Algebra • Core Pillar

Partial Fraction Decomposition Calculator

The definitive algebraic partial fraction decomposition calculator for expanding complex rational functions, solving coefficients with Heaviside cover-up methods, and computing calculus antiderivatives.

Reviewed by Sanjay Samanta & Academic Review Board
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Last Updated: September 2026
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Verified Mathematical Solution
Factor: (x − 2)
Factor: (x + 1)
Preset Examples:
Decomposition Result Proper Rational Fraction (deg P < deg Q)
Partial Fraction Expansion
2 / (x - 2) + 3 / (x + 1)
A = 2, B = 3
Antiderivative ∫ [P(x)/Q(x)] dx:
2 ln|x - 2| + 3 ln|x + 1| + C

Step-by-Step Algebraic Derivation Heaviside Method & Equating Coefficients

Direct Answer & Overview
Verified Educational Guide

How to Decompose a Rational Fraction into Partial Fractions

Partial fraction decomposition reverses common-denominator addition by splitting a proper rational fraction P(x)/Q(x) into a sum of simpler fractions: P(x)/[(x - r_1)(x - r_2)] = A/(x - r_1) + B/(x - r_2). Coefficients A and B are found using the Heaviside cover-up method or equating numerator coefficients, enabling straightforward calculus integration into logarithmic terms.

Primary Mathematical Formula Standard Mathematical Model
Standard Equation
ƒ(x)
Q.E.D.
P(x) / [(x - r_1)(x - r_2)] = A / (x - r_1) + B / (x - r_2) | ∫ [A / (x - r)] dx = A ln|x - r| + C
Evaluated with exact mathematical formulation • Rigorously verified
Exact Formula
Input Parameters
Required
1
Numerator P(x): Polynomial with degree strictly less than denominator
2
Denominator Factors Q(x): Factored linear binomials (x - r) or irreducible quadratics
Expected Outputs
Calculated
Partial Fraction Decomposition: Expanded sum of elementary rational fractions
Coefficient Values (A, B, C): Exact numerical constants
Indefinite Integral: Term-by-term calculus antiderivative with ln and arctan terms
Worked Numerical Example
Instant Verification
Decompose (5x - 4) / [(x - 2)(x + 1)]
→ Heaviside cover-up: A = (5(2) - 4)/(2 - (-1)) = 6/3 = 2 | B = (5(-1) - 4)/(-1 - 2) = -9/-3 = 3
2 / (x - 2) + 3 / (x + 1)

Anatomy of Partial Fraction Decomposition & Proper Rationals

In elementary algebra, students learn to combine separate fractions by finding a common denominator:

2 / (x − 2) + 3 / (x + 1) = [2(x + 1) + 3(x − 2)] / [(x − 2)(x + 1)] = (5x − 4) / (x² − x − 2)

Partial fraction decomposition is the exact reverse operation: starting with the condensed fraction (5x − 4)/(x² − x − 2), it breaks it back down into 2/(x − 2) + 3/(x + 1).

The Four Denominator Factorization Cases

The algebraic form of the decomposition depends strictly on the factor types of Q(x):

1. Distinct Linear Factors
A / (x − r₁) + B / (x − r₂)

Each distinct linear root gets a simple constant numerator.

2. Repeated Linear Factors
A / (x − r) + B / (x − r)²

Ascending powers up to the multiplicity of the root.

3. Irreducible Quadratic
(Ax + B) / (x² + c)

Quadratic factor with negative discriminant requires linear numerator.

4. Repeated Quadratic
(Ax + B)/(x² + c) + (Cx + D)/(x² + c)²

Ascending powers with linear numerators for each power.

The Heaviside Cover-Up Shortcut Method

For distinct linear factors, the Heaviside Cover-Up Method eliminates the need for solving matrix systems:

  1. To find A above (x − 2), cover up (x − 2) in the original expression.
  2. Evaluate the remaining fraction at x = 2: A = [5(2) − 4] / (2 + 1) = 6/3 = 2.
  3. To find B above (x + 1), cover up (x + 1) and evaluate at x = -1: B = [5(-1) − 4] / (-1 − 2) = -9/-3 = 3.

Calculus Applications: Analytical Indefinite Integration

The main application of partial fractions is solving integrals of rational expressions:

∫ [A / (x − r)] dx = A ln|x − r| + C
∫ [B / (x − r)²] dx = −B / (x − r) + C
∫ [(Ax + B) / (x² + a²)] dx = (A/2) ln(x² + a²) + (B/a) arctan(x/a) + C

Step-by-Step Worked Examples (Distinct, Repeated, Quadratic)

Standard Distinct Linear Decomposition Level: Intermediate

Decompose (5x − 4) / [(x − 2)(x + 1)].

1. Template: (5x − 4) / [(x − 2)(x + 1)] = A / (x − 2) + B / (x + 1).

2. Cover-up for A (x = 2): A = [5(2) − 4] / (2 + 1) = 6/3 = 2.

3. Cover-up for B (x = -1): B = [5(-1) − 4] / (-1 − 2) = −9/−3 = 3.

Result: 2 / (x − 2) + 3 / (x + 1).

Common Decomposition Pitfalls & Degree Validation Errors

Pitfall 1: Improper Rational Fraction

Applying partial fractions when deg P ≥ deg Q without performing polynomial long division first produces invalid constant coefficients.

Pitfall 2: Missing Repeated Factor Terms

Writing only A/(x − 1)² instead of A/(x − 1) + B/(x − 1)² lacks the necessary degrees of freedom to match numerator coefficients.

Fact-Checked & Verified • Computational Accuracy Standards
Updated July 2026 • Editorial Policy
Authored By
Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By
Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

Found an error or have an improvement suggestion? Report a calculation issue

Frequently Asked Questions

What is partial fraction decomposition?
Partial fraction decomposition is an algebraic technique used to reverse the process of finding a common denominator. It splits a single complex rational expression P(x)/Q(x) into a sum of simpler rational fractions whose denominators are linear or irreducible quadratic factors of Q(x).
When can partial fraction decomposition be applied directly?
It can only be applied directly to strictly proper rational functions where the degree of the numerator is strictly less than the degree of the denominator (deg P < deg Q). If deg P ≥ deg Q, polynomial long division must be performed first to extract the polynomial quotient.
What is the Heaviside Cover-Up Method?
The Heaviside Cover-Up Method is a rapid mental shortcut for finding coefficients of non-repeated linear factors (x - r). By covering up the (x - r) factor in the original denominator and evaluating the remaining expression at x = r, the coefficient A is calculated instantly without solving systems of equations.
How do you handle repeated linear factors like (x - r)^k?
For a factor (x - r)^k repeated k times in the denominator, you must include k separate terms in the decomposition with ascending powers: A_1/(x - r) + A_2/(x - r)^2 + ... + A_k/(x - r)^k.
How do you handle irreducible quadratic factors like (x^2 + c)?
For an irreducible quadratic factor (ax^2 + bx + c) with negative discriminant (b^2 - 4ac < 0), the numerator of its partial fraction term must be linear: (Ax + B) / (ax^2 + bx + c).
Why is partial fraction decomposition crucial in calculus and engineering?
It converts impossible rational function integrals into simple standard forms that integrate into logarithms (ln|x - r|) and inverse tangents (arctan(x/a)). It is also essential in control systems engineering for computing Inverse Laplace Transforms.

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