How do I find the partial fraction decomposition of a rational expression?

Put the top polynomial in Numerator and the bottom polynomial in Denominator. Results update as you type. For 1 over x^2 - 1, the answer is 1/(2(x - 1)) - 1/(2(x + 1)). Use an example button to load a complete fraction, or Copy to copy the result and its recombination check.

What form do the partial fractions take?

A linear factor takes a constant numerator, so a denominator (x - 1)(x + 1) has the form A/(x - 1) + B/(x + 1). Each repeated factor contributes a term at every power up to its multiplicity. An irreducible quadratic takes a numerator of degree at most one, such as Ax + B. The calculator solves for these coefficients and omits terms whose numerator is 0.

Where can I see the partial fraction coefficients?

The Every part, one by one table lists each factor, its power and its exact signed numerator. A coefficient such as 1/2 stays an exact fraction in that column even when the displayed result clears it into the denominator as 1/(2(x - 1)).

What happens with a repeated factor like (x - 1) squared?

Type x + 3 over (x - 1)^2 to get 1/(x - 1) + 4/(x - 1)^2. The lower-power term can be 0: 1 over (x - 1)^2 stays 1/(x - 1)^2.

How does it handle an irreducible quadratic?

Irreducible here means over the rationals. A quadratic with no rational root stays whole and takes a numerator of degree at most one. For 3x + 5 over (x - 1)(x^2 + 1), the result is 4/(x - 1) - (4x + 1)/(x^2 + 1). The calculator does not expand quadratics into irrational real roots or complex roots.

Does it use long division for an improper fraction?

Yes. When the numerator degree is at least the denominator degree after cancellation, the polynomial quotient is separated before the remaining fraction is decomposed. For x^3 over x^2 - 1, the result is x + 1/(2(x - 1)) + 1/(2(x + 1)), and the Quotient readout shows x.

Why does it say unsupported instead of giving a decomposition?

This solver supports rational linear and irreducible quadratic factors within a finite factor-search budget. A remaining higher-degree factor such as x^3 + x + 1, or a search that reaches its step limit, produces an unsupported result. This is a limit of this calculator; partial fraction methods can also use other factor types.

Does cancellation change where my original fraction is defined?

No. Type x - 1 over x^2 - 1 to get 1/(x + 1), but the original fraction still excludes every point where x^2 - 1 is 0. Cancelling a factor does not restore its excluded roots, even when cancellation leaves a polynomial result.

What input formats and limits does it support?

Use one letter consistently for the variable, numbers, + - * / ^ and brackets. Multiplication may be implicit, as in 2x or (x - 1)(x + 2). Division inside either box must be by a number. Comma-separated coefficients run from the highest power down. Fractions such as 2/3, decimals such as 0.75, superscript powers and a typographic minus sign are accepted. Each box allows up to 500 characters and degree 8, including intermediate products and powers.

Does it show every step or use the cover-up method?

It shows the factors, solved numerators, polynomial quotient when needed, and a recombination line whose difference from the input numerator is 0. It uses exact coefficient solving rather than a cover-up workflow. It does not display the coefficient-matching equations, elimination row operations or a full step-by-step derivation.

Can it integrate partial fractions or compute Laplace and Z transforms?

It computes the algebraic decomposition only. It does not calculate an antiderivative, a definite integral, a Laplace or inverse Laplace transform, or a Z transform. A decomposition alone does not complete those operations.

Can I download the result as a PDF or another file?

The Copy button puts a text result and its check on the clipboard when the browser allows it. If clipboard access is unavailable, select and copy the visible text. There is no result-file download or PDF export control.

Partial Fraction Calculator

Enter the numerator and denominator separately to find a partial fraction decomposition, also called a partial fraction expansion. The calculator shows exact coefficients, any polynomial quotient, and a symbolic recombination check for supported rational factors.

rational expressiondegree 1 over degree 3

      3x + 5      
------------------
 (x - 1)(x^2 + 1) 

proper: numerator degree < denominator

decompositionexact

   4         4x + 1
-------  -  ---------
 x - 1       x^2 + 1

recombination
  4(x^2 + 1) - (4x + 1)(x - 1) = 3x + 5
  difference from your numerator   0

Numerator

Denominator

Terms: 2

Denominator: linear × irreducible quadratic

Quotient: None, proper

Check: Exact

x^2 + 1 has no rational root, so it stays whole and its part carries a numerator of degree at most one, including a constant when the linear coefficient is 0.

1/(x² - 1)two linear factors
1/(2(x - 1)) - 1/(2(x + 1))
(x + 3)/(x - 1)²a repeated linear factor
1/(x - 1) + 4/(x - 1)^2
x³/(x² - 1)improper, divided first
x + 1/(2(x - 1)) + 1/(2(x + 1))

Accuracy. Decomposition over rational linear and irreducible quadratic factors, with repeated factors kept at every multiplicity. An improper fraction is divided first and its polynomial quotient is shown separately rather than folded in. The page recombines the parts symbolically and checks the difference against your input is zero before showing them, and a denominator it cannot factor inside its degree bound is returned unchanged and labelled unsupported.

Every part, one by one

A linear factor takes a constant on top. An irreducible quadratic takes a numerator of degree at most one, because that is the most a numerator can be and still sit below it. A factor that divides the denominator more than once takes one part at every power up to its multiplicity, and the numerator column is the exact rational the page solved for, before any fraction was cleared into the denominator.

