Partial Fraction Calculator
Decompose a rational function into simpler fractions and see the coefficient-matching steps.
Accepted expressions
The numerator may be any polynomial through degree 8. The denominator must already be written as a product of factors, for example (x-1)(x+2), (x-3)^2(x+1), or (x^2+1)(x-2). Each factor may be linear or quadratic, powers may be 1, 2 or 3, and the total denominator degree is limited to 6.
How partial fraction decomposition works
If the numerator degree is at least the denominator degree, polynomial division runs first. For every linear factor (ax+b)^m, the decomposition includes A₁/(ax+b) through Aₘ/(ax+b)^m. For every quadratic factor q(x)^m, it includes (B₁x+C₁)/q(x) through (Bₘx+Cₘ)/q(x)^m. The calculator multiplies by the common denominator, matches polynomial coefficients and solves the resulting linear system with partial pivoting.
Checked examples
| Rational function | Decomposition |
|---|---|
| (2x+3)/[(x-1)(x+2)] | (5/3)/(x-1) + (1/3)/(x+2) |
| (3x+5)/[(x^2+1)(x-2)] | (11/5)/(x-2) + [(-11/5)x+(2/5)]/(x^2+1) |
| (x^3+2x^2+3x+4)/[(x-1)(x+1)] | x+2 + 5/(x-1) – 1/(x+1) |
| 1/[(x-1)(x+1)] | (1/2)/(x-1) – (1/2)/(x+1) |
| x/[(x-1)^2] | 1/(x-1) + 1/(x-1)^2 |
| 1/(x^2+1) | 1/(x^2+1) |
Limits
- This calculator accepts one variable, x, and real decimal coefficients.
- The denominator must be supplied in factored form; it does not factor arbitrary polynomials.
- A zero factor, malformed exponent or singular coefficient system is rejected rather than guessed.
- Displayed fractions are rational approximations of coefficients solved numerically; the verification samples avoid denominator zeros.
Related tools: use the Fraction Calculator for arithmetic fractions, the Polynomial Calculator for polynomial operations, or the System of Equations Calculator for linear systems.
Solving a rational equation instead? Use the Solve for X Fraction Calculator for proportions and linear or quadratic fraction equations.