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Enter the rational function into the partial fractions calculator to decompose the partial fractions.
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The Partial Fraction Decomposition Calculator helps you break down complex rational expressions into a sum of simpler fractions. It is particularly useful for solving integrals, differential equations, and other mathematical operations involving rational functions.
Partial fraction decomposition is the process of expressing a complex rational expression as a sum of simpler fractions. This technique, also called partial fraction expansion, is essential when integrating rational functions.
For a rational function:
S = p(x) / q(x)
Here, p(x) is the numerator polynomial and q(x) is the denominator polynomial in variable x.
Type 1: Unrepeated linear factor
A / (ax + b)
Type 2: Repeated linear factor
B / (ax + b)m, where m ≥ 2
Type 3: Unrepeated quadratic factor
(Cx + D) / (ax² + bx + c)
Type 4: Repeated quadratic factor
(Ex + F) / (ax² + bx + c)n, where n ≥ 2
To decompose a rational function into partial fractions:
Decompose the rational function:
(5x + 10) / [(x + 1)(x + 6)]
Solution:
(5x + 10)/[(x + 1)(x + 6)] = A/(x + 1) + B/(x + 6)
(x + 1)(x + 6)·(RHS) = A(x + 6) + B(x + 1)
5x + 10 = A(x + 6) + B(x + 1)
5x + 10 = (A + B)x + (6A + B)
(5x + 10)/[(x + 1)(x + 6)] = 1/(x + 1) + 4/(x + 6)
Newcastle University: Partial Fractions
Wikipedia: Partial Fraction Decomposition
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