Partial Fraction Decomposer (Rational Functions)

Partial Fraction Decomposer (Rational Functions) | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Math & Numbers
Input types: text, number
Output type: text
Sample coverage: 4
API ready: Yes

Overview

The Partial Fraction Decomposer breaks down rational functions N(x)/D(x) into sums of simpler terms, including distinct linear factors, repeated roots, and irreducible quadratic expressions. It automatically executes polynomial long division for improper fractions, determines denominator roots, solves the linear system for coefficients, and verifies numerical residuals at test probe points.

When to use

Preparing rational expressions for term-by-term symbolic integration in calculus.
Deriving time-domain responses by computing inverse Laplace transforms of transfer functions in engineering.
Simplifying higher-degree rational algebraic fractions into sums of linear and quadratic components.

How it works

1Enter polynomial coefficients for numerator N(x) and denominator D(x) as comma-separated values, starting with the constant term.
2If the numerator degree equals or exceeds the denominator degree, the tool applies polynomial long division to separate the polynomial quotient from the strictly proper remainder.
3The solver computes denominator roots and classifies them into distinct real roots, repeated factors, or complex conjugate pairs.
4A linear system is solved to find decomposition coefficients for terms of the form A/(x-r)^j and (Bx+C)/((x-α)²+β²), followed by residual verification at sample points.

Use cases

Calculus integration: Converting rational functions into standard logarithmic and arctangent integrands.
Control systems engineering: Splitting s-domain transfer functions into standard forms for table-based inverse Laplace transformations.
Algebraic simplification: Expanding rational expressions with distinct, repeated, or complex factors into elementary components.

Examples

1. Proper Fraction with Distinct Real Roots

Calculus Student

Background

A student needs to integrate (3x + 5)/(x² + 3x + 2) by splitting it into distinct linear denominators.

Problem

Determine the partial fraction decomposition coefficients for the factored denominator (x + 1)(x + 2).

How to use

Enter 5,3 in Numerator Coefficients, 2,3,1 in Denominator Coefficients, and set Decimal Places to 4.

numCoeffs = "5,3", denCoeffs = "2,3,1", decimalPlaces = 4

Outcome

Outputs 2/(x + 1) + 1/(x + 2) with roots at -1 and -2 and residual verification confirming zero error.

2. Improper Fraction with Complex Conjugate Roots

Engineering Student

Background

An engineer is analyzing an improper rational function (x³ + 2x)/(x² + 1) involving quadratic factors.

Problem

Extract the polynomial quotient and decompose the remaining fraction over the conjugate root pair.

How to use

Input 0,2,0,1 for Numerator Coefficients and 1,0,1 for Denominator Coefficients.

FAQ

How should polynomial coefficients be formatted?

Enter comma-separated numbers in ascending order of power, starting with the constant term (e.g., 5,3 represents 3x + 5).

What polynomial degree limits apply to the inputs?

The numerator N(x) supports degrees up to 8, and the denominator D(x) supports degrees from 1 to 6.

How does the tool handle improper rational functions?

Polynomial long division runs automatically to isolate the polynomial quotient Q(x) before decomposing the remaining proper fraction R(x)/D(x).

Are complex and repeated roots supported?

Yes, repeated roots produce powers A/(x-r)^j and complex conjugate roots produce real irreducible quadratic terms (Bx+C)/((x-α)²+β²).

What is the residual verification step?

It computes the numerical difference between the original fraction and the decomposed form at algebraic test points to verify mathematical equivalence.

numCoeffs = "0,2,0,1", denCoeffs = "1,0,1", decimalPlaces = 4

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Proper Fraction with Distinct Real Roots

2. Improper Fraction with Complex Conjugate Roots

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