Partial Fraction Decomposition Calculator — Step-by-Step Solutions

Decompose rational expressions into partial fractions instantly. Free online partial fraction calculator with step-by-step breakdown, supporting distinct linear, repeated linear, and quadratic denominator factors.

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Partial Fraction Decomposition Calculator

Enter the numerator coefficients and denominator roots to decompose the rational expression into partial fractions.

Comma-separated numbers. Constant term first.
Each root r corresponds to factor (x - r). Use r^m for repeated roots.
Enter coefficients and roots, then click Decompose to see the partial fraction expansion.

Partial Fraction Decomposition Formula Explained

Partial fraction decomposition expresses a rational function P(x)/Q(x) as a sum of simpler rational expressions. The form depends on the denominator's factorization:

Distinct Linear Factors

P(x) / [(x - r₁)(x - r₂)...(x - rₙ)] = A₁/(x - r₁) + A₂/(x - r₂) + ... + Aₙ/(x - rₙ)

Where each coefficient Aᵢ is found using the cover-up method: Aᵢ = P(rᵢ) / Π_{j≠i}(rᵢ - rⱼ).

Repeated Linear Factors

For (x - r)^m: A₁/(x - r) + A₂/(x - r)² + ... + Aₘ/(x - r)^m

Irreducible Quadratic Factors

For (x² + bx + c): (Ax + B)/(x² + bx + c)

How to Perform Partial Fraction Decomposition

Follow these steps to decompose any proper rational expression into partial fractions:

  1. Ensure the fraction is proper — The degree of the numerator must be less than the degree of the denominator. If not, perform polynomial long division first.
  2. Factor the denominator completely — Break Q(x) into linear factors (x - r) and irreducible quadratic factors (x² + bx + c).
  3. Write the general form — Set up unknown coefficients for each factor type according to the rules above.
  4. Solve for coefficients — Use the cover-up method for distinct linear factors, and solve a linear system for repeated or quadratic factors.
  5. Write the final decomposition — Substitute the computed coefficients back into the general form.

Partial Fraction Decomposition Examples

Example 1: Distinct Linear Factors

Decompose (5x - 3) / [(x - 1)(x + 2)]

Form: A/(x - 1) + B/(x + 2)
Cover-up A: (5·1 - 3)/(1 + 2) = 2/3
Cover-up B: (5·(-2) - 3)/(-2 - 1) = -13/-3 = 13/3
Result: 2/[3(x - 1)] + 13/[3(x + 2)]

Example 2: Repeated Linear Factor

Decompose (3x + 1) / [(x - 2)²]

Form: A/(x - 2) + B/(x - 2)²
B = (3·2 + 1) = 7
A = 3 (from comparing coefficients)
Result: 3/(x - 2) + 7/(x - 2)²

Example 3: With Quadratic Factor

Decompose (2x² + 3x + 1) / [(x - 1)(x² + 1)]

Form: A/(x - 1) + (Bx + C)/(x² + 1)
A = (2+3+1)/(1+1) = 6/2 = 3
Solving: B = -1, C = 2
Result: 3/(x - 1) + (-x + 2)/(x² + 1)

Real-World Partial Fraction Applications

  • Calculus Integration: Breaking rational integrands into simpler fractions that integrate to logarithms and arctangents.
  • Laplace Transforms: Inverting complex s-domain expressions in control systems and differential equations.
  • Signal Processing: Decomposing transfer functions for digital filter design and Z-transform inversion.
  • Control Theory: Analyzing system responses by expanding transfer functions into first and second-order terms.
  • Chemical Kinetics: Solving rate equations for consecutive reactions using partial fraction expansions.
  • Probability Theory: Simplifying moment-generating functions and characteristic functions for distribution analysis.

