Partial Fraction Decomposition Calculator

About the Partial Fraction Decomposition Calculator

**Partial fraction decomposition** rewrites a single complicated fraction of polynomials as a sum of simpler fractions. This technique is indispensable in calculus for integrating rational functions, in differential equations for inverse Laplace transforms, and in discrete mathematics for solving recurrence relations.

Given a rational expression P(x)/Q(x) where deg(P) < deg(Q), the method factors Q(x) and writes one partial-fraction term per factor (and per power of repeated factors). Distinct linear factors produce constants in the numerator; irreducible quadratic factors produce linear expressions. If the fraction is improper (deg P ≥ deg Q), polynomial long division is performed first.

This calculator accepts numerator and denominator polynomials as comma-separated coefficients (highest degree first), factors the denominator, writes the decomposition template, and solves for the unknown constants using the cover-up (Heaviside) method for distinct-linear cases. A reference table of rules, preset examples covering every factor type, and toggleable step-by-step output make this an ideal study companion for calculus students.

When This Page Helps

Partial fraction decomposition is one of the most important techniques in a calculus or engineering-mathematics toolkit. Without it, integrating rational functions, computing inverse Laplace transforms, and expanding generating functions for recurrences would require far more complex methods.

This calculator automates the tedious algebra: it factors the denominator (finding rational roots automatically), writes the decomposition template, and — for distinct-linear cases — solves coefficients with the cover-up method. Students can toggle step-by-step output to follow the reasoning, and a reference table summarizes the rules for every factor type. Presets cover the full spectrum: distinct linear, repeated linear, irreducible quadratic, and mixed cases.

How to Use the Inputs

1. Enter numerator polynomial coefficients (highest degree first) separated by commas, e.g. "3,2" for 3x + 2.
2. Enter denominator polynomial coefficients the same way, e.g. "1,0,-1" for x² − 1.
3. Or click a preset to load a common example.
4. Read the output cards: proper/improper status, factorization, number of terms.
5. Review the decomposition template showing the form of each partial fraction.
6. If factors are all distinct linear, the cover-up method solves for coefficients automatically.
7. Toggle "Show steps" to see or hide the step-by-step solution procedure.

Example Calculation

Tips & Best Practices

• Enter coefficients from highest to lowest degree: "1,0,-1" means x² − 1, not x − 1.
• Use the (3x+2)/(x²−1) preset to see the cover-up method in action on a textbook example.
• If the fraction is improper, the quotient polynomial is shown separately — add it back to the partial fractions for the full answer.
• Repeated-root cases (like 2x/(x−1)²) require differentiation or coefficient matching to solve; the page shows the decomposition form and the factors.
• Verify your answer by recombining the partial fractions over a common denominator — you should recover the original fraction.
• The rules-reference table at the bottom is handy during exams — bookmark this page as a quick-reference sheet.

Partial Fractions and Integration

The primary application of partial fraction decomposition is integrating rational functions. The integral of A/(x − r) is A·ln|x − r| + C. For repeated linear factors, ∫A/(x − r)ⁿ dx = A/(1 − n)·(x − r)^(1−n) + C. Irreducible quadratics lead to arctan terms after completing the square. By decomposing first, every rational function reduces to a sum of these elementary integrals.

Laplace Transforms and Control Theory

In engineering, the inverse Laplace transform of a transfer function Y(s) = P(s)/Q(s) requires partial fraction decomposition. Each term maps to a known inverse: A/(s − a) ↔ Ae^{at}, A/(s − a)² ↔ Ate^{at}, and (Bs + C)/(s² + ω²) ↔ B·cos(ωt) + (C/ω)·sin(ωt). Partial fractions make the transition from frequency domain to time domain mechanical.

Computational Algebra and CAS Systems

Computer algebra systems (Maple, Mathematica, SymPy) implement partial fraction decomposition as a core routine. The algorithm involves: (1) factoring the denominator over the rationals or a specified field, (2) setting up the ansatz, (3) solving the resulting linear system. Over the reals, irreducible quadratics stay as single factors; over the complexes, every factor is linear, yielding even simpler partial fractions.

Frequently Asked Questions

• What is partial fraction decomposition?

  It is the process of rewriting a rational expression P(x)/Q(x) as a sum of simpler fractions whose denominators are factors of Q(x). The technique reduces complex integrals and transforms to manageable pieces.

• When do I need polynomial long division first?

  When the degree of the numerator is greater than or equal to the degree of the denominator (an improper fraction). Long division produces a polynomial quotient plus a proper-fraction remainder, and you decompose the remainder.

• What is the cover-up (Heaviside) method?

  For distinct linear factors, you can find each coefficient by "covering up" its factor in the denominator and evaluating the remaining expression at the root. It is quick but only works when all linear factors are distinct.

• How do I handle repeated factors?

  A factor (ax + b)^n requires n separate terms with increasing powers in the denominator: A₁/(ax+b) + A₂/(ax+b)² + … + Aₙ/(ax+b)ⁿ. You solve the coefficients by equating coefficients or using differentiation.

• How do I handle irreducible quadratic factors?

  An irreducible quadratic factor (ax² + bx + c) with discriminant < 0 gets a term (Bx + C)/(ax² + bx + c) — a linear numerator, not a constant. Repeated irreducible quadratics get one such term per power.

• Why is this useful for integration?

  Each partial-fraction term integrates easily: A/(ax+b) → (A/a)ln|ax+b|, and (Bx+C)/(ax²+bx+c) integrates via completing the square and arctan. The original integral of P/Q, which may have no obvious antiderivative, becomes a sum of elementary integrals.