Partial Fraction Decomposition Calculator
Break any proper rational expression into a sum of simpler partial fractions — enter the numerator and denominator polynomials and get the full decomposition instantly.
Enter polynomials using standard notation (e.g. x^2 + 3x + 2). The degree of the numerator must be less than the degree of the denominator.
Partial Fraction Decomposition Calculator
Break any proper rational expression into a sum of simpler partial fractions — enter the numerator and denominator polynomials and get the full decomposition instantly.
About the Partial Fraction Decomposition Calculator
Partial fraction decomposition is an algebraic technique that rewrites a rational expression — a fraction whose numerator and denominator are both polynomials — as a sum of simpler fractions. The method is the reverse of combining fractions over a common denominator: instead of adding fractions together, you split one complicated fraction apart. The result is a set of terms whose individual integrals, inverse Laplace transforms, or other operations are straightforward to compute.
The fundamental theorem of algebra guarantees that any polynomial over the real numbers can be written as a product of linear factors (x − r) for real roots r, and irreducible quadratic factors (x² + px + q) for complex conjugate roots. Partial fraction decomposition works by factoring the denominator and then writing the original fraction as a sum where each term has one of those factors in its denominator. For a distinct linear factor (x − r), the corresponding term is A/(x − r) for some constant A. For a repeated linear factor (x − r)ⁿ, you need n terms: A₁/(x − r) + A₂/(x − r)² + … + Aₙ/(x − r)ⁿ. For an irreducible quadratic factor (x² + px + q), the term is (Ax + B)/(x² + px + q).
The constants are determined by the method of undetermined coefficients: multiply both sides of the decomposition equation by the denominator, then either substitute convenient values of x (such as the roots) or compare coefficients of like powers of x to build a system of equations. Solving that system yields the exact values of all constants.
Partial fractions are indispensable in integral calculus. The integral of 1/(x − r) is ln|x − r| and the integral of 1/(x − r)² is −1/(x − r), both computable with elementary formulas. Without decomposition, integrating something like (5x − 4)/(x² − x − 2) would require recognizing an obscure substitution; with decomposition the same expression becomes 2/(x − 2) + 3/(x + 1), each piece integrable by inspection.
Beyond calculus, partial fractions appear in control engineering when performing inverse Laplace transforms to find time-domain responses of systems described by transfer functions; in signal processing for the analysis of z-transform representations of digital filters; and in algebra for simplifying complex rational expressions before further manipulation. Understanding how to set up and solve the system of equations for the unknown constants is the core skill, and this calculator shows every step so you can follow the reasoning and build that intuition.
Partial Fraction Decomposition Examples
Worked examples showing distinct linear factors, cubic denominators, and constant numerators.
| Rational expression | Decomposition | Key observation |
|---|---|---|
| (5x − 4) / (x² − x − 2) | 2/(x − 2) + 3/(x + 1) | Denominator factors as (x − 2)(x + 1). Two distinct linear factors; cover-up gives A = 2, B = 3. |
| (x² + 12x + 12) / (x³ − 4x) | −3/x + 2/(x − 2) + 2/(x + 2) | Denominator = x(x − 2)(x + 2). Substitute roots x = 0, 2, −2 to find constants. |
| 1 / (x² + x) | 1/x − 1/(x + 1) | Denominator = x(x + 1). Constant numerator; A = 1, B = −1 by substitution. |
| (8x² − 3x + 10) / (x³ − 2x² + 4x − 8) | 3/(x − 2) + (5x + 2)/(x² + 4) | Denominator = (x − 2)(x² + 4). Linear + irreducible quadratic factor. |
How to Use the Partial Fraction Decomposition Calculator
- Enter the numerator polynomial in the Numerator P(x) field using standard notation, e.g. 5x - 4 or x^2 + 3.
- Enter the denominator polynomial in the Denominator Q(x) field, e.g. x^2 - x - 2.
- Verify that the degree of the numerator is strictly less than the degree of the denominator; if not, perform polynomial long division first.
- Click Calculate. The calculator factors the denominator and uses the Heaviside cover-up method to solve for all constants.
- Click Reset to clear both fields and start a new decomposition.
Partial Fraction Decomposition FAQ
What is partial fraction decomposition?
Partial fraction decomposition rewrites a rational expression P(x)/Q(x) as a sum of simpler fractions whose denominators are the factors of Q(x). It is the reverse of adding fractions over a common denominator, and it makes the expression much easier to integrate or inverse-transform.
When can I use partial fractions?
You can use partial fractions when the expression is a proper rational function — meaning the degree of the numerator is strictly less than the degree of the denominator. If the expression is improper (numerator degree ≥ denominator degree), first divide to get a polynomial plus a proper remainder, then decompose only the remainder.
How do I find the constants A, B, C?
Multiply both sides by the factored denominator to clear all fractions, then solve for the constants. The fastest method is substituting the roots of each linear factor into x (each root zeroes out all terms except one). For irreducible quadratic factors, expand and compare coefficients of like powers of x.
What if the denominator has repeated factors?
A repeated linear factor (x − r)ⁿ requires n separate terms: A₁/(x − r) + A₂/(x − r)² + … + Aₙ/(x − r)ⁿ. Each power introduces an independent unknown constant that must be solved for, usually by expanding and matching coefficients.
Why do irreducible quadratic factors get a linear numerator (Ax + B)?
An irreducible quadratic factor x² + px + q cannot be factored into real linear factors. The partial fraction term for it must be a fraction whose numerator has degree one less than the denominator, giving the form (Ax + B)/(x² + px + q) with two unknown constants A and B.
What is the main application of partial fractions?
The most common application is integration in calculus: simple fractions of the form A/(x − r) integrate to A·ln|x − r|, making otherwise difficult integrals tractable. Partial fractions are equally important in engineering for computing inverse Laplace transforms of transfer functions and inverse z-transforms of digital filters.