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Partial Fraction Decomposition Calculator with Steps

Decompose a rational function N(x)/D(x) into partial fractions: factor the denominator, set up A/(x − r) terms (repeated and quadratic factors too), solve for the constants exactly.

Numerator coefficients
Denominator coefficients
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How it works

N/D with deg N < deg D · each factor (x − r) → A/(x − r) · repeated (x − r)ᵏ → A₁/(x − r) + … + Aₖ/(x − r)ᵏ · irreducible quadratic q(x) → (Bx + C)/q(x) · constants from N(x) = Σ (constant × D(x)/factor)

Partial fraction decomposition splits a rational function into a sum of simpler fractions, the key step for integrating rational functions and for inverse Laplace and z-transforms. If the numerator’s degree is not lower than the denominator’s, polynomial long division comes first. The denominator is factored (here using the rational root theorem); each linear factor gets a constant over it, a repeated factor gets one term per power, and an irreducible quadratic gets a linear numerator. Multiplying through by the denominator and matching coefficients gives a linear system that is solved exactly with fractions.

Worked example

Decompose (x + 5)/(x² + x − 2). The denominator factors as (x + 2)(x − 1), so write A/(x + 2) + B/(x − 1). Multiplying through gives x + 5 = A(x − 1) + B(x + 2). Matching coefficients (or setting x = 1 and x = −2) gives A = −1 and B = 2, so the result is −1/(x + 2) + 2/(x − 1).

Sources: Partial fraction decomposition — Wikipedia · Paul’s Online Notes — Partial fractions

Frequently asked questions

What is the cover-up method?

For a non-repeated linear factor (x − r), cover that factor in the original fraction and substitute x = r into what remains. For (x + 5)/((x + 2)(x − 1)), covering (x − 1) and putting x = 1 gives 6/3 = 2 — the constant over (x − 1).

How are repeated factors handled?

A factor (x − r)ᵏ needs k terms: A₁/(x − r) + A₂/(x − r)² + … + Aₖ/(x − r)ᵏ. For example 1/(x(x − 1)²) needs A/x + B/(x − 1) + C/(x − 1)².

What about x² + 1 in the denominator?

A quadratic that cannot be factored over the rationals gets a linear numerator: (Bx + C)/(x² + 1). This calculator supports one such quadratic factor alongside any rational linear factors.

Why do I need long division first?

Partial fractions only work for proper fractions (numerator degree lower than denominator degree). Long division splits off a polynomial and leaves a proper fraction to decompose.

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This calculator is for educational purposes. Double-check important results before acting on them.