OnlineTool.me

Partial Fraction Decomposition Calculator

Enter a linear numerator and quadratic denominator to get the real partial-fraction form and factor type.

Decompose a Proper Rational Expression

Enter coefficients for (mx + n) / (ax² + bx + c). The real-number form updates automatically.

(x + ) / (x² + x + )

Scope: a linear numerator over a quadratic denominator. The numerator must already be proper.

How to Use Partial Fraction Decomposition Calculator

1

Enter the Numerator

Provide m and n for the linear numerator mx+n.

2

Enter the Denominator

Provide a, b, and c for ax²+bx+c, with a not equal to zero.

3

Check the Decomposition

Review the real factor type and combine the displayed terms to verify the original expression.

Partial Fraction Decomposition Calculator Features

  • Distinct and repeated real quadratic factors
  • Immediate coefficient-based decomposition
  • Irreducible-real result when no real roots exist
  • Visible scope and algebra verification anchor

Common Use Cases

Check algebra homework

Compare a hand decomposition for a proper linear-over-quadratic expression.

Prepare an integral

Rewrite a supported rational expression into simpler real fractions before integrating.

Limits and Important Notes

Result type: Exact calculation

Calculation Details

  • Calculate the quadratic discriminant b² − 4ac.
  • Use two linear terms for distinct real roots or first- and second-power terms for a repeated root.
  • Evaluate the numerator at each root to solve the decomposition coefficients.
  • Report an irreducible quadratic when the discriminant is negative over the real numbers.

Formula

For distinct roots r₁ and r₂: (mx+n)/(a(x−r₁)(x−r₂)) = A/(x−r₁)+B/(x−r₂), where A=(mr₁+n)/(a(r₁−r₂)) and B=(mr₂+n)/(a(r₂−r₁))

Rounding

Displayed coefficients use up to eight significant digits; calculations retain JavaScript floating-point precision.

Default Assumptions

  • The denominator has a nonzero x-squared coefficient.
  • The numerator degree is lower than the denominator degree, so polynomial division is unnecessary.
  • The requested decomposition is over the real numbers.

Verified Example

Inputs: x / (x² − 5x + 6).

Calculation: The roots are 3 and 2; cover-up substitution gives coefficients 3 and −2.

Result: 3/(x − 3) − 2/(x − 2).

Sources and Version

Version: Proper degree-1 numerator over degree-2 denominator, real-number decomposition

Last verified: 2026-08-11

Update responsibility: OnlineTool.me

  • This focused calculator supports only (mx+n)/(ax²+bx+c); it does not accept higher-degree polynomials or an improper rational expression.
  • A negative discriminant is reported as irreducible over the reals rather than decomposed with complex roots.
  • Decimal output may be a rounded representation of an irrational coefficient or root.

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Frequently Asked Questions