Partial Fraction Calculator
Break a rational expression into simpler partial fractions. Enter the numerator coefficients and the two roots of the denominator, and the calculator finds A and B so that (px + q) / ((x - r1)(x - r2)) = A/(x - r1) + B/(x - r2). For example, (5x + 3) / ((x - 1)(x + 2)) decomposes to 8/3/(x - 1) + 7/3/(x + 2).
What Are Partial Fractions?
Partial fraction decomposition rewrites a complex fraction as a sum of simpler fractions. It is one of the core techniques in calculus, especially for integration.
The Method (Distinct Linear Factors)
Given (px + q) / ((x - r1)(x - r2)), we want to find constants A and B such that:
(px + q) / ((x - r1)(x - r2)) = A/(x - r1) + B/(x - r2)
How to Find A and B
The cover-up method is the fastest approach:
- To find A, plug x = r1 into the numerator and divide by the remaining factor: A = (p*r1 + q) / (r1 - r2)
- To find B, plug x = r2: B = (p*r2 + q) / (r2 - r1)
Worked Example
Decompose (3x + 5) / ((x - 2)(x + 1)):
- A = (3(2) + 5) / (2 - (-1)) = 11/3
- B = (3(-1) + 5) / ((-1) - 2) = 2/(-3) = -2/3
- Result: (11/3)/(x - 2) + (-2/3)/(x + 1)
Why This Matters
Partial fractions make otherwise difficult integrals easy to evaluate. For instance, integrating 1/((x-1)(x+1)) directly is hard, but integrating (1/2)/(x-1) - (1/2)/(x+1) is easy since each term gives a natural logarithm.