# Partial Fractions Decomposition Calculator

Decompose rational expressions with distinct or repeated linear-factor denominators into partial fractions, shown in clear step-by-step form.

---

- **Canonical URL:** https://dothecalculation.com/calculators/partial-fractions-calculator
- **Category:** Math calculators
- **Publisher:** Do The Calculation (https://dothecalculation.com)
- **Cost:** Free, no account or sign-up required
- **Privacy:** Runs entirely in the browser; inputs are never sent to a server
- **Methodology:** https://dothecalculation.com/methodology

---

## Partial Fraction Decomposition Calculator

Decompose rational functions into simpler partial fraction sums to evaluate calculus integrals, Laplace transforms, and complex polynomial fractions.

- Distinct & repeated linear factor decomposition
- Root-substitution constant solving with full step-by-step work
- Antiderivative (integration) result shown alongside the decomposition

## Partial Fraction Decomposition Fundamentals

Partial Fraction Decomposition (PFD) is an essential algebraic technique in calculus and engineering mathematics that breaks down a complex rational function \(\frac{P(x)}{Q(x)}\) (where \(P(x)\) and \(Q(x)\) are polynomials) into a sum of simpler fractions called partial fractions.

Before applying partial fraction decomposition, the rational function must be proper: the degree of the numerator \(P(x)\) must be strictly less than the degree of the denominator \(Q(x)\). If the function is improper, polynomial long division must be performed first: $$\frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}$$ where \(S(x)\) is the quotient polynomial and \(R(x)\) is the remainder polynomial with a strictly smaller degree than \(Q(x)\).

Once a proper rational function is established, the denominator \(Q(x)\) is factored completely into linear factors \((ax + b)\) and irreducible quadratic factors \((ax^2 + bx + c)\) over the real numbers. Each factor type dictates a specific partial fraction term structure. Partial fraction decomposition is indispensable for evaluating challenging calculus integrals, finding inverse Laplace transforms in control engineering, solving differential equations, and analyzing Z-transforms in digital signal processing.

## Case 1 & Case 2: Distinct Linear Factors & Repeated Linear Factors

Decomposition rules depend on the algebraic nature of the denominator \(Q(x)\)'s factorization. This calculator handles both of these cases directly.

**Case 1: Distinct Linear Factors.** If \(Q(x) = (a_1 x + b_1)(a_2 x + b_2) \dots (a_k x + b_k)\) with non-repeating linear factors, the decomposition assigns a single constant numerator \(A_i\) to each factor: $$\frac{P(x)}{Q(x)} = \frac{A_1}{a_1 x + b_1} + \frac{A_2}{a_2 x + b_2} + \dots + \frac{A_k}{a_k x + b_k}$$

**Case 2: Repeated Linear Factors.** If \(Q(x)\) contains a repeated linear factor \((ax + b)^m\) of multiplicity \(m\), the decomposition requires \(m\) separate partial fraction terms with increasing powers in the denominator: $$\frac{P(x)}{(ax + b)^m} = \frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + \dots + \frac{A_m}{(ax + b)^m}$$

For example, decomposing \(\frac{5x + 1}{(x - 1)(x + 2)^2}\) (a mix of both cases) yields: $$\frac{5x + 1}{(x - 1)(x + 2)^2} = \frac{A}{x - 1} + \frac{B}{x + 2} + \frac{C}{(x + 2)^2}$$ The unknown constants \(A, B, C\) are found by equating numerators after clearing denominators and substituting strategic \(x\)-values (\(x=1\), \(x=-2\)) to isolate each one.

## Case 3 & Case 4: Irreducible Quadratic Factors (Reference Only)

This calculator solves Case 1 and Case 2 (distinct and repeated linear factors) directly. The quadratic cases below are included as reference material for denominators that also contain irreducible quadratic terms, which you would need to decompose by hand using the coefficient-matching method shown here.

When the denominator \(Q(x)\) contains a quadratic factor that cannot be factored into real linear factors (discriminant \(b^2 - 4ac < 0\)), a linear numerator must be assigned instead of a constant.

