Partial Decomposition Calculator
Partial Fraction Decomposition Calculator works out Partial fraction decomposition and Polynomial part (from long division) from Numerator N(x), Denominator D(x) and Variable. The default figures on this page give partial fraction decomposition of -1/(x + 2) + 4/(x - 3); change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
6 steps
- N(x) = 3x + 11, D(x) = x^2 - x - 6 (degree 2).
- The fraction is already proper, so no long division is needed.
- Factor the denominator: (x + 2)(x - 3)
- Write one unknown per power of each factor (2 unknowns for a degree-2 denominator).
- Clear denominators and match coefficients of x^0 ... x^1, then solve the 2 x 2 system.
- Decomposition: -1/(x + 2) + 4/(x - 3)
Termstable
| denominator | numerator | term |
|---|---|---|
| (x + 2) | -1 | -1/(x + 2) |
| (x - 3) | 4 | 4/(x - 3) |
Coefficients in term ordertable
Its 4 worked examples are re-run on every release, last on 15 September 2026. What changed
Sources and references
2 sources
What the Partial Fraction Decomposition Calculator calculates
Decomposes a rational function into a sum of partial fractions, handling improper fractions, repeated linear factors and irreducible quadratic factors.
How to use this Partial Fraction Decomposition Calculator
Enter Numerator N(x), Denominator D(x) and Variable. Partial fraction decomposition updates as you type, and “Use a worked example” loads checked figures you can adjust. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Partial fraction decomposition, Polynomial part (from long division) and Terms.
The method follows James Stewart, Calculus: Early Transcendentals, section 7.4: Integration of Rational Functions by Partial Fractions and OpenStax, Calculus Volume 2, section 3.4: Partial Fractions.
Distinct linear factors: (3x + 11)/(x^2 - x - 6) example
With Numerator N(x): 3x + 11, Denominator D(x): x^2 - x - 6, the calculator returns Coefficients in term order: see result table, Polynomial part (from long division): 0, Factored denominator: (x + 2)(x - 3). This example is checked against an independent source in the test suite.
What you can compare
Change one input at a time to see how it affects Partial fraction decomposition, Polynomial part (from long division) and Terms. The inputs you can vary are Numerator N(x), Denominator D(x) and Variable.
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
See something wrong? Report a problem with this calculator
Frequently asked questions
How to find a partial fraction decomposition?
Decomposes a rational function into a sum of partial fractions, handling improper fractions, repeated linear factors and irreducible quadratic factors. Open “How this is worked out” beside the result to see each step for your own figures.
Can you provide a calculator for expressing partial fractions?
Decomposes a rational function into a sum of partial fractions, handling improper fractions, repeated linear factors and irreducible quadratic factors. Put your own figures in above and the working updates beside the result. For scale, the default figures on this page give -1/(x + 2) + 4/(x - 3).
How hard is partial fraction decomposition?
It needs Numerator N(x), Denominator D(x) and Variable. Nothing else is required, and the figures stay in your browser.
How to know when to use partial fraction decomposition?
The result gives Partial fraction decomposition, Polynomial part (from long division), Terms, Coefficients in term order, Factored denominator and Degree of the denominator, each labelled with its unit so the figure can be read on its own.
What information do I need for the Partial Fraction Decomposition Calculator?
You need Numerator N(x), Denominator D(x) and Variable. The page opens with default figures that give partial fraction decomposition of -1/(x + 2) + 4/(x - 3), so you can see the scale before entering your own.
What does this Partial Fraction Decomposition Calculator show?
It shows Partial fraction decomposition, Polynomial part (from long division), Terms, Coefficients in term order, Factored denominator, Degree of the denominator, The input fraction was proper and Largest check error at sample points, with the working and any warnings for your entries.