Partial Fraction Decomposition Calculator

How to use the partial fraction decomposition calculator

What this calculator does

This calculator rewrites a rational function P(x)/Q(x) as a sum of simpler rational terms after the denominator's factor pattern is known. It supports two distinct linear factors, three distinct linear factors, one repeated linear factor with one distinct factor, and one linear factor multiplied by an irreducible quadratic. The result is an algebraic identity, not a numerical approximation of the function itself. It is useful for preparing rational functions for integration, inverse Laplace transforms, recurrence work, and coefficient checks. It does not factor an arbitrary polynomial denominator, and it does not replace polynomial long division for numerator degrees above the supported quadratic form.

When to use it

Use the calculator when you have already factored Q(x), when you want to check hand-solved constants A, B, and C, when an integration problem requires simpler denominators, or when a transform problem needs terms that match a table of standard forms. The OpenStax Calculus treatment of partial fractions explains why decomposed rational terms are easier to integrate.

How to calculate

1. The calculator opens with a ready-to-use demonstration: (2x² + 3x + 4)/((x – 1)(x + 2)(x – 3)). Its finite result is already displayed, and Download Excel is immediately available.
2. Choose Denominator structure. The relevant factor controls appear below the numerator controls.
3. Replace the three numerator coefficients and the active roots or quadratic coefficients. Values use a decimal point; standard comma grouping such as 1,000 is accepted, but decimal-comma forms such as 1,5 are rejected to avoid ambiguity.
4. Read Decomposed form, then review Coefficient set, Partial-fraction terms, and Numerical identity verification. Updates occur live.
5. Select Download Excel to export the current validated controls, typed coefficients, terms, and verification rows as a real .xlsx workbook. Select Reset to clear the demonstration and all numeric inputs. Reset may disable Download Excel until a complete valid state is entered again.

Input guide

Denominator structure is required and determines the algebraic template. Select the pattern matching the already factored denominator; choosing a different pattern changes the meaning of every root field. Numerator x² coefficient p₂, Numerator x coefficient p₁, and Numerator constant p₀ are required real decimals. For example, 2, 3, and 4 produce 2x² + 3x + 4. Enter 0 for a missing power rather than leaving it blank. Higher or lower values change the solved constants linearly, but very large magnitudes can make results harder to interpret.

For distinct-linear modes, Root r₁, Root r₂, and, when shown, Root r₃ are required real roots in factors of the form x – r. Thus r₂ = – 2 means the displayed factor is x + 2. Distinct roots cannot equal one another because repeated denominators require a different decomposition template. In repeated mode, Repeated root r belongs to (x – r)² and Distinct root s belongs to x – s; r and s must differ. In linear-quadratic mode, Linear root r defines x – r, while Quadratic x coefficient b and Quadratic constant c define x² + bx + c. The calculator requires b² – 4c < 0, so the quadratic is irreducible over the real numbers; for example, b = 2 and c = 5 gives a discriminant of – 16. A common mistake is entering the signed constant from a factor incorrectly: x + 3 corresponds to root – 3, not 3.

Output guide

Structure, Denominator degree, and Partial terms summarize the active template. Excel reports whether the current workbook has passed the package validator. Original form restates the numerator and factorized denominator. Decomposed form is the primary exact identity at the calculator's displayed precision. Polynomial part is normally 0; it can be nonzero only in the two-linear mode because a quadratic numerator and quadratic denominator may require a constant long-division quotient. Coefficient set lists the solved constants. Identity check compares reconstructed numerator coefficients with p₂, p₁, and p₀. Verification points gives the number of safe x-values used in the numerical cross-check.

The Partial-fraction terms table lists each term's numerator, denominator, and solved coefficient value. For an irreducible quadratic denominator, the numerator is linear, Bx + C, rather than a single constant. The Numerical identity verification table shows x, Original value, Decomposed value, and Absolute difference. A tiny difference near floating-point precision is expected; a denominator zero is never evaluated. These results represent an algebraic decomposition under the selected factor pattern, not a recommendation or statistical estimate.

Worked example

For the startup values p₂ = 2, p₁ = 3, p₀ = 4 and roots r₁ = 1, r₂ = – 2, r₃ = 3, the denominator is (x – 1)(x + 2)(x – 3). The cover-up value at r₁ gives A = P(1)/((1 + 2)(1 – 3)) = 9/( – 6) = – 1.5. At r₂, B = P( – 2)/(( – 2 – 1)( – 2 – 3)) = 6/15 = 0.4. At r₃, C = P(3)/((3 – 1)(3 + 2)) = 31/10 = 3.1. Therefore the first-open result is – 1.5/(x – 1) + 0.4/(x + 2) + 3.1/(x – 3), exactly matching the displayed result and exported workbook.