# Partial Fraction Decomposer (Rational Functions) Decompose N(x)/D(x) into A/(x−r)^j + (Bx + C)/((x−α)² + β²) with long division for improper fractions and numeric residual verification. > Canonical page: https://elysiatools.com/en/tools/partial-fraction-decomposer - **Category:** Math & Numbers - **Keywords:** partial fractions, partial fraction decomposition, rational function, polynomial long division, durand-kerner, complex roots, repeated roots, conjugate pairs, algebra, integral calculus ## 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. ## Inputs - **Numerator Coefficients N(x) (constant first)** (text): Comma-separated numerator coefficients, constant term first, degree ≤ 8, e.g. 5,3 → 3x + 5. - **Denominator Coefficients D(x) (constant first)** (text): Comma-separated denominator coefficients, constant term first, degree 1–6, e.g. 2,3,1 → x² + 3x + 2. - **Decimal Places** (number) ## 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 - Enter polynomial coefficients for numerator N(x) and denominator D(x) as comma-separated values, starting with the constant term. - If 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. - The solver computes denominator roots and classifies them into distinct real roots, repeated factors, or complex conjugate pairs. - A 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. ## Frequently asked questions ### 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? 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