General | Home › Math & Conversions › Core Math & Algebra › Partial Fraction Decomposition Calculator 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 Numerator polynomial \( P(x) \) Use x as the variable, ^ for powers, and standard algebraic notation: examples: x^2+3x-4 , -0.5x^3 + 2x . Denominator in factored form \( Q(x) \) 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. Decimal places Example: (2x+3)/[(x-1)(x+2)] Example: improper (x^2+1)/[(x-1)(x+1)] Clear inputs Decompose Clear results Requires distinct linear factors in the denominator 2. Results Decomposition overview Original rational function: Expanded denominator \( Q(x) \): Classification: Final form Polynomial part \( Q(x) \): Proper fraction remainder: Partial fractions (plain text): Partial fractions (TeX): TeX form can be pasted into LaTeX documents or CAS systems that accept LaTeX input. Coefficients summary Coefficients are rounded to the selected number of decimal places for display. Internally, the linear system is solved in double precision. Step-by-step breakdown Copy as plain text 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 Why do I need the denominator in factored form? + Factoring arbitrary polynomials is a non-trivial task and can introduce numerical or symbolic ambiguities. By asking you to provide the denominator as a product of linear factors, the calculator can focus on the partial fraction step itself and give you a transparent, step-by-step solution based on a clean linear system. What happens if the degree of the numerator is too large? + If \( \deg P \geq \deg Q \), the calculator performs polynomial long division to split the function into a polynomial part plus a proper fraction. Only the proper fraction is decomposed into partial fractions. The polynomial part is displayed explicitly. Can I use this as a teaching aid in class? + Yes. The step-by-step breakdown shows the expanded denominator, the long division (when needed), and the linear system used to solve for the coefficients. You can copy the full working as plain text or TeX and adapt it to slides or handouts. Core Math & Algebra tools Bisection Method Monte Carlo Simulation Lottery Odds Dot Product Matrix Determinant Partial Fraction Decomposition – you are here Trapezoid Area Combination and Permutation Torus Volume Matrix Transpose Surface Area Related calculus tools Integral Calculator Derivative Calculator Limit Calculator