Part	Over	Power	Numerator	Factor
+ 4/(x - 1)	x - 1	1	4	linear, root 1
- (4x + 1)/(x^2 + 1)	x^2 + 1	1	-4x - 1	irreducible quadratic, discriminant -4

How this is worked out

Coefficients are held as whole-number pairs, a numerator over a denominator, so nothing is ever rounded and a fraction you type survives to the answer. Anything dividing both the top and the bottom comes out first. If the top is not of lower degree than the bottom, the fraction is improper and is divided: the polynomial part is shown on its own and only what is left goes over the denominator. The denominator is then factored over the rationals, every rational root by the rational root theorem and every quadratic by a search inside a bound on the size of a root. Both searches are finite, and one that runs out of steps says so rather than calling what is left irreducible.

One unknown polynomial is set up for each factor and each power of it, of degree below that factor, and the whole system is solved exactly by elimination over the rationals. Then the parts are put back over the common denominator, added up and subtracted from the numerator you typed. That difference has to be the zero polynomial or nothing is shown at all, which is why the check cell reads Exact rather than quoting a percentage.

This solver supports rational linear and irreducible quadratic factors. A remaining higher-degree factor or a factor search that reaches its step limit returns an unsupported result and names what stopped it. Higher-degree rational factors and expansions into real or complex roots need a different method.

What you can type

One letter for the variable, whole numbers, fractions such as 2/3, decimals such as 0.75, the signs + - * / ^ and brackets. Multiplication can be left out where it is obvious, so 2x and (x - 1)(x + 2) read the way they are written, and pasted maths works too: superscripts such as x² and a typographic minus are read as the powers and signs they stand for. Both boxes take a power up to 8 and are read up to 500characters. A list separated by commas and holding no letter, such as 1, 0, -1, is read as coefficients from the highest power down. Dividing inside a box only works by a number: the fraction itself belongs in the two boxes, one part in each.

Common questions

How do I find the partial fraction decomposition of a rational expression?: Put the top polynomial in Numerator and the bottom polynomial in Denominator. Results update as you type. For 1 over x^2 - 1, the answer is 1/(2(x - 1)) - 1/(2(x + 1)). Use an example button to load a complete fraction, or Copy to copy the result and its recombination check.
What form do the partial fractions take?: A linear factor takes a constant numerator, so a denominator (x - 1)(x + 1) has the form A/(x - 1) + B/(x + 1). Each repeated factor contributes a term at every power up to its multiplicity. An irreducible quadratic takes a numerator of degree at most one, such as Ax + B. The calculator solves for these coefficients and omits terms whose numerator is 0.
Where can I see the partial fraction coefficients?: The Every part, one by one table lists each factor, its power and its exact signed numerator. A coefficient such as 1/2 stays an exact fraction in that column even when the displayed result clears it into the denominator as 1/(2(x - 1)).
What happens with a repeated factor like (x - 1) squared?: Type x + 3 over (x - 1)^2 to get 1/(x - 1) + 4/(x - 1)^2. The lower-power term can be 0: 1 over (x - 1)^2 stays 1/(x - 1)^2.
How does it handle an irreducible quadratic?: Irreducible here means over the rationals. A quadratic with no rational root stays whole and takes a numerator of degree at most one. For 3x + 5 over (x - 1)(x^2 + 1), the result is 4/(x - 1) - (4x + 1)/(x^2 + 1). The calculator does not expand quadratics into irrational real roots or complex roots.
Does it use long division for an improper fraction?: Yes. When the numerator degree is at least the denominator degree after cancellation, the polynomial quotient is separated before the remaining fraction is decomposed. For x^3 over x^2 - 1, the result is x + 1/(2(x - 1)) + 1/(2(x + 1)), and the Quotient readout shows x.
Why does it say unsupported instead of giving a decomposition?: This solver supports rational linear and irreducible quadratic factors within a finite factor-search budget. A remaining higher-degree factor such as x^3 + x + 1, or a search that reaches its step limit, produces an unsupported result. This is a limit of this calculator; partial fraction methods can also use other factor types.
Does cancellation change where my original fraction is defined?: No. Type x - 1 over x^2 - 1 to get 1/(x + 1), but the original fraction still excludes every point where x^2 - 1 is 0. Cancelling a factor does not restore its excluded roots, even when cancellation leaves a polynomial result.
What input formats and limits does it support?: Use one letter consistently for the variable, numbers, + - * / ^ and brackets. Multiplication may be implicit, as in 2x or (x - 1)(x + 2). Division inside either box must be by a number. Comma-separated coefficients run from the highest power down. Fractions such as 2/3, decimals such as 0.75, superscript powers and a typographic minus sign are accepted. Each box allows up to 500 characters and degree 8, including intermediate products and powers.
Does it show every step or use the cover-up method?: It shows the factors, solved numerators, polynomial quotient when needed, and a recombination line whose difference from the input numerator is 0. It uses exact coefficient solving rather than a cover-up workflow. It does not display the coefficient-matching equations, elimination row operations or a full step-by-step derivation.
Can it integrate partial fractions or compute Laplace and Z transforms?: It computes the algebraic decomposition only. It does not calculate an antiderivative, a definite integral, a Laplace or inverse Laplace transform, or a Z transform. A decomposition alone does not complete those operations.
Can I download the result as a PDF or another file?: The Copy button puts a text result and its check on the clipboard when the browser allows it. If clipboard access is unavailable, select and copy the visible text. There is no result-file download or PDF export control.

Decomposition over rational linear and irreducible quadratic factors, with repeated factors kept at every multiplicity. An improper fraction is divided first and its polynomial quotient is shown separately rather than folded in. The page recombines the parts symbolically and checks the difference against your input is zero before showing them, and a denominator it cannot factor inside its degree bound is returned unchanged and labelled unsupported.