Partial Fraction Decomposition Calculator
Split a proper rational function with distinct linear factors into simpler fractions, with coefficients found from the factor roots.
Comma-separated, e.g. 3, 5 for 3x + 5.
Distinct real roots of each linear factor, e.g. 1, -2 for (x - 1)(x + 2).
Result
2.6667/(x - 1) + 0.3333/(x + 2)
The partial fraction decomposition is 2.6667/(x - 1) + 0.3333/(x + 2)
Quick Answer
A fraction decomposition calculator rewrites one rational function as a sum of simpler fractions. For distinct linear factors, substitute each root into the cover-up formula to find its coefficient. For (3x + 5) ÷ ((x − 1)(x + 2)), the coefficients are 8 ÷ 3 and 1 ÷ 3, or 2.6667 and 0.3333.
How A Partial Fraction Calculator Works Step By Step
A fraction decomposition calculator handles the case taught first in calculus. The numerator degree must be lower than the denominator degree. The denominator must split into real linear factors. No root can occur more than once. Each factor has the form x − r. The tool finds one fixed value for each factor. Set x equal to one root to solve for that value. This makes all other terms drop out. Add the fractions to get the first form back. The new sum is easier to use in integrals. “Partial decomposition calculator” and “find partial fraction calculator” point to this tool. Use long division first when the top degree is too high. Use a fuller method for a repeated root or a quadratic part.
How Distinct Partial Fraction Coefficients Are Found
The cover-up method provides a direct algebraic way to resolve coefficients for distinct linear factors.
- N(x) ÷ Π(x − rᵢ) = Σ Aᵢ ÷ (x − rᵢ).
- Aᵢ = N(rᵢ) ÷ Πⱼ≠ᵢ(rᵢ − rⱼ).
N(x): the numerator polynomial.
rᵢ: one distinct real root of the denominator.
Aᵢ: the coefficient above the factor x − rᵢ.
Π: multiplication across all listed factors.
Σ: addition across all partial-fraction terms.
How To Use A Fraction Decomposition Calculator In 4 Steps
Inputs
- Numerator coefficients: enter the numerator coefficients descending in degree, separated by commas.
- Denominator roots: enter distinct real roots of the base, separated by commas.
Steps
- Check that the numerator degree is lower than the denominator degree.
- Split the denominator into distinct linear factors.
- Enter the numerator coefficients and denominator roots.
- Calculate and recombine to check the numerator.
See Partial Fraction Decomposition Applied To Numbers
Numerator N(x) is 3x + 5, denominator roots r₁ is 1 and r₂ is −2.
- Coefficient A₁ for factor (x − 1): N(1) ÷ (1 − (−2)) = (3 × 1 + 5) ÷ 3 = 8 ÷ 3 = 2.6666667.
- Coefficient A₂ for factor (x + 2): N(−2) ÷ ((−2) − 1) = (3 × (−2) + 5) ÷ (−3) = 1 ÷ 3 = 0.3333333.
- Write the new decomposed sum: (8 ÷ 3) ÷ (x − 1) + (1 ÷ 3) ÷ (x + 2).
- Perform a common-base check: (8/3)(x + 2) + (1/3)(x − 1) = (9x + 15)/3 = 3x + 5.
(3x + 5) ÷ ((x − 1)(x + 2)) = 2.6667 ÷ (x − 1) + 0.3333 ÷ (x + 2), rounded to four decimal places. Exact values are 8 ÷ 3 and 1 ÷ 3.
Is Partial Fraction Decomposition Right For Your Case?
Use this calculator before integrating a proper fraction with distinct real roots. Run long division when the numerator degree is not lower. Use a fuller method for repeated roots or quadratic parts with no real roots.
For other algebraic factoring and partial fraction tools, visit the math calculators hub and look up concepts in the glossary.
Assumptions
- The numerator degree is lower than the denominator degree.
- Roots are real and distinct.
- Each root r gives one factor x − r.
- All terms use the same variable.
- Full precision is kept.
Limitations
- Repeated linear factors are outside the current scope.
- Irreducible quadratic factors are not supported.
- The calculator does not factor the denominator.
- An improper fraction needs long division first.
- Decimal output can hide exact fractions.
In Practice
The most common mistake is entering a factor with the wrong root sign. The factor x + 2 has root −2, not 2. Write every factor as x − r before entering roots. Recombine the result to catch a sign error.
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Fraction Decomposition Calculator Questions And Answers
What Is Partial Fraction Decomposition Used For?
Partial fraction decomposition turns one hard fraction into a sum of small ones. Each small term has one linear factor in its base. That form makes many calculus integrals easier to work by hand. The same sum also gives a clear way to check roots and fixed values.
Why Must The Numerator Degree Be Lower First?
The top degree must be lower for the cover-up form to work on its own. When the top degree is as high as the base degree, divide the two polynomials first. The leftover fraction is then proper and can use the root method.
How Do I Check Partial Fraction Coefficients?
Check the coefficients by putting all small terms over the first common base. Multiply each top value by the factors missing from its term, then add the new tops. The sum must match N(x). A wrong sign or root will stop the match.
Can This Calculator Handle Repeated Linear Factors?
No, the current fraction decomposition calculator does not handle any repeated linear factors. A repeated factor such as (x − 1)² needs one term for 1 ÷ (x − 1) and another for 1 ÷ (x − 1)². That case is outside the current tool.
Is The Cover-Up Method The Same As Substitution?
For distinct roots, the cover-up method is the same as strategic substitution. Set x equal to one root after clearing the fractions. All other terms fall to zero, so one fixed value stays. Put the terms back over a common base to check the sum.
Sources
Last updated: . Reviewed for accuracy against the formula shown above.