Enter a rational expression by typing your numerator and denominator as polynomials, and this Partial Fraction Decomposition Calculator breaks it down into its simplest fractional components. Supports linear, repeated, and quadratic factors — results display the decomposed form along with the solved constants (A, B, C…) for each term.
Partial Fraction Decomposition
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Constant A
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Constant B
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Number of Partial Fractions
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Verification
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Partial fraction decomposition is a technique that rewrites a complex rational expression (a fraction of polynomials) as a sum of simpler fractions. It is especially useful in calculus for integrating rational functions and in engineering for inverse Laplace transforms.
It applies when you have a proper rational function — meaning the degree of the numerator is strictly less than the degree of the denominator. If the degree of the numerator is equal to or greater than the denominator, you must first perform polynomial long division to get a proper fraction before decomposing.
For a repeated linear factor like (x + 2)^2, you must include a separate partial fraction term for each power: A/(x+2) + B/(x+2)^2. The calculator automatically detects and handles repeated factors.
An irreducible quadratic factor like (x^2 + 4) that cannot be factored over the reals produces a numerator of the form (Ax + B) in the corresponding partial fraction: (Ax + B)/(x^2 + 4). This is different from linear factors, which only require a constant numerator.
Combine all partial fraction terms back over a common denominator and simplify. The result should be identical to the original rational expression. You can also substitute specific numeric values of x into both sides of the equation to confirm equality.
Enter polynomials using standard notation: use ^ for exponents (e.g. x^2), * for multiplication (or just write it as 3x), and + or - for addition and subtraction. For example, enter x^2 + 3x + 2 or (x+1)(x+2) for the denominator.
It transforms difficult integrals of rational functions into sums of simpler integrals that have standard forms, such as ∫A/(x+a) dx = A·ln|x+a| + C. Without decomposition, many rational function integrals would be extremely difficult or impossible to evaluate analytically.
This calculator focuses on real-coefficient decompositions. When the denominator has irreducible quadratic factors (with negative discriminant), it handles those with the (Ax+B)/quadratic form rather than splitting into complex roots, which keeps the result in real-number form.