Partial Fraction Decomposition Calculator
Break a rational function into simpler fractions, see the polynomial part, and review the factor and coefficient steps used to build the final answer.
Partial fractions simplify rational functions
A partial fraction decomposition calculator rewrites a rational expression as simpler fractions. Enter the numerator and denominator, factor the denominator, then solve for constants. For example, (3x + 5)/(x^2 + x - 2) decomposes into A/(x + 2) + B/(x - 1).
Enter the numerator and denominator separately: Use polynomial expressions such as 3x^2 + 5x - 2 and (x - 1)(x + 2).
Supported forms: The calculator handles improper fractions, repeated rational linear factors, and one remaining irreducible quadratic factor.
Exact coefficients: Integer, decimal, and fractional coefficients are converted to exact rational arithmetic where possible.
Check the denominator: Partial fraction decomposition works only when the denominator is not zero and can be factored into supported pieces.
Partial Fraction Decomposition
For --
Polynomial Part
--
This appears when the numerator degree is at least the denominator degree.
Solved Coefficients
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Constants are found by matching polynomial coefficients.
Denominator Factors
--
Repeated factors create repeated partial fraction terms.
Check Method
Recombine Terms
Put terms over the original denominator and verify the numerator.
Step-by-step notes
How to Use This Calculator
- Enter the numerator: Type the polynomial above the fraction bar, such as 2x + 7.
- Enter the denominator: Type the polynomial below the fraction bar, such as x^2 - x - 2 or (x - 2)(x + 1).
- Choose the variable: Match the symbol used in your expression, usually x.
- Calculate the decomposition: The calculator divides first if needed, factors the denominator, then solves for the unknown coefficients.
- Check the result: Recombine the partial fractions over the original denominator to confirm the numerator matches.
Partial Fraction Rules of Thumb
Partial fraction decomposition rewrites one rational expression as a sum of simpler rational expressions. It is most useful when the denominator can be split into linear or quadratic factors.
If the rational function is improper, first use polynomial division. Then decompose only the proper remainder. Repeated denominator factors need a separate term for every power of the repeated factor.
- Linear factor: A factor like x - a gets a constant numerator, such as A/(x - a).
- Repeated linear factor: A factor like (x - a)^3 gets A/(x - a) + B/(x - a)^2 + C/(x - a)^3.
- Quadratic factor: An irreducible quadratic like x^2 + 1 gets a linear numerator, such as (Ax + B)/(x^2 + 1).
- Coefficient matching: Multiply both sides by the full denominator, expand, and match coefficients of equal powers.
This calculator uses exact rational arithmetic for the algebraic coefficient solve, so answers such as 1/3 stay as fractions instead of being rounded decimals.
Source: OpenStax Algebra and Trigonometry: Partial Fractions
Common Partial Fraction Forms
| Denominator Factor | Partial Fraction Form | Numerator Type | Notes |
|---|
Tip: The degree of each partial fraction numerator should be lower than the degree of its denominator factor.
Step-by-Step Method
The safest way to decompose a rational expression is to keep the algebra organized. The structure below is the same method the calculator uses internally.
1. Divide if Needed
If the numerator degree is greater than or equal to the denominator degree, split off the polynomial quotient first.
2. Factor the Denominator
Write one partial fraction term for each linear factor power and one linear-numerator term for each quadratic factor power.
3. Match Coefficients
Clear denominators, expand the right side, and solve the linear system created by equal powers of the variable.
Further reading: Paul's Online Math Notes: Partial Fractions
Where Partial Fractions Are Used
Partial fractions are a standard bridge between algebra and calculus. They turn complicated rational functions into smaller pieces that are easier to integrate, invert, or transform.
Calculus: Rational integrals often become a sum of logarithm and arctangent terms after decomposition.
Differential equations: Laplace transform problems use partial fractions to find inverse transforms.
Signals and systems: Transfer functions can be rewritten into simpler first-order or second-order components.
How to Check the Answer
The most reliable check is to recombine every term over the original denominator. If the resulting numerator matches the original numerator, the decomposition is correct.
Original numerator = expanded sum of each partial numerator times the missing denominator factors
For example, A/(x - 1) + B/(x + 2) becomes A(x + 2) + B(x - 1) after multiplying by (x - 1)(x + 2).
- Check degrees first: Make sure the proper fraction remainder has a numerator degree lower than the denominator degree.
- Substitute easy values: Roots of linear factors can quickly confirm many coefficients.
- Expand only at the end: Avoid unnecessary expansion until you are ready to match coefficients or verify the final expression.
Interesting Fact
Partial fraction ideas go back to early calculus and algebra work on rational functions. The technique became especially important because it turns many rational integrals into a small library of familiar antiderivatives, which is why it still appears in calculus, engineering math, and transform methods.
Frequently Asked Questions
What does a partial fraction decomposition calculator do with a rational expression?
A partial fraction decomposition calculator rewrites one rational expression as a sum of simpler fraction terms. In algebra and calculus, that form makes the equation easier to analyze, transform, or use during integration because each piece has a smaller denominator.
When do I need polynomial division before using the solver?
Use polynomial division first when the numerator degree is greater than or equal to the denominator degree. The solver separates the polynomial quotient, then applies partial fraction decomposition to the proper remainder.
How does the calculator handle a repeated factor in the denominator?
If the denominator has a repeated factor, include a term for every power of that factor. Each term gets its own constant coefficient; for example, (x - 2)^2 needs both A/(x - 2) and B/(x - 2)^2.
What happens with a linear factor or quadratic factor?
A linear factor gets a constant numerator, such as A. An irreducible quadratic factor gets a linear numerator such as Ax + B, where x is the selected variable.
Why did denominator factoring fail in this algebra solver?
The built-in parser is focused on common classroom algebra and engineering math cases. It looks for rational linear factors and can keep one irreducible quadratic factor. Higher-degree irreducible factors or products of several irreducible quadratics may require a full computer algebra system.
Can I use decimals or fractions for each coefficient?
Yes. You can type a coefficient as a decimal or fraction, such as 0.5x, 1/2x, or (3/4)x^2. The calculator converts decimal values to exact fractions based on the digits you enter.
How do I verify the final decomposition equation?
Place every partial fraction over the original denominator, combine the numerators, and simplify. This turns the result back into one rational expression, and the combined numerator should equal the original numerator after any polynomial quotient is included.
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Disclaimer: This partial fraction decomposition calculator is an educational algebra tool. Always verify symbolic results against your course, textbook, or computer algebra system requirements, especially for high-degree or unusual denominator factors.
Last updated: May 5, 2026