**Case 3: Distinct Irreducible Quadratic Factors.** For each non-repeating factor \((ax^2 + bx + c)\), assign a linear numerator \((Ax + B)\): $$\frac{P(x)}{ax^2 + bx + c} = \frac{Ax + B}{ax^2 + bx + c}$$

**Case 4: Repeated Irreducible Quadratic Factors.** If an irreducible quadratic factor is repeated with multiplicity \(m\), assign linear numerators to increasing powers: $$\frac{P(x)}{(ax^2 + bx + c)^m} = \frac{A_1 x + B_1}{ax^2 + bx + c} + \frac{A_2 x + B_2}{(ax^2 + bx + c)^2} + \dots + \frac{A_m x + B_m}{(ax^2 + bx + c)^m}$$

For example, decomposing \(\frac{x^2 + 2x + 3}{(x - 1)(x^2 + 4)}\) requires: $$\frac{x^2 + 2x + 3}{(x - 1)(x^2 + 4)} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + 4}$$ Equating coefficients for the \(x^2\), \(x^1\), and \(x^0\) terms generates a \(3 \times 3\) linear system to solve for \(A, B, C\).

## How to Use This Calculator

Choose Distinct Linear Factors or Repeated Linear Factors, enter the numerator coefficients \(a\) and \(b\) for \(ax + b\), and enter the root(s) of the denominator. The calculator solves for constants \(A\) and \(B\) using root substitution and shows the decomposed form plus its integral.

## Worked Example: (3x + 5) / [(x − 2)(x + 1)]

Using the calculator's default distinct-factor inputs, \(a=3\), \(b=5\), \(r_1=2\), \(r_2=-1\):

$$A = \frac{3(2) + 5}{2 - (-1)} = \frac{11}{3} \approx 3.6667$$

$$B = \frac{3(-1) + 5}{-1 - 2} = \frac{2}{-3} \approx -0.6667$$

$$\frac{3x + 5}{(x-2)(x+1)} = \frac{11/3}{x - 2} + \frac{-2/3}{x + 1}$$

## Related Calculators

To find the roots that go into the denominator of a quadratic factor, use the [quadratic solver](/calculators/quadratic-solver). For evaluating the resulting integral numerically instead of symbolically, see the [integral solver](/calculators/integral-solver).

## Heaviside Cover-Up Method: A Shortcut for Distinct Linear Factors

The Heaviside cover-up method is a rapid shortcut for finding numerator constants corresponding to distinct, non-repeated linear factors, without solving simultaneous linear equations.

To find the constant \(A_k\) corresponding to distinct linear factor \((x - r_k)\) in \(\frac{P(x)}{Q(x)}\): cover up (remove) \((x - r_k)\) from the denominator, then evaluate the remaining expression at \(x = r_k\).

For example, to decompose \(\frac{7x - 1}{(x - 2)(x + 3)} = \frac{A}{x - 2} + \frac{B}{x + 3}\): to find \(A\), cover \((x - 2)\) and evaluate at \(x = 2\): \(A = \frac{7(2) - 1}{2 + 3} = \frac{13}{5}\). To find \(B\), cover \((x + 3)\) and evaluate at \(x = -3\): \(B = \frac{7(-3) - 1}{-3 - 2} = \frac{-22}{-5} = \frac{22}{5}\). Thus: $$\frac{7x - 1}{(x - 2)(x + 3)} = \frac{13/5}{x - 2} + \frac{22/5}{x + 3}$$

This is the same substitution logic this calculator's distinct-factor mode automates for you. The cover-up method is exceptionally fast for distinct linear factors, but it cannot directly solve repeated linear or quadratic factor cases — equating polynomial coefficients remains necessary there.

## Applications in Calculus Integration & Differential Equations

In integral calculus, partial fraction decomposition converts otherwise difficult rational function integrals into simple logarithmic and inverse trigonometric antiderivatives. Standard partial fraction antiderivative building blocks include:

$$\int \frac{A}{x - a}\, dx = A \ln|x - a| + C$$

$$\int \frac{A}{(x - a)^n}\, dx = \frac{-A}{(n-1)(x - a)^{n-1}} + C \quad (n > 1)$$

$$\int \frac{Bx + C}{x^2 + a^2}\, dx = \frac{B}{2} \ln(x^2 + a^2) + \frac{C}{a} \arctan\left(\frac{x}{a}\right) + C$$

In differential equations, partial fractions solve non-linear first-order separable equations, such as the logistic population growth equation \(\frac{dP}{dt} = kP\left(1 - \frac{P}{K}\right)\) — decomposing \(\frac{1}{P(K-P)}\) via partial fractions is the key step that yields the classic logistic sigmoid growth curve.

## Applications in Control Engineering: Inverse Laplace & Z-Transforms

In electrical engineering, control systems, and signal processing, partial fraction decomposition is the primary tool used to compute inverse Laplace transforms and inverse Z-transforms.

Continuous-time control system transfer functions \(H(s) = \frac{Y(s)}{X(s)}\) are expressed as rational functions in the complex frequency domain \(s\). Finding the time-domain response requires decomposing \(H(s)\) into partial fractions matching standard Laplace lookup pairs, such as \(\mathcal{L}^{-1}\left\{\frac{1}{s + a}\right\} = e^{-at}\). The [Laplace transform calculator](/calculators/laplace-transform-calculator) automates exactly this kind of inversion.

