Partial Fraction Decomposition Calculator

Q: Does this partial fraction calculator show steps?

Yes. It shows the original restriction, cancellation or polynomial division where needed, factor forms, coefficient equations, exact elimination and recombination verification.

Q: Does it support repeated factors?

Yes, for supported rational linear and quadratic factor shapes. Each repeated power gets a numerator of degree below its factor.

Q: Can the numerator degree equal or exceed the denominator degree?

Yes, within the input degree limits. Polynomial division first produces a quotient plus a proper rational remainder.

Q: Why are canceled denominator zeros still excluded?

Cancellation changes the formula but does not define the original function at its excluded inputs. The original denominator must remain nonzero.

Q: Why should I enter a denominator in factored form?

Explicit parenthesized linear and quadratic factors can guide exact factorization when the bounded automatic root search cannot split an expanded polynomial.

Q: Does it split irrational roots, complex factors or integrate?

No. It decomposes over rational coefficients; quadratics without rational roots remain quadratic. Complex coefficients, radical-linear splitting and integration are outside scope.

What is the Partial Fraction Decomposition Calculator?

This partial fraction decomposition calculator rewrites a supported rational function P(x)/Q(x) as a polynomial plus simpler rational terms. It uses exact rational coefficients, accounts for repeated factors, and checks the result by recombining polynomials rather than sampling decimal values.

The existing Rational Expression Calculator simplifies and combines algebraic fractions. This page performs a different operation: decomposing a rational function into a sum of fractions. Original denominator restrictions remain in force even when a common factor cancels.

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Optional scientific calculators (paid link) can help check small numerical arithmetic, but a few decimal evaluations cannot prove a symbolic identity. No purchase is required.

How to use the partial fraction calculator with steps

Enter the numerator polynomial P(x) and denominator polynomial Q(x) in separate fields.
Use x, ordinary numbers, parentheses and +, −, *, /, ^. Constant fractions such as 1/2 are allowed.
Keep variable denominators out of the individual polynomial fields.
For a difficult denominator, supply explicit parenthesized rational linear or quadratic factors, such as (x-1)^2*(x^2+1).
Calculate, then review factorization, coefficient equations, exact verification and the original domain.

Each field accepts 1–110 characters. Numeric literals are at most 1,000,000 with up to six decimal places; scientific notation and equations are not supported. Literal integer powers are limited to 0–6; products may create a numerator through degree 10 or a denominator through degree 8.

Optional calculus workbooks (paid link) can provide practice with the decomposition and its later use in integration. This calculator does not integrate the answer or add an integration constant.

Partial fractions with repeated linear and quadratic factors

Linear factor (x − a)^m:
A1/(x − a) + A2/(x − a)^2 + … + Am/(x − a)^m
Quadratic factor q(x)^m:
(B1x + C1)/q(x) + … + (Bmx + Cm)/q(x)^m
Improper rational function: P/Q = polynomial quotient + proper remainder/Q

The calculator first records the original denominator, cancels common polynomial factors while retaining that original restriction, and performs polynomial division if needed. It then identifies repeated factor multiplicities and assigns a numerator of degree below each supported factor, including every power of a repeated factor.

After clearing denominators, it equates polynomial coefficients and solves the resulting rational linear system by exact row elimination. Zero numerator terms are omitted from the final expression. The output is checked by rebuilding the exact common-denominator numerator, so the final verification is an algebraic identity.

Partial fraction decomposition examples

For P(x) = 3x and Q(x) = x^2 − x − 2, the denominator is (x + 1)(x − 2). The result is 1/(x + 1) + 2/(x − 2), valid only where x^2 − x − 2 ≠ 0, which excludes x = −1 and x = 2.

For P(x) = 2x + 3 and Q(x) = (x − 1)^2, both repeated powers are needed: 2/(x − 1) + 5/(x − 1)^2. Combining the numerators gives 2(x − 1) + 5 = 2x + 3. The original exclusion x = 1 remains.

For P(x) = x^2 + 3x + 1 and Q(x) = x^2 − 4, division first gives a polynomial term. The result is 1 + (1/4)/(x + 2) + (11/4)/(x − 2), with x ≠ −2 and x ≠ 2.

For P(x) = 1 and Q(x) = (x^2 + 1)(x^2 + 2), the result is 1/(x^2 + 1) − 1/(x^2 + 2). Explicit parentheses provide supported quadratic factor hints. For a separate quotient and remainder calculation, use the Polynomial Long Division Calculator.

Original domain and rational-coefficient decomposition

Every real zero of the original denominator is excluded. The result displays that exact polynomial nonzero condition, rather than promising a complete numerical root list. If (x^2 − 1)/(x − 1) reduces to x + 1, x = 1 is still excluded because the original function was undefined there.

Factorization and decomposition are over rational coefficients. A quadratic with no rational roots stays quadratic, including a quadratic with two irrational real roots. For example, x^2 − 2 is not split into x ± √2. This is an exact rational-coefficient decomposition, not a claim of a fully split real-linear or complex decomposition.

Optional algebra reference books (paid link) can help review polynomial factors, domain restrictions and coefficient comparisons. The existing Polynomial Factoring Calculator is a separate factorization workflow; supported scopes may differ.

Supported polynomials, factor search and limitations

The reduced denominator must split into supported rational linear factors and rational quadratic factors. Repeated factors are handled by exact square-free multiplicities. Automatic rational-root search is bounded: expanded primitive integer coefficients must be within magnitude 1 billion, and at most 20,000 candidates are tried per search. Explicit parenthesized linear/quadratic factors can guide cases that automatic expansion cannot split.

An unresolved factor above degree two produces an explanation and no decomposition. For example, an irreducible cubic is outside this tool’s supported factor shapes. The search limit is not a proof that a factorization is impossible. A product of two quadratic factors may need to be entered as that product instead of as an expanded quartic.

No trigonometric functions, integration, complex coefficients, arbitrary roots or numerical approximation of coefficients are included. Exact new intermediate fractions are limited to 512 digits, and the reused polynomial parser retains its 256-digit arithmetic guard. Unsupported inputs are rejected rather than silently approximated.

Optional graph paper notebooks (paid link) can help organize a manual coefficient comparison. For an actual supported antiderivative calculation, use the existing Integral Calculator and review that tool’s own limits.