What this calculator does
Partial fraction decomposition reverses adding fractions. A single expression over a product of factors becomes a sum of simpler fractions, each with one factor in its denominator, which is what makes it integrable.
The cover-up method makes it quick. For (x + 1) over (x − 1)(x − 2) the decomposition is −2.000 over (x − 1) plus 3.000 over (x − 2), found by substituting each root in turn.
The formula
For distinct roots, each numerator is the original numerator evaluated at that root, divided by the difference of the roots. A repeated root needs a different form with both a linear and a squared denominator.
| Term | Meaning |
|---|---|
| Partial fractions | A sum of simple fractions equivalent to one complicated one. |
| Cover-up method | Finding each numerator by substituting the corresponding root. |
| Repeated root | When both factors are the same, requiring a squared term in the decomposition. |
The inputs explained
| Field | What to enter |
|---|---|
| Numerator: coefficient of x (a) | Coefficient of x in the numerator. |
| Numerator: constant (b) | Constant term in the numerator. |
| Denominator root p (x−p) | First denominator root, giving the factor (x − p). |
| Denominator root q (x−q) | Second denominator root. Setting it equal to p gives the repeated-root case. |
When to use it
Preparing an integral
Rational functions become integrable once split into partial fractions.
Inverse Laplace transforms
The standard method requires decomposing into recognisable pieces first.
Checking algebra
Recombining the fractions should return the original expression.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do the numerators change with the roots?
The same numerator with three second roots.
Questions
Why does a repeated root need a different form?
Because two copies of the same fraction would just add together into one, which cannot reproduce the original. The squared denominator supplies the extra independent piece the decomposition needs.
What is the cover-up method?
Substituting each root into the numerator while ignoring its own factor in the denominator. It works because that factor vanishes at its own root, leaving only one term, and it is considerably faster than equating coefficients.
What if the numerator degree is too high?
Divide first. Partial fractions require the numerator degree to be below the denominator degree, so a polynomial division step comes first and leaves a proper fraction to decompose.
What about irreducible quadratic factors?
They need a linear numerator rather than a constant, in the form (Ax + B) over the quadratic. This calculator handles the two-linear-factor case, which covers most introductory work.
For dividing polynomials, see the synthetic division calculator. For solving quadratics, see the quadratic calculator.