Online Partial Fraction Decomposition Calculator

December 7, 2025 • 2 min read • Math

Online Free Partial Fraction Decomposition Calculator

Quickly break down complex rational expressions with our Partial Fraction Decomposition Calculator! Ideal for students, engineers, and anyone working with integrals, Laplace transforms, or advanced algebra—complete with clear, step-by-step solutions.

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Partial Fraction Decomposition
Professional calculator with support for all cases: linear factors, repeated factors, irreducible quadratics, and polynomial division
📐 Distinct Linear Factors
P(x)
(x – a)(x – b)…
💡 Examples:
• (5x+7) / [(x-1)(x+3)] → A/(x-1) + B/(x+3)
• (2x-5) / [(x+2)(x-4)] → A/(x+2) + B/(x-4)
🔄 Repeated Linear Factors
P(x)
(x – a)²(x – b) or (x – a)³…
💡 Examples:
• (3x-1) / [(x-2)²(x+1)] → A/(x-2) + B/(x-2)² + C/(x+1)
• (x+5) / [(x+1)³] → A/(x+1) + B/(x+1)² + C/(x+1)³
📈 Irreducible Quadratic Factors
P(x)
(x – a)(x² + bx + c)
💡 Examples:
• (2x²+3x+1) / [(x-1)(x²+1)] → A/(x-1) + (Bx+C)/(x²+1)
• (x²-5) / [(x+2)(x²+4)] → A/(x+2) + (Bx+C)/(x²+4)
🎯 Mixed & Complex Cases
P(x)
Any combination of factors
💡 Advanced Features:
✓ Automatic polynomial long division for improper fractions
✓ Support for repeated quadratic factors
✓ Three solving methods available
✓ Step-by-step solution with verification
✨ Partial Fraction Decomposition ✨
(3x + 11) / [(x-3)(x+2)] = 4/(x-3) + (-1)/(x+2)
🔬 Method Used
Cover-Up Method (Heaviside): A fast technique for distinct linear factors where we substitute root values directly.
📊 Step-by-Step Solution
Numerator Degree
1
Denominator Degree
2
Fraction Type
Proper
Number of Terms
2
✓ Verification & Integration
Verified: Adding the partial fractions returns the original expression.
∫ [4/(x-3) – 1/(x+2)] dx = 4ln|x-3| – ln|x+2| + C
ℹ️ Complete Partial Fraction Guide
All Cases Supported: (1) Distinct linear factors: A/(x-a) + B/(x-b) [web:112]; (2) Repeated factors: A/(x-a) + B/(x-a)² + … [web:106][web:107]; (3) Irreducible quadratics: (Ax+B)/(x²+bx+c) [web:111][web:114]; (4) Mixed cases with polynomial division for improper fractions [web:115]. Essential for calculus integration, Laplace transforms, and solving differential equations.

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