? ▾
Each distinct linear factor (x − a) contributes A/(x − a). A factor repeated k times contributes one term for each power up to k. Each irreducible quadratic contributes a linear numerator (Bx + C) over it, again one term per power if it repeats.
What do I do if the numerator's degree is too high? ▾
Divide first. If the numerator's degree is greater than or equal to the denominator's, polynomial long division produces a polynomial plus a proper fraction, and only the proper part gets decomposed. The calculator does this automatically.
What is the Heaviside cover-up method? ▾
A shortcut for distinct linear factors: to find the constant over (x − a), cover that factor in the original expression and substitute x = a into what is left. It only works for non-repeated linear factors, but it is very fast when it applies.
When is a quadratic irreducible? ▾
When its discriminant b² − 4ac is negative, so it has no real roots and cannot be factored further over the reals. Such a factor keeps a linear numerator Bx + C rather than a constant.
Why is partial fraction decomposition useful? ▾
Integration and inverse Laplace transforms both have simple rules for 1/(x − a) and for terms over a quadratic, but no rule for a general rational function. Decomposing first turns one hard problem into several easy ones.
About this Calculator
Free partial fraction decomposition calculator with steps. Break any rational function into simple fractions — distinct, repeated and irreducible quadratic factors all handled, with exact fractions and the full coefficient solve.