Partial Fractions Calculator
Decompose proper rational functions into partial fractions with a practical, step-by-step solver. Use expression mode for a single rational expression or structured factor mode when the denominator is already factored.
Use one top-level slash. The numerator and denominator should each be polynomials in x.
Supported Partial Fraction Forms
Distinct linear factors
A factor like (x - 2)(x + 3) becomes A/(x - 2) + B/(x + 3). Each linear factor gets one constant coefficient.
Repeated linear factors
A repeated factor like (x - 1)^3 becomes A/(x - 1) + B/(x - 1)^2 + C/(x - 1)^3 so every power is represented explicitly.
Irreducible quadratics
A factor like x^2 + 1 or 2x^2 + 3x + 5 uses a linear numerator, such as (Ax + B)/(x^2 + 1), because quadratics need two coefficients.
Why This Page Is Safer Than a Generic CAS
Honest support boundaries
If the denominator cannot be split into supported factors, the calculator reports that the input is unsupported instead of inventing a decomposition.
Coefficient-solving visibility
The solver shows the template, the coefficient-matching equations, and the final values so the decomposition is auditable.
Frequently Asked Questions
What types of denominators does this calculator support?
It supports proper rational functions whose denominators can be written as distinct linear factors, repeated linear factors, and irreducible quadratic factors. If the denominator cannot be factored into those patterns reliably, the tool will say so instead of guessing.
Does the calculator handle repeated factors?
Yes. A repeated linear factor like (x - 1)^2 expands into A/(x - 1) + B/(x - 1)^2. A repeated irreducible quadratic like (x^2 + 1)^2 expands into (Ax + B)/(x^2 + 1) + (Cx + D)/(x^2 + 1)^2.
Why does an irreducible quadratic use Ax + B in the numerator?
A quadratic denominator needs a linear numerator so the decomposition has enough degrees of freedom to match the original polynomial after multiplying through. That is the standard partial-fractions form for irreducible quadratic factors.
What if my rational function is improper?
This page is designed for proper rational functions only, where the numerator degree is smaller than the denominator degree. If your expression is improper, divide first and then use the calculator on the proper remainder.
Can I enter the denominator already in factored form?
Yes. The structured factor mode accepts factor lists such as x - 1, x - 1, x + 2, x^2 + 1. It is the best option when you already know the denominator factors or want to skip automatic factoring.
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