People Also Ask About Partial Fractions

Partial fraction decomposition is primarily used in calculus to integrate rational functions, in differential equations for Laplace transform inversion, and in signal processing for analyzing system transfer functions. It breaks complex rational expressions into simpler terms that are easier to work with.
For distinct linear factors (x - rᵢ) in the denominator, the cover-up method finds coefficient Aᵢ by: covering the factor (x - rᵢ) in the denominator, then evaluating the remaining expression at x = rᵢ. This gives Aᵢ = P(rᵢ) divided by the product of (rᵢ - rⱼ) for all j ≠ i.
When the rational expression is improper (numerator degree ≥ denominator degree), perform polynomial long division first to obtain a polynomial quotient plus a proper remainder. Then apply partial fraction decomposition only to the proper remainder part.
Yes. When a quadratic factor has complex conjugate roots (negative discriminant), it is treated as an irreducible quadratic factor (x² + bx + c). The corresponding partial fraction term has the form (Ax + B)/(x² + bx + c), and the coefficients A and B are real numbers.
Yes, for a given proper rational function with a fully factored denominator, the partial fraction decomposition is unique. The coefficients are uniquely determined by the numerator and the denominator's factorization, which is why methods like cover-up and linear systems always produce the same result.

Partial Fraction Calculator Frequently Asked Questions

Yes. Toggle to "Repeated Linear Factors" mode and use the caret notation (e.g., 2^3 for (x-2)³) to specify multiplicities. The calculator sets up the correct form with terms for each power of the repeated factor and solves for all coefficients.
Enter coefficients in ascending order of powers, separated by commas. The first number is the constant term, the second is the coefficient of x, the third is the coefficient of x², and so on. For example, "3, -2, 1" represents 3 - 2x + x².
Toggle to "Include Quadratic Factor" mode. An additional input field appears where you enter the b and c values for the quadratic factor x² + bx + c. For example, entering "0, 1" adds the factor (x² + 1) to the denominator.
Absolutely. This calculator is designed specifically to support integration by partial fractions. Once you have the decomposition, each term integrates easily: A/(x-r) → A·ln|x-r|, and (Ax+B)/(x²+bx+c) → combination of ln and arctan.
This calculator expects the denominator in factored form (you provide the roots). If you have an expanded denominator polynomial, you may need to factor it first using a root-finding method or factoring calculator before using this partial fraction decomposition tool.

Partial Fraction Glossary

Partial Fraction

A simpler rational expression that is part of a decomposition. Each partial fraction has a denominator that is a power of a linear or irreducible quadratic factor.

Cover-Up Method

A quick technique for finding coefficients of distinct linear factors by covering the factor and evaluating at its root. Also called the Heaviside method.

Proper Rational Function

A rational expression where the numerator's degree is strictly less than the denominator's degree. Partial fraction decomposition requires a proper fraction.

Irreducible Quadratic

A quadratic polynomial x² + bx + c with negative discriminant (b² - 4ac < 0) that cannot be factored into real linear factors.

Multiplicity

The number of times a factor appears in the denominator. A root r with multiplicity m produces m partial fraction terms with denominators (x-r), (x-r)², ..., (x-r)^m.

Polynomial Long Division

A prerequisite step when the rational function is improper. It separates the polynomial part from the proper remainder before decomposition.

Linear System

A set of equations used to solve for unknown coefficients when the cover-up method is insufficient, such as for repeated or quadratic factors.

Residue

In complex analysis, the coefficient of 1/(z - z₀) in a Laurent series. For simple poles, residues correspond to partial fraction coefficients.

Editorial Review & Methodology

This partial fraction decomposition calculator was built and reviewed by the NumbrWiz Editorial Team. The decomposition algorithm implements the standard Heaviside cover-up method for distinct linear factors and Gaussian elimination for repeated and quadratic factors, verified against standard calculus and algebra textbooks including Stewart's Calculus and Anton's Elementary Linear Algebra.

  • Algorithm verification: Cross-checked decomposition results against symbolic algebra systems and standard reference problems.
  • Edge case testing: Tested with improper fractions, repeated roots, complex conjugate quadratic factors, and zero coefficients.
  • UX review: Designed with clear input formatting, error validation, and detailed step-by-step display.

Transparency note: All calculations run client-side in your browser. No data is ever collected, stored, or transmitted. Results are for educational purposes; verify critical calculations independently.

Page last reviewed: May 2026 · NumbrWiz Editorial Team