Partial Fraction Decomposition

Summary

This tool decomposes a proper rational function P(x)/Q(x) into a sum of simpler fractions. It handles denominators that factor into distinct linear terms and repeated linear terms up to multiplicity 3. Enter your numerator coefficients and denominator roots to get the full expansion with all coefficients.

How it works

1. Enter the numerator polynomial coefficients (highest degree first).
2. Enter each denominator factor as a root value (the value r where the factor is (x - r)).
3. Specify the multiplicity of each factor (1 for distinct, 2 or 3 for repeated).
4. Click Decompose — the tool solves the linear system to find each coefficient A, B, C.
5. 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