Partial fraction decomposition calculator

A partial fraction decomposition calculator splits a fraction of polynomials into a sum of simpler fractions, one for each factor of the denominator. Type a fraction such as (5x + 1)/((x − 1)(x + 2)) and our calculator factorises the bottom, sets up the right form, finds every constant exactly and shows each step: here the answer is 2/(x − 1) + 3/(x + 2).

It handles distinct linear factors, repeated linear factors and irreducible quadratic factors, and divides first when the top is too big. It opens on (3x + 5)/((x + 1)(x + 2)).

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Partial Fractions Calculator

Split rational expressions into partial fractions step by step

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How to use the partial fraction decomposition calculator

1. Type the fraction in the box. The / key builds a fraction: type the top, press /, then type the bottom. Use brackets for factors, such as (x-1)(x+2).
2. The bottom does not have to be factorised: x^2-x-6 works as well as (x-3)(x+2).
3. Read the answer, then the readouts: the factors of the bottom and the value of each constant.
4. Follow the working: factorise, write the form, multiply through by the denominator, then find the constants by substituting values of x or comparing coefficients.

The graph shows each partial fraction with the original fraction it adds up to. Constants are exact, so 7/5 stays 7/5. For practice, Practice and help sets random two-factor fractions where you find A or B, with the worked solution.

Which form to use

The form depends on the factors of the denominator. Each factor gets its own fraction, and the top of each is one degree lower than its bottom.

A quadratic factor is irreducible when it has no real roots, such as x² + 4. Check the form before you start:

Worked examples you can check in the calculator

Example 1: two distinct linear factors

Split (7x − 1)/(x² − x − 6). Factorise the bottom: x² − x − 6 = (x − 3)(x + 2). So write:

Multiply both sides by (x − 3)(x + 2) to clear the fractions:

Put x = 3, which makes the B term vanish: 20 = 5A, so A = 4. Put x = −2, which makes the A term vanish: −15 = −5B, so B = 3. The answer is 4/(x − 3) + 3/(x + 2), and the calculator shows the same substitutions.

Example 2: a repeated factor

Split (2x + 9)/(x + 3)². A repeated factor needs a fraction for each power:

Put x = −3: B = 3. There is no other value to substitute that removes a term, so compare coefficients of x on both sides: A = 2. The answer is 2/(x + 3) + 3/(x + 3)². The calculator uses both methods here and says which it is doing at each step.

Example 3: an irreducible quadratic factor

Split (2x² + 5)/((x − 1)(x² + 4)). The factor x² + 4 has no real roots, so its top is Bx + C:

Put x = 1: 7 = 5A, so A = 7/5. Compare the x² terms: A + B = 2, so B = 3/5. Compare the constants: 4A − C = 5, so C = 3/5. The answer is 7/(5(x − 1)) + 3(x + 1)/(5(x² + 4)), the form the calculator writes it in, with the 3/5 taken out of the second top.

Example 4: divide first

Split (x² + 2x + 3)/((x + 1)(x + 2)). The top and bottom are both degree 2, so divide first. The bottom is x² + 3x + 2, which goes in once, leaving a remainder of −x + 1:

The calculator does the division as its first step. For long division of polynomials on its own, use the Polynomial Division Calculator.

Partial fractions for integration

The main reason to split a fraction is to integrate it. Each partial fraction with a linear bottom integrates to a log. Splitting 1/(x² − 1) gives 1/(2(x − 1)) − 1/(2(x + 1)), so:

Our Integral Calculator does the split and the integration together, and names partial fractions as the method. The integral calculator guide has more examples. Partial fractions are also used to expand fractions as series and in Laplace transforms.

Common mistakes

• Forgetting to divide first. If the top's degree is at least the bottom's, divide before splitting, or the constants come out wrong.
• Giving a repeated factor one fraction. (x − a)² needs both A/(x − a) and B/(x − a)².
• Giving a quadratic factor a constant top. An irreducible quadratic needs Bx + C on top.
• Not factorising the bottom fully. x² − 1 is (x − 1)(x + 1), so it splits into two fractions, not one.
• Substituting before clearing the fractions. Multiply through by the whole denominator first, then put in values of x.
• Not checking. Add the partial fractions back over a common denominator, or put x = 0 into both sides.

Practise partial fractions in a short timed paper

Partial fractions come up in our College Calculus II & III daily papers, and in A-level Maths: Algebra & Functions papers in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics, so partial fractions will not come up in every one. The Calculus course covers the integration they lead to.

Read next

Maths · AlgebraPartial fractions questions with worked answersPartial fractions questions with full answers: the two methods for finding constants, repeated factors, integrating with partial fractions, and practice.