Partial Decomposition Calculator

Partial Fraction Decomposition Documentation

What Is It?

A mathematical tool that decomposes rational functions (P(x)/Q(x)) into simpler fractions:

• Handles proper/improper fractions
• Supports linear/repeated/quadratic factors
• Shows step-by-step solutions
• Visualizes coefficient magnitudes

Key Formulas

1. Linear Factors:

   P(x)/((x-a)(x-b)) = A/(x-a) + B/(x-b)

2. Repeated Factors:

   P(x)/(x-a)^n = A₁/(x-a) + A₂/(x-a)² + ... + Aₙ/(x-a)ⁿ

3. Quadratic Factors:

   P(x)/(ax²+bx+c) = (Ax + B)/(ax²+bx+c)

How to Use

1. Basic Operation:

   • Enter numerator (e.g., 3x + 5)
   • Enter denominator (e.g., (x+1)(x+2))
   • Choose real/complex decomposition
   • View results with steps

2. Special Cases:

   • For repeated roots: (x+1)^2
   • For quadratics: (x^2+4)
   • Improper fractions: Do polynomial division first

FAQs

Q: Why use partial fractions?
A: Essential for integration, Laplace transforms, and solving differential equations

Q: What’s the difference between proper/improper?
A: Proper: numerator degree < denominator degree. Must convert improper fractions first

Q: How to handle complex roots?
A: Either keep as complex conjugates or combine into real quadratic terms

Terminology

• Rational Function: Ratio of two polynomials P(x)/Q(x)
• Proper Fraction: Degree of P(x) < Degree of Q(x)
• Irreducible Quadratic: Can’t be factored into real linear terms (e.g., x²+1)
• Residues: Another term for the coefficients (A, B, C…)

Technical Sources

Methods based on:

• Heaviside cover-up method for linear factors
• Algebra of polynomial equations
• Complex analysis techniques
• Standard calculus textbook approaches

Important Notes

1. Always factor denominator completely first
2. For repeated roots, include all powers up to multiplicity
3. Complex mode shows exact roots (e.g., √-1)
4. Numerical precision affects complex results
5. Graph shows absolute coefficient values