What types of denominators are supported?

Distinct linear factors (x - r) and repeated linear factors (x - r)^2 and (x - r)^3. Quadratic and irreducible factors are not yet supported.

Does the degree of the numerator matter?

Yes. The numerator degree must be strictly less than the denominator degree (proper rational function). If your fraction is improper, do polynomial long division first, then decompose the remainder.

How are coefficients entered?

Enter coefficients from the highest power to the constant term, separated by spaces. For 3x^2 + 2x + 1, enter "3 2 1".

Can two factors share the same root?

If you need (x - r)^2, add a single factor with root r and multiplicity 2 instead of adding the same root twice as distinct factors.

What does the tool output?

It outputs the partial fraction form such as A/(x-r1) + B/(x-r2) and the exact decimal values of all unknown coefficients.

Partial Fraction Decomposition

Name: Partial Fraction Decomposition
Availability: InStock
Author: Nham Vu

Enter a rational function and get its partial fraction decomposition with step-by-step coefficients.

Enter Rational Function

Numerator coefficients (highest degree first, space-separated)

Numerator: x

Denominator factors

Denominator: —

Quick Examples

Enter a function and click Decompose

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Summary

Enter a rational function and get its partial fraction decomposition with step-by-step coefficients.

How it works

Enter the numerator polynomial coefficients (highest degree first).
Enter each denominator factor as a root value (the value r where the factor is (x - r)).
Specify the multiplicity of each factor (1 for distinct, 2 or 3 for repeated).
Click Decompose — the tool solves the linear system to find each coefficient A, B, C.
The result shows the full partial fraction form and individual terms.

Use cases

Compute partial fractions before performing inverse Laplace transforms.
Integrate rational functions by hand or check your textbook work.
Verify decomposition results when studying for calculus exams.
Solve differential equations that require partial fraction expansions.
Check coefficient values when working control-system transfer functions.

Frequently Asked Questions

Last updated: 2026-07-24 · Reviewed by Nham Vu