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Partial fraction decomposition splits one fraction of polynomials into a sum of simpler fractions. For example, \(\frac{5x – 4}{x^2 – x – 2} = \frac{2}{x – 2} + \frac{3}{x + 1}\). The simpler pieces are much easier to integrate.
How to Use the Partial Fraction Calculator
- Type a fraction of two polynomials, like
(5x-4)/(x^2-x-2). Put brackets around the whole top and the whole bottom. - Press Solve.
- Read the steps: factor the bottom, set up the fractions, and find the constants. The answer is checked by adding the pieces back together.
The Setup for Each Kind of Factor
| Factor in the denominator | Partial fraction terms |
|---|---|
| Linear: \((x – a)\) | \(\frac{A}{x – a}\) |
| Repeated linear: \((x – a)^2\) | \(\frac{A}{x – a} + \frac{B}{(x – a)^2}\) |
| Irreducible quadratic: \(x^2 + bx + c\) | \(\frac{Bx + C}{x^2 + bx + c}\) |
One rule comes first: if the top has the same or higher degree than the bottom, do polynomial long division before anything else.
Worked Examples
Example 1: \(\frac{5x – 4}{x^2 – x – 2}\)
Factor the bottom: \(x^2 – x – 2 = (x – 2)(x + 1)\). Set up \(\frac{5x – 4}{(x-2)(x+1)} = \frac{A}{x – 2} + \frac{B}{x + 1}\).
Multiply both sides by \((x-2)(x+1)\): \(5x – 4 = A(x + 1) + B(x – 2)\).
Plug in \(x = 2\): \(6 = 3A\), so \(A = 2\). Plug in \(x = -1\): \(-9 = -3B\), so \(B = 3\).
Answer: \(\frac{2}{x – 2} + \frac{3}{x + 1}\).
Example 2: \(\frac{1}{x^2 – 1}\)
\(x^2 – 1 = (x – 1)(x + 1)\). Solving gives \(A = \frac12\) and \(B = -\frac12\), so \(\frac{1}{x^2-1} = \frac{1}{2(x – 1)} – \frac{1}{2(x + 1)}\).
Example 3: A repeated factor
For \(\frac{3x + 5}{(x – 1)(x + 2)^2}\), the setup is \(\frac{A}{x – 1} + \frac{B}{x + 2} + \frac{C}{(x + 2)^2}\). Solving gives \(A = \frac89\), \(B = -\frac89\), and \(C = \frac13\).
Why It Matters: Integration
Each simple piece has an easy antiderivative. From Example 1:
\[\int \frac{5x – 4}{x^2 – x – 2}\,dx = 2\ln|x – 2| + 3\ln|x + 1| + C.\]
The integral calculator does this automatically and shows the partial fraction step. Learn the log rule in the antiderivative of 1/x.
Common Mistakes
- Skipping long division. If the top’s degree is not smaller than the bottom’s, divide first.
- Missing a term for a repeated factor. \((x + 2)^2\) needs both \(\frac{B}{x+2}\) and \(\frac{C}{(x+2)^2}\).
- Using a constant over a quadratic. An irreducible quadratic needs \(Bx + C\) on top, not just \(B\).
Learn the Math Behind the Partial Fraction Calculator
Partial fractions split a rational function into simple pieces that integrate to logarithms and arctangents.
Frequently Asked Questions
What is the cover-up method?
For a distinct linear factor \((x – a)\), cover that factor in the original fraction and plug \(x = a\) into what is left. The result is the constant on top of \(\frac{A}{x – a}\). It is a quick way to do Example 1.
Does the fraction have to be proper?
For the decomposition itself, yes. If it is not, the calculator divides first and decomposes the remainder.
Can I use this for inverse Laplace transforms?
Yes. Partial fractions is the standard way to split a transform before using a table. See the Laplace transform calculator.