Partial Fraction Decomposition Calculator
Break down a rational function into simpler fractions. Enter the linear numerator (ax + b) and the two roots of the denominator, and this tool finds the partial fraction coefficients A and B.
How Partial Fraction Decomposition Works
Partial fraction decomposition reverses the process of adding fractions. When you add 3/(x−1) + 2/(x−2), you get a single fraction with a combined denominator. Decomposition does the opposite: given a fraction like (5x + 2)/[(x−1)(x−2)], it finds the A and B such that the fraction equals A/(x−1) + B/(x−2).
The method starts by setting up the equation (5x + 2)/[(x−1)(x−2)] = A/(x−1) + B/(x−2). Multiply both sides by (x−1)(x−2) to clear denominators: 5x + 2 = A(x−2) + B(x−1). Expand and equate coefficients of like terms, or substitute strategic values of x (like x = 1 and x = 2) to solve for A and B directly.
For distinct linear factors, the algebra is straightforward. More complex denominators (repeated roots, irreducible quadratics) require additional terms and more sophisticated coefficient matching, but the principle remains the same: express one complicated fraction as a sum of simpler ones.
Applications in Calculus and Beyond
Partial fractions are indispensable in calculus. Integrating a rational function directly is often difficult, but integrating A/(x−r) is immediate: A·ln|x−r|. Decomposing a complex rational function into partial fractions turns one hard integral into several easy ones.
In differential equations, solving linear constant-coefficient equations via Laplace transforms often yields solutions in the s-domain as rational functions. Taking the inverse Laplace transform requires partial fractions to break the expression into terms whose inverses are known (exponentials, sines, cosines).
Signal processing uses partial fractions to analyze transfer functions of linear systems. Decomposing H(s) reveals the system's poles and helps predict time-domain behavior. Even in complex analysis, residue theory for evaluating integrals relies on partial fraction techniques. The method bridges algebra and analysis, unlocking solutions across mathematics and engineering.
Tips for Manual Decomposition
When working by hand, start by factoring the denominator completely. Ensure it's a product of linear factors (or irreducible quadratics if necessary). If the numerator degree is too high, perform polynomial long division first to reduce the fraction to a polynomial plus a proper fraction.
Write the general form with unknowns A, B, C, etc., one for each factor. Clear the denominators by multiplying through, then expand and simplify. You'll get an equation in x. To solve for the unknowns, either equate coefficients of each power of x (forming a system of equations) or substitute convenient x values that zero out terms (the cover-up method).
The cover-up method is fast for distinct roots: to find A in A/(x−r₁), cover up (x−r₁) in the original denominator and evaluate the numerator at x = r₁. Repeat for each coefficient. Check your answer by combining the partial fractions back into a single fraction and verifying it matches the original. Consistent practice makes the process quick and reliable.
Frequently Asked Questions
What is partial fraction decomposition?
Partial fraction decomposition expresses a rational function (a ratio of polynomials) as a sum of simpler fractions. Each term has a lower-degree denominator, making integration, inverse transforms, and other operations easier.
When do I use partial fractions?
Use partial fractions in calculus for integrating rational functions, in differential equations for inverse Laplace transforms, and in signal processing for breaking down transfer functions. Anytime a complex fraction blocks progress, decomposition simplifies it.
What if the denominator has repeated roots?
This calculator handles distinct linear roots. For repeated roots, the decomposition includes terms like A/(x−r) + B/(x−r)². For irreducible quadratics, terms like (Cx + D)/(x² + bx + c) appear. These cases require more advanced techniques.
How do I find the coefficients A and B?
Set the original fraction equal to A/(x−r₁) + B/(x−r₂), multiply both sides by the common denominator to clear fractions, then solve for A and B by substituting convenient values of x or equating coefficients. This calculator automates that algebra.
Can I decompose improper fractions?
If the numerator degree ≥ denominator degree, first perform polynomial long division to get a polynomial plus a proper fraction. Then decompose the proper fraction. This ensures the decomposition is valid.