Partial Fractions Calculator
Decompose a rational expression into partial fractions from numerator coefficients and denominator roots.
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Request a ToolHow to Use the Partial Fractions Calculator
Enter the numerator as polynomial coefficients (highest degree first, comma-separated) and the denominator as its roots. The calculator computes the partial fraction decomposition.
- Enter numerator coefficients. List coefficients from highest to lowest degree, separated by commas. For x^2 - 1, enter "1, 0, -1".
- Enter denominator roots. List the roots of the denominator polynomial. For (x-1)(x+1)(x-2), enter "1, -1, 2".
- Read the result. The decomposition shows each partial fraction term with its coefficient.
This calculator assumes distinct linear factors in the denominator (no repeated roots).
About Partial Fraction Decomposition
Partial fraction decomposition rewrites a rational expression as a sum of simpler fractions. It is a key technique in calculus for integrating rational functions, in control theory for inverse Laplace transforms, and in signal processing. The method works by expressing N(x)/D(x) as A1/(x-r1) + A2/(x-r2) + ... where r1, r2, etc. are the roots of D(x).
Frequently Asked Questions
What is partial fraction decomposition?
It is a technique that breaks a complex rational expression into a sum of simpler fractions, each with a linear denominator. This makes integration and other operations much easier.
When do you use partial fractions?
Partial fractions are most commonly used in calculus to integrate rational functions, and in engineering for inverse Laplace transforms in control systems.
Does this work with repeated roots?
This calculator handles distinct linear factors only. Repeated roots require additional terms (A/(x-r) + B/(x-r)^2 + ...) which are not supported here.