Partial Fractions Calculator
Decompose a rational function into simpler fractions. Enter numerator and denominator to get partial fractions and see calculation steps clearly.
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Helping Notes
Only two inputs are required: the numerator and the denominator polynomials in x. Use ^ for powers (e.g., x^3).
Below, the decomposition is shown in text and also typeset exactly like your screenshot using a rendered fraction.
Results
Partial Fraction Decomposition
Denominator Factorization
Degrees & Properness
What Is a Partial Fractions Calculator?
A Partial Fractions Calculator rewrites a rational function P(x)/Q(x) as a sum of simpler fractions whose denominators are linear or irreducible quadratic factors. This decomposition makes integration, inverse Laplace transforms, and algebraic manipulation much easier. The calculator first ensures the input is proper (degree numerator < degree denominator); if not, it performs polynomial long division to separate out a polynomial part plus a proper remainder. It then factors the denominator, sets up the correct template for distinct linear, repeated linear, and irreducible quadratic factors, and solves for the unknown coefficients by equating coefficients or using cover‑up/residue shortcuts where valid. Results are displayed clearly with responsive formulas so you can verify every step and copy the final decomposition directly into your work.
About the Partial Fractions Calculator
The engine accepts integer or real coefficients and works over the reals by default. For distinct real roots, it applies the cover‑up formula; for repeated roots, it builds a ladder of powers; for irreducible quadratics, it uses linear numerators. The solver multiplies both sides by the full denominator and collects like powers of x to obtain a linear system for the unknown constants. Optional exact mode preserves rational numbers; decimal mode rounds to a chosen precision. A verification stage recombines the found fractions to confirm they equal the original input and, if requested, differentiates/integrates term‑wise to assist with calculus tasks.
Properness via division: P/Q = S(x) + R/Q; deg R < deg Q
Distinct linear factors: P/∏(x−ri) = Σ Ai/(x−ri)
Cover‑up (simple roots): Ai = P(ri)/Q′(ri)
Repeated linear factor (multiplicity m): P/(x−r)m = Σk=1..m Ak/(x−r)k
Irreducible quadratic: P/(ax²+bx+c) = (Bx + C)/(ax²+bx+c)
Repeated quadratic: P/(q(x))m = Σ (Bkx + Ck)/(q(x))k
Coefficient matching: Multiply by Q(x), equate coefficients
How to Use This Partial Fractions Calculator
- Enter numerator P(x) and denominator Q(x). Choose exact (rational) or decimal mode.
- Click decompose. If improper, the tool performs division to return S(x) + R(x)/Q(x) first.
- Review the template built from the factorization and the solved coefficients. Copy the final sum of terms.
- Optionally verify by recombining; for calculus, integrate term‑by‑term or export steps to your notes.
Examples
- Distinct linear factors: with ⇒ .
- Repeated linear: ⇒ .
- Irreducible quadratic term: ⇒ ⇒ .
- Improper case (division first): .
Formula Snippets Ready for Rendering
FAQs
Why use partial fractions?
They simplify integration and inverse transforms by turning a complex rational function into a sum of easy‑to‑handle terms.
What if the numerator degree is ≥ denominator degree?
Perform polynomial long division first to obtain a polynomial plus a proper remainder, then decompose the proper part.
Do I always factor over the reals?
Usually yes for calculus. If complex roots are allowed, factors become linear and the method is analogous.
How do I handle repeated roots?
Include a full ladder of terms: A₁/(x−r) + A₂/(x−r)² + … up to the multiplicity.
What goes over an irreducible quadratic?
A linear numerator (Bx + C). For repeated quadratics, include one such term for each power.
What is the cover‑up rule?
A shortcut for distinct linear factors: Aᵢ = P(rᵢ)/Q′(rᵢ), evaluating the numerator at each root and dividing by Q′ at that root.
Can I use this for Laplace transforms?
Yes. Decompose F(s) and then invert each term using standard tables (e.g., 1/(s+a) ↔ e^{−at}).
How are coefficients actually solved?
Multiply both sides by the denominator and equate coefficients, or plug in strategic x‑values to form a solvable linear system.
What if factoring is hard?
Use numerical factorization or leave irreducible quadratics as is; the method still works with the template shown.
Will rounding affect results?
Exact mode keeps rationals exact. Decimal mode rounds to your chosen precision; verification recombines terms to check accuracy.
Can this detect input errors?
Yes, the calculator validates degrees, nonzero denominator, and factor consistency before solving for coefficients.