Decompose rational expressions into partial fractions. Handles distinct/repeated linear and irreducible quadratic factors with step-by-step solutions.
**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.
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 instantly using 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.
For Q(x) = ∏(aᵢx + bᵢ)^nᵢ · ∏(aⱼx² + bⱼx + cⱼ)^mⱼ: each linear factor (ax+b)^n yields terms A₁/(ax+b) + … + Aₙ/(ax+b)^n; each irreducible quadratic (ax²+bx+c)^m yields (B₁x+C₁)/(…) + … + (Bₘx+Cₘ)/(…)^m.
Result: 5/2 / (x−1) + 1/2 / (x+1)
The denominator x²−1 factors as (x−1)(x+1). Writing (3x+2)/((x−1)(x+1)) = A/(x−1) + B/(x+1), the cover-up method gives A = (3·1+2)/(1+1) = 5/2 and B = (3·(−1)+2)/(−1−1) = 1/2.
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.
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.
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.
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 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.
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.
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.
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.
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.