Evaluating pole locations (roots of the denominator) via partial fraction decomposition determines system stability: poles in the left half of the s-plane represent stable exponential decay, while right-half-plane poles signal unstable exponential growth.

## Frequently asked questions

### What is Partial Fraction Decomposition?

Partial fraction decomposition is an algebraic process that breaks a complex rational function into a sum of simpler fractions with lower-degree denominators.

### When can Partial Fraction Decomposition be applied?

It applies to proper rational functions where the numerator's degree is strictly less than the denominator's degree. Improper functions require long division first.

### What is the Heaviside Cover-Up Method?

The Heaviside cover-up method is a shortcut technique for quickly finding partial fraction numerator constants for distinct, non-repeating linear factors — the same substitution method this calculator uses.

### How do you handle repeated linear factors in partial fractions?

Repeated linear factors (x − a)^m require m separate partial fraction terms with denominators increasing in power from (x − a)¹ up to (x − a)^m. This calculator's Repeated Linear Factors mode handles the (x − a)² case.

### Does this calculator handle irreducible quadratic factors?

No — it solves distinct and repeated linear-factor denominators only. Quadratic denominator factors (ax² + bx + c) are covered as reference material in the content above, but aren't computed directly by this tool.

### Why is partial fraction decomposition used in calculus?

It transforms otherwise difficult-to-integrate rational functions into simple sum terms that can be integrated using basic natural logarithm and inverse tangent formulas.

### How are partial fractions used in Laplace Transforms?

Control engineers use partial fractions to break complex transfer functions into standard terms that are easily converted back to time-domain signals via inverse Laplace tables.

### What is an Improper Rational Function?

A rational function P(x)/Q(x) where the degree of the numerator P(x) is greater than or equal to the degree of the denominator Q(x).

### How do you check if a quadratic factor is irreducible?

Calculate the discriminant b² − 4ac. If it's less than zero, the quadratic factor cannot be factored into real linear factors.

### What is the connection between partial fractions and logistic growth?

Solving the logistic differential equation requires partial fraction decomposition of 1/[P(K − P)] to integrate and derive the sigmoid population curve.

## Related concepts

- **Rational Function** — A mathematical function defined as the ratio of two polynomial expressions.
- **Inverse Laplace Transform** — The mathematical operation converting complex s-domain transfer functions back into time-domain signals.
- **Polynomial Long Division** — An algorithmic process for dividing polynomials to convert improper rational functions into proper forms.

## Related guides

- [Fraction Operations Guide: Add, Subtract, Multiply, Divide](https://dothecalculation.com/blog/math/fraction-operations-guide) — Learn the rules for adding, subtracting, multiplying, and dividing fractions, with simplification steps and live DTC calculator workflows.

## Related calculators

- [Prime Factorization & Divisors Calculator](https://dothecalculation.com/calculators/prime-factorization-calculator) — Decompose a number into its prime factors and get its total divisor count and Euler totient, with clear step-by-step results instantly.
- [Fourier Series Coefficient Synthesizer](https://dothecalculation.com/calculators/fourier-series-calculator) — Decompose periodic waveforms like square, triangle, and sawtooth into sine and cosine Fourier series coefficients instantly.
- [Fraction to Decimal Calculator](https://dothecalculation.com/calculators/fraction-to-decimal-calculator) — Convert fractions to decimals with step-by-step long division and repeating decimal bar notation shown clearly and instantly.
- [Laplace & Inverse Laplace Transform Solver](https://dothecalculation.com/calculators/laplace-transform-calculator) — Compute Laplace transforms for time-domain functions and inverse Laplace transforms with clear step-by-step solutions shown.
- [Mixed Number Fraction Calculator](https://dothecalculation.com/calculators/mixed-number-calculator) — Add, subtract, multiply, divide, and simplify mixed number fractions with clear step-by-step conversions and instant results.
- [Modular Arithmetic & Congruence Solver](https://dothecalculation.com/calculators/modular-arithmetic-calculator) — Perform modular addition, multiplication, and exponentiation, and solve linear modular congruence equations with clear steps.

---

_This mathematical calculator is designed for educational, scientific, and engineering computations. Always verify matrix dimensions, boundary conditions, and floating-point precision limits when applying results to mission-critical engineering or physics systems._

---

_Source: [Do The Calculation](https://dothecalculation.com/calculators/partial-fractions-calculator). Quote freely with attribution and a link to this page._