Partial Fraction Decomposition Calculator

Decompose rational functions into simpler partial fractions

Enter Your Rational Function
Examples: (x^2+3x+2)/(x^2+2x+1), (2x+1)/(x^2-4), (x^3+1)/(x^2-1)
Decomposition Result
Click "Decompose" to see result
Step-by-Step Solution

Enter a rational function and click "Decompose" to see the step-by-step solution.

What is Partial Fraction Decomposition?

Partial fraction decomposition breaks down a rational function into a sum of simpler fractions. It's essential for:

  • 📚 Calculus integration
  • 🧮 Solving differential equations
  • 📈 Laplace transforms
  • 🔧 Engineering mathematics

Common Cases

  • Linear Factors: A/(x-a) + B/(x-b)
  • Repeated Linear: A/(x-a) + B/(x-a)²
  • Quadratic Factors: (Ax+B)/(x²+px+q)
  • Improper Fractions: Polynomial division first

Input Format Guide

  • Use ^ for exponents: x^2 + 2x + 1
  • Use * for multiplication: 2*x
  • Factor format: (x+1)(x+2)
  • Examples: x^2 + 3x + 2, (x+1)(x+2)
Frequently Asked Questions
What is partial fraction decomposition used for?

Partial fraction decomposition is primarily used in calculus to integrate rational functions. It's also essential for solving differential equations, performing Laplace transforms, and simplifying complex algebraic expressions in engineering and physics.

How do I know which partial fraction form to use?

The form depends on the denominator's factors:

  • Linear factors (x-a): Use A/(x-a)
  • Repeated linear (x-a)²: Use A/(x-a) + B/(x-a)²
  • Irreducible quadratic (x²+px+q): Use (Ax+B)/(x²+px+q)
  • Repeated quadratic (x²+px+q)²: Use (Ax+B)/(x²+px+q) + (Cx+D)/(x²+px+q)²
What if the numerator degree is greater than the denominator?

If the degree of numerator ≥ degree of denominator, you must first perform polynomial long division to get a polynomial plus a proper fraction. Then decompose the proper fraction.

Can this calculator handle complex roots?

Yes, the calculator can handle irreducible quadratic factors (complex roots). It will decompose them into terms of the form (Ax+B)/(quadratic).

What are the different cases for partial fraction decomposition?
  • Case 1: Denominator has distinct linear factors
  • Case 2: Denominator has repeated linear factors
  • Case 3: Denominator has irreducible quadratic factors
  • Case 4: Denominator has repeated quadratic factors
  • Case 5: Improper rational functions (degree numerator ≥ degree denominator)
How accurate is the decomposition?

The calculator uses standard algebraic methods to find coefficients A, B, C, etc. It's accurate for proper rational functions and provides step-by-step explanations for educational purposes.