Partial Fraction Decomposition Calculator

This partial fraction decomposition calculator rewrites a rational function \( \frac{P(x)}{Q(x)} \) as a sum of simpler fractions with factored denominators. It is aimed at students and professionals working with integration, Laplace transforms, and algebraic manipulation of rational functions.

Enter a numerator polynomial \( P(x) \) and a denominator in factored form built from distinct linear factors, such as \( (x-1)(x+2)(2x-3) \). The tool performs polynomial long division if needed and then solves for the partial fraction coefficients using a linear system of equations.

1. Define the rational function

Use x as the variable, ^ for powers, and standard algebraic notation: examples: x^2+3x-4, -0.5x^3 + 2x.

Enter a product of distinct linear factors of the form (a x + b). Examples: (x-1)(x+2), (2x+3)(x-4)(x+5). Repeated factors like (x-1)^2 and quadratic factors (x^2+1) are not supported in this version.

Requires distinct linear factors in the denominator

Partial fraction decomposition – core idea

Given a rational function \[ \frac{P(x)}{Q(x)}, \] where \( P(x) \) and \( Q(x) \) are polynomials and \( \deg P < \deg Q \), a partial fraction decomposition rewrites it as a sum of simpler fractions whose denominators are factors of \( Q(x) \).

In the simplest case where the denominator factors into distinct linear factors, \[ Q(x) = (x - r_1)(x - r_2)\cdots(x - r_n), \] we search for constants \( A_1,\dots,A_n \) such that \[ \frac{P(x)}{Q(x)} = \frac{A_1}{x - r_1} + \frac{A_2}{x - r_2} + \cdots + \frac{A_n}{x - r_n}. \]

From polynomial long division to partial fractions

If \( \deg P \geq \deg Q \), the rational function is improper and we start with polynomial long division:

\[ \frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}, \]

where \( S(x) \) is a polynomial (the quotient) and \( R(x) \) is the remainder satisfying \( \deg R < \deg Q \). Only the proper fraction \( R(x)/Q(x) \) needs to be decomposed into partial fractions.

Solving for the coefficients of the partial fractions

Suppose the denominator is factored into distinct linear factors: \[ Q(x) = (a_1 x + b_1)(a_2 x + b_2)\cdots(a_n x + b_n). \] The calculator assumes a decomposition of the form

\[ \frac{R(x)}{Q(x)} = \frac{A_1}{a_1 x + b_1} + \frac{A_2}{a_2 x + b_2} + \cdots + \frac{A_n}{a_n x + b_n}. \]

Multiplying both sides by \( Q(x) \) we obtain a polynomial identity:

\[ R(x) = A_1 \prod_{j\neq 1}(a_j x + b_j) + A_2 \prod_{j\neq 2}(a_j x + b_j) + \cdots + A_n \prod_{j\neq n}(a_j x + b_j). \]

Expanding each product on the right-hand side and matching coefficients of powers of \( x \) yields a linear system in the unknowns \( A_1,\dots,A_n \). The calculator solves this system using Gaussian elimination, which is numerically stable for moderate degrees.

Typical use cases

  • Integration: partial fractions reduce rational integrals to sums of basic integrals of the forms \( \int \frac{1}{x-a}\,dx \) and \( \int \frac{1}{(x-a)^n}\,dx \).
  • Laplace transform inversion: decomposing rational transfer functions into elementary terms whose inverse Laplace transforms are tabulated.
  • Control and signal processing: analyzing poles of transfer functions and breaking them into contributions of simple first-order components.
  • Algebraic simplification: rewriting expressions to isolate singularities or analyze asymptotic behaviour near poles.

Limitations of this calculator

  • Only distinct linear factors in the denominator are supported: terms like \( (x-1)^2 \) or \( (x^2+1) \) are intentionally rejected with a clear warning.
  • All coefficients are treated as real numbers; complex partial fractions are not handled.
  • For better numerical stability, choose moderate coefficient sizes and avoid highly ill-conditioned factorizations if exact arithmetic is important.

Partial fraction decomposition – FAQ