1. Make it proper
If the numerator degree is at least the denominator degree, divide first. The quotient becomes the polynomial part.
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Enter a fraction to see its original domain restrictions.
Algebraic Fraction Decomposition Calculator — Partial Fractions with Steps
Enter the numerator and denominator separately. The calculator reduces common factors, divides improper fractions, builds the correct partial-fraction template, solves its coefficients, and verifies the identity.
Enter an algebraic fraction
Partial fraction result
Decomposition
Polynomial part—
Denominator factors—
Partial numerators—
Verification—
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Step-by-step decomposition
Factoring, template, coefficient, and verification steps will appear here.
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How partial fraction decomposition works
2. Factor the denominator
Factor completely and retain multiplicities. A repeated factor needs one term for every power up to its multiplicity.
3. Choose numerators
Use a constant over a linear factor and a general linear polynomial over a quadratic factor.
4. Match coefficients
Clear denominators, collect like powers of x, and solve the resulting linear equations.
General forms
For a repeated linear factor (x−a)^m, include A₁/(x−a) + A₂/(x−a)² + … + Aₘ/(x−a)^m.
For a repeated quadratic factor q(x)^m, include (B₁x+C₁)/q(x) + … + (Bₘx+Cₘ)/q(x)^m.
Partial fraction examples
Distinct linear factors
(5x+3)/(x²−1) has denominator (x−1)(x+1), so its decomposition is 4/(x−1) + 1/(x+1).
Repeated linear factor
A denominator containing (x+1)² needs both A/(x+1) and B/(x+1)²; omitting either term makes the template incomplete.
Irreducible quadratic
A factor such as x²+4 receives a numerator of the form Bx+C, not just a constant.
Improper fraction
For N(x)/D(x) with degree N ≥ degree D, first write N/D = Q + R/D, then decompose R/D.
Calculation method and limits
Parsing, polynomial division, Gaussian elimination, and verification use reduced BigInt fractions. Decimal inputs are converted to exact base-10 fractions; the result is not produced from floating-point fitting.
The calculator supports one-variable polynomials through degree 8 when the reduced denominator can be factored over the rational numbers into linear and quadratic factors. If a rational cubic or higher-degree factor remains, the calculator stops with a clear message instead of returning an incomplete decomposition. Original denominator restrictions are retained after cancellation. Last reviewed: 4 August 2026.
Partial fractions FAQs
What is partial fraction decomposition?
It rewrites a rational function as a polynomial plus a sum of simpler rational terms whose denominators are powers of the factored original denominator.
What if the algebraic fraction is improper?
Divide the numerator by the denominator first. Keep the polynomial quotient and decompose only the proper remainder fraction.
How are repeated linear factors handled?
For (x−a)^3, include separate terms over x−a, (x−a)², and (x−a)³.
What numerator belongs above a quadratic factor?
Use a general linear numerator such as Bx+C above every power of a quadratic factor.
Does cancellation remove domain restrictions?
No. A canceled factor changes the displayed formula but not the original function’s excluded values.
What denominators does this calculator support?
It supports degree-8-or-lower denominators that factor over the rational numbers into linear and quadratic factors, including repeated factors.