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Partial Fraction Decomposition Calculator
Decompose rational functions into simpler fractions
📐Types of Partial Fractions
Distinct Linear Factors
A/(x-a) + B/(x-b)
Each factor gets one fraction
Repeated Linear Factors
A/(x-a) + B/(x-a)²
One fraction per power
Irreducible Quadratic
(Ax+B)/(x²+bx+c)
Linear numerator needed
Improper Fractions
Divide first, then decompose
When deg(num) ≥ deg(den)
💡Common Examples
Distinct Linear Factors
(3x+5)/[(x-1)(x+2)] = A/(x-1) + B/(x+2)
Result: A = 2, B = 1
Repeated Factor
(2x+1)/(x-1)² = A/(x-1) + B/(x-1)²
Result: A = 2, B = 3
Quadratic Factor
x/[(x-1)(x²+1)] = A/(x-1) + (Bx+C)/(x²+1)
Solve for A, B, and C
💼Applications
Calculus
• Integration
• Laplace transforms
• Inverse transforms
Engineering
• Control systems
• Signal processing
• Transfer functions
Mathematics
• Differential equations
• Complex analysis
• Rational functions
🔢Decomposition Steps
1. Check if fraction is proper (degree of numerator < degree of denominator)
2. Factor the denominator completely
3. Write the partial fraction form based on factors
4. Multiply both sides by the common denominator
5. Expand and collect like terms
6. Set up system of equations by comparing coefficients
7. Solve for unknown coefficients
8. Write final decomposed form