Keep the original domain
Zeros of the original denominator are excluded even when algebraic cancellation produces a simpler expression.
The result card lists real excluded points separately from the simplified formula.
Rational functions · Verified recombination
Partial Fraction Decomposition Calculator with Steps
Enter a one-variable rational expression. The calculator performs long division when needed, factors the denominator, solves the coefficients and checks the reconstructed identity.
Original domain retainedRepeated factors supportedIdentity recombined
Partial-fraction decomposition can prepare a rational expression for later integration. Continue with the integral calculator only when the resulting terms are supported. All integral tools
Turn one quotient into simpler pieces
How partial fraction decomposition works
A rational expression is decomposed only after its numerator degree is lower than its denominator degree. Every denominator factor determines the required numerator template.
Divide improper fractions first
If the numerator degree is at least the denominator degree, polynomial long division separates a polynomial quotient and a proper remainder.
Only the proper remainder is decomposed into simple fractions.
Match the factor structure
Distinct and repeated linear factors use constant numerators. Irreducible quadratic factors use linear numerators.
Repeated factors require one term for every power up to the recorded multiplicity.
Verify by recombination
After solving the linear coefficient system, the fractions are brought over the original denominator.
A result is marked verified only when every reconstructed numerator coefficient matches.
A reliable decomposition workflow
Start with the original denominator restrictions. Perform long division for an improper rational expression, then factor the denominator over the real numbers. Give each repeated linear factor a constant term for every power and each irreducible quadratic factor a linear numerator. Multiply through by the denominator, match polynomial coefficients, and solve the resulting linear system. Finally recombine the terms and retain every exclusion from the original denominator. The current deterministic tool supports one variable, rational coefficients and denominators through degree six when they factor into the documented linear and irreducible quadratic forms.
Check the structure before the arithmetic
Count the unknown coefficients before solving: their number should equal the degree of the proper denominator. Substitute convenient values for quick checks, then use coefficient matching for a complete verification.
Supported structures
Try distinct, repeated and quadratic factors
Distinct linear factors give 2/(x+1)+1/(x+2), with x≠−1,−2.
A repeated linear factor needs terms over both (x−1) and (x−1)².
The irreducible quadratic receives a linear numerator, producing 1/x−x/(x²+1).
Current deterministic scope
Supported input is a one-variable rational expression with rational coefficients and a denominator of degree at most six. Multivariable, transcendental and unsupported high-degree factorization requests are reported honestly.
FAQ
Why perform polynomial division first?+
Partial fraction templates apply to a proper remainder whose degree is lower than the denominator degree.
Are canceled denominator zeros still excluded?+
Yes. The original expression is undefined at those points, so the exclusions remain part of the answer.
How are repeated factors handled?+
The decomposition includes one numerator term for every power of the repeated factor.
How is the answer verified?+
The terms are recombined and the resulting numerator is compared coefficient by coefficient with the original numerator.