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Partial Fraction Calculator

Free online fraction decomposition calculator — Instant results with step-by-step solutions

Enter Rational Function

Use this partial fraction calculator to decompose any rational expression. Enter in format: (numerator)/(denominator)

Supports x as variable, ^ for powers, * for multiplication. Use parentheses around numerator and denominator.
Invalid expression format

Partial Fraction Calculator: The Ultimate Fraction Decomposition Tool

Welcome to the most powerful partial fraction calculator available online. This fraction decomposition calculator instantly breaks down any rational function into simpler partial fractions with complete step-by-step explanations. Whether you're a student, teacher, or professional, our tool handles all cases with precision.

From basic linear factors to complex repeated quadratics, our partial fraction calculator delivers accurate results in seconds. No software installation required — completely free and accessible from any device.

"Divide et impera" — Divide and conquer the complexity of rational functions

How the Partial Fraction Calculator Works

Our fraction decomposition calculator follows the standard algebraic method:

  1. Input Validation: Ensures proper rational expression format
  2. Polynomial Division: For improper fractions (degree of numerator ≥ denominator)
  3. Factorization: Complete factorization of denominator
  4. Template Setup: Correct partial fraction form based on factor types
  5. Coefficient Solving: Heaviside cover-up or substitution method
  6. Verification: Reconstruct to confirm accuracy

All Partial Fraction Cases Supported

This partial fraction calculator handles every standard decomposition case:

Case 1: Distinct Linear Factors

\frac{3x+5}{(x-1)(x+2)(x-3)} = \frac{A}{x-1} + \frac{B}{x+2} + \frac{C}{x-3}

Result: A = -4/3, B = -1/15, C = 7/5

Case 2: Repeated Linear Factors

\frac{2x+1}{(x-1)^3} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{(x-1)^3}

Case 3: Irreducible Quadratic Factors

\frac{x+2}{x(x^2+1)} = \frac{A}{x} + \frac{Bx+C}{x^2+1}

Case 4: Improper Fractions

\frac{x^3+2x^2+x+1}{x^2+1} = x+2 + \frac{-x+1}{x^2+1}

Why Choose This Fraction Decomposition Calculator?

Unlike basic calculators, our partial fraction calculator offers:

  • 100% Free & No Sign-up
  • Step-by-Step Solutions
  • All Factorization Cases
  • Input Validation
  • Mobile Optimized
  • Instant Results

Applications of Partial Fraction Decomposition

  • Integration: Simplify complex rational integrals
  • Laplace Transforms: Essential for inverse transforms
  • Control Systems: Analyze transfer functions
  • Signal Processing: Design and analyze filters
  • Differential Equations: Solve linear systems

Input Guidelines for Best Results

  • Use x as the variable
  • Use ^ for powers: x^3
  • Use * for multiplication or implicit: 2x
  • Always use parentheses: (2x+3)/(x^2-1)
  • No spaces in expressions
  • Currently supports up to quadratic denominators

Frequently Asked Questions

What is partial fraction decomposition?

Partial fraction decomposition is a technique to express a rational function as a sum of simpler fractions. This partial fraction calculator automates the entire process with detailed steps.

Can this handle improper fractions?

Yes! Our fraction decomposition calculator automatically performs polynomial long division when the numerator degree is greater than or equal to the denominator.

Does it show work?

Absolutely. Every calculation includes complete step-by-step solutions showing factorization, coefficient calculation, and verification.

Start Using the Partial Fraction Calculator Now

The most accurate and user-friendly partial fraction calculator for students and professionals worldwide.