Partial fractions, explained

By David Mashiah · 01.11.2026

Partial fractions split one complicated fraction into a sum of simple ones. Each simple piece has a known integral and a known inverse Laplace transform, which is why the method shows up in calculus, ODEs and control alike.

\frac{A s + B}{( s - r _{1} ) ( s - r _{2} )} = \frac{C _{1}}{s - r _{1}} + \frac{C _{2}}{s - r _{2}}

\frac{A s + B}{( s - r ) ^{2}} = \frac{C _{1}}{s - r} + \frac{C _{2}}{( s - r ) ^{2}}

\frac{A s + B}{s ^{2} + p s + q}, p^{2} < 4 q : \frac{C ( s + α ) + D}{( s + α ) ^{2} + β ^{2}}

C_{1} = \frac{A s + B}{s - r _{2}}_{s = r_{1}}

Before you start

The numerator’s degree must be lower than the denominator’s. If it is not, divide the polynomials first and decompose only the remainder. Then factor the denominator as far as possible over the real numbers.

Three kinds of factors

A distinct linear factor (s − r) gets one term C/(s − r). A repeated factor (s − r)² gets two terms, one for each power. A quadratic with no real roots stays whole, with a linear numerator Cs + D, usually rewritten by completing the square so it matches a sine and cosine.

Finding the coefficients

For distinct linear factors, the cover-up method is fastest: to find the coefficient of 1/(s − r₁), cover that factor and evaluate the rest at s = r₁. For repeated and quadratic factors, multiply out and compare the coefficients of each power of s.

Worked example

(s + 3)/(s² + 3s + 2) = (s + 3)/((s + 1)(s + 2)). Cover-up at s = −1: (−1 + 3)/(−1 + 2) = 2. At s = −2: (−2 + 3)/(−2 + 1) = −1.

So the fraction is 2/(s + 1) − 1/(s + 2). As an integral in x, that gives 2 ln|x + 1| − ln|x + 2| + C. As an inverse Laplace transform, 2e^(−t) − e^(−2t). These are the calculator’s default inputs.

Partial fractions calculator

(A·s + B) / (s² + p·s + q)

A B p q

Denominator–

Decomposition–

Common mistakes

Skipping the polynomial division when the numerator’s degree is too high.
Writing one term for a repeated factor instead of one per power.
Putting a constant over an irreducible quadratic. It needs a linear numerator, Cs + D.