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Partial Fraction Calculator

Decompose rational expressions

Expression Input

Input Format:

  • • Use x as the variable
  • • Use ^ for exponents (e.g., x^2)
  • • Use + and - for operations
  • • Coefficients can be integers or decimals

Solution

Original Expression:
(x + 1)/(x^2 - 1)

Long Division:

Quotient: x + 2
Remainder: 3x + 1

Factored Form:

1x

Partial Fraction Decomposition:

A/(x + 1) + B/(x - 1)

Solution Steps:

  1. 1Perform polynomial long division first
  2. 2Factor the denominator
  3. 3Set up partial fraction decomposition
  4. 4Find coefficients by solving system of equations

Understanding Partial Fractions

When to Use

  • • Integration of rational functions
  • • Simplifying complex fractions
  • • Finding inverse Laplace transforms
  • • Solving differential equations

Prerequisites

  • • Proper/improper fractions
  • • Polynomial factoring
  • • Linear equations
  • • Polynomial long division

Decomposition Types

Linear Factors

  • • A/(x + a)
  • • Simple terms
  • • One coefficient
  • • Basic integration

Repeated Factors

  • • A/(x + a)^n
  • • Multiple terms
  • • Higher powers
  • • More complex

Irreducible Quadratic

  • • (Ax + B)/(x² + px + q)
  • • Cannot factor
  • • Two coefficients
  • • Special methods