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PartialFractionElements[expr,x]

gives the data elements related to a partial fraction decomposition of the rational function expr in x.

PartialFractionElements[expr,x,prop]

gives the data elements for property prop.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
See Also
Related Guides
History
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PartialFractionElements

PartialFractionElements[expr,x]

gives the data elements related to a partial fraction decomposition of the rational function expr in x.

PartialFractionElements[expr,x,prop]

gives the data elements for property prop.

Details and Options

  • Partial fraction decompositions, or partial fraction expansions, have applications in the integration of rational functions and other calculus operations.
  • Let be a rational function of with numerator and denominator . Then the complete partial fraction decomposition of has the form:
  • r(x)=sum_(i=1)^pgamma_i TemplateBox[{x, i}, Superscript]+sum_(i=1)^msum_(j=1)^(k_i)(a(i,j))/((x-beta_i)^j)
  • where sum_(i=1)^pgamma_i TemplateBox[{x, i}, Superscript] is the polynomial part of the decomposition, are the poles of (i.e. the roots of ) with multiplicities , and is the fractional part of the decomposition.
  • PartialFractionElements returns an Association with data related to the above partial fraction decomposition of , as determined by PartialFractions. The returned data is of the following two kinds:
  • Data related to the algebraic expression for the partial fraction decomposition:
  • "Decomposition"
    "FractionalPart"
    "PolynomialPart"
  • Data related to the coefficients appearing in the partial fraction decomposition:
  • "FractionalCoefficients"
    "PoleMultiplicities"
    "Poles"
    "PolynomialCoefficients"

Examples

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Basic Examples  (2)

Compute the partial fraction elements for a rational function:

Wolfram Language code: PartialFractionElements[x ^ 3 / (x ^ 2 + 3x + 2), x]

Obtain the polynomial part of a decomposition:

Wolfram Language code: PartialFractionElements[x ^ 3 / (x + 2), x, "PolynomialPart"]

Obtain the fractional part of the decomposition:

Wolfram Language code: PartialFractionElements[x ^ 3 / (x + 2), x, "FractionalPart"]

Compare with the result from PartialFractions:

Wolfram Language code: PartialFractions[x ^ 3 / (x + 2), x]

Scope  (1)

Compute all elements for a partial fraction decomposition:

Wolfram Language code: PartialFractionElements[x ^ 2 / (x ^ 2 + x - 2), x]

Compute just the decomposition:

Wolfram Language code: pfd = PartialFractionElements[x ^ 2 / (x ^ 2 + x - 2), x, "Decomposition"]

Check that it is equivalent to the rational function input:

Wolfram Language code: Together[pfd - x ^ 2 / (x ^ 2 + x - 2)]

Get the polynomial part of the decomposition:

Wolfram Language code: PartialFractionElements[x ^ 2 / (x ^ 2 + x - 2), x, "PolynomialPart"]

Get the fractional part:

Wolfram Language code: PartialFractionElements[x ^ 2 / (x ^ 2 + x - 2), x, "FractionalPart"]

Compute the poles:

Wolfram Language code: PartialFractionElements[x ^ 2 / (x ^ 2 + x - 2), x, "Poles"]

Get their multiplicities:

Wolfram Language code: PartialFractionElements[x ^ 2 / (x ^ 2 + x - 2), x, "PoleMultiplicities"]

Obtain the lists of {pole,mult,coeffs} with the coefficients ordered from degree 1 to the full multiplicity for each pole:

Wolfram Language code: PartialFractionElements[x ^ 2 / (x ^ 2 + x - 2), x, "FractionalCoefficients"]

Similarly, obtain the coefficient list for the polynomial part:

Wolfram Language code: PartialFractionElements[x ^ 2 / (x ^ 2 + x - 2), x, "PolynomialCoefficients"]

Properties & Relations  (1)

Obtain a decomposition:

Wolfram Language code: rat = (1 + x ^ 2 - x ^ 4 + 4 * x ^ 7) / ((x ^ 2 + 2 x + 1) * (x ^ 2 - 1) ^ 2);
Wolfram Language code: PartialFractionElements[rat, x, "Decomposition"]

The decomposition is also the result of PartialFractions:

Wolfram Language code: PartialFractions[rat, x]

Apart will also give the same result when a denominator fully factors over the rationals:

Wolfram Language code: Apart[rat, x]

Now use a denominator that does not factor over the rationals:

Wolfram Language code: rat2 = (1 + x ^ 2 - x ^ 4 + 4 * x ^ 7) / ((x ^ 2 + 2 x + 1) * (x ^ 2 + 1) ^ 2); PartialFractionElements[rat2, x, "Decomposition"]

This time, Apart gives a different result:

Wolfram Language code: Apart[rat2, x]
Wolfram Research (2026), PartialFractionElements, Wolfram Language function, https://reference.wolfram.com/language/ref/PartialFractionElements.html.

Text

Wolfram Research (2026), PartialFractionElements, Wolfram Language function, https://reference.wolfram.com/language/ref/PartialFractionElements.html.

CMS

Wolfram Language. 2026. "PartialFractionElements." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PartialFractionElements.html.

APA

Wolfram Language. (2026). PartialFractionElements. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PartialFractionElements.html

BibTeX

@misc{reference.wolfram_2026_partialfractionelements, author="Wolfram Research", title="{PartialFractionElements}", year="2026", howpublished="\url{https://reference.wolfram.com/language/ref/PartialFractionElements.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_partialfractionelements, organization={Wolfram Research}, title={PartialFractionElements}, year={2026}, url={https://reference.wolfram.com/language/ref/PartialFractionElements.html}, note=[Accessed: 12-June-2026]}