Partial Fraction Calculator
Decompose rational functions into partial fractions with distinct linear factors.
Input
Numerator Coefficients (highest power first)
x^1
x⁰
Denominator Roots: (x - r₁)(x - r₂)...
r1
r2
r3
Original Fraction
x / (x - 1)(x - 2)(x - 3)
Partial Fraction Decomposition
0.5000/(x - 1) - 2.0000/(x - 2) + 1.5000/(x - 3)
Coefficients
A1 = 0.500000
for factor (x - 1)
A2 = -2.000000
for factor (x - 2)
A3 = 1.500000
for factor (x - 3)
Calculation Details
A1 = P(1) / Π(r1 - rⱼ) = 1.0000 / 2.0000 = 0.500000
A2 = P(2) / Π(r2 - rⱼ) = 2.0000 / -1.0000 = -2.000000
A3 = P(3) / Π(r3 - rⱼ) = 3.0000 / 2.0000 = 1.500000
Verification at x = 4
Original: 0.66666667
Partial fractions: 0.66666667
Partial Fractions
Distinct Linear Factors
P(x)/((x-r₁)...(x-rₙ)) = A₁/(x-r₁) + ... + Aₙ/(x-rₙ)
Applications
- Integration of rational functions
- Inverse Laplace transforms
- Solving differential equations