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ApartSquareFree[expr]

rewrites a rational expression as a sum of terms whose denominators are powers of square-free polynomials.

ApartSquareFree[expr,var]

treats all variables other than var as constants.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Advanced Uses  
Options  
Modulus  
Trig  
Properties & Relations  
See Also
Related Guides
History
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ApartSquareFree

ApartSquareFree[expr]

rewrites a rational expression as a sum of terms whose denominators are powers of square-free polynomials.

ApartSquareFree[expr,var]

treats all variables other than var as constants.

Details and Options

  • ApartSquareFree gives the square-free partial fraction decomposition of a rational expression.
  • ApartSquareFree takes the following options:
  • Modulus 0modulus to assume for integers
    Trig Falsewhether to do trigonometric as well as algebraic transformations
  • ApartSquareFree[expr,Trig->True] treats trigonometric functions as rational functions of exponentials, and manipulates them accordingly.
  • ApartSquareFree automatically threads over lists in expr, as well as equations, inequalities, and logic functions.

Examples

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Basic Examples  (1)

Decompose into partial fractions using square-free factorization of the denominator:

Wolfram Language code: ApartSquareFree[(x ^ 2 - 2) / (x ^ 4 - 2x ^ 2 + 1)]

Decompose into partial fractions using full factorization of the denominator:

Wolfram Language code: Apart[(x ^ 2 - 2) / (x ^ 4 - 2x ^ 2 + 1)]

Scope  (7)

Basic Uses  (5)

Decompose a rational function into partial fractions using square-free factorization:

Wolfram Language code: ApartSquareFree[24(x ^ 2 - 1) / ((x ^ 4 - 2x ^ 2 + 1)(x + 5))]

ApartSquareFree can handle symbolic parameters:

Wolfram Language code: ApartSquareFree[(x ^ 2 - a) / (x ^ 3 - 2a x ^ 2 + a ^ 2x)]

Treat as the main variable and as a constant:

Wolfram Language code: ApartSquareFree[(x ^ 2 + x y + y ^ 2) / (x ^ 4 - 2x ^ 2y ^ 2 + y ^ 4), x]

Treat as the main variable and as a constant:

Wolfram Language code: ApartSquareFree[(x ^ 2 + x y + y ^ 2) / (x ^ 4 - 2x ^ 2y ^ 2 + y ^ 4), y]

Here ApartSquareFree picks as the main variable and treats as a constant:

Wolfram Language code: ApartSquareFree[(x ^ 2 + x y + y ^ 2) / (x ^ 4 - 2x ^ 2y ^ 2 + y ^ 4)]

ApartSquareFree can handle non-polynomial expressions:

Wolfram Language code: ApartSquareFree[(Exp[x] ^ 2 - 2) / (Exp[4 x] - 2Exp[2x] + 1)]

ApartSquareFree threads over equations and inequalities:

Wolfram Language code: ApartSquareFree[1 < (x ^ 2 - 2) / (x ^ 4 - 2x ^ 2 + 1) < 2]

Advanced Uses  (2)

Compute the square-free partial fraction representation over the integers modulo :

Wolfram Language code: ApartSquareFree[(x ^ 2 - 2) / (x ^ 4 - 2x ^ 2 + 1), Modulus -> 5]

Compute the square-free partial fraction representation following common trigonometric identities:

Wolfram Language code: ApartSquareFree[15 / ((Sin[x] ^ 2 - 1) (Sin[x] + 4)), Trig -> True]

Options  (2)

Modulus  (1)

Partial fraction decomposition over the rationals:

Wolfram Language code: ApartSquareFree[(1 - x ^ 2) / (x ^ 4 + 2x ^ 2 + 1)]

Partial fraction decomposition over the integers modulo 2:

Wolfram Language code: ApartSquareFree[(1 - x ^ 2) / (x ^ 4 + 2x ^ 2 + 1), Modulus -> 2]

Trig  (1)

Partial fraction decomposition of a trigonometric expression:

Wolfram Language code: ApartSquareFree[Cos[x] / (1 + Sin[x]), Trig -> True]

Properties & Relations  (1)

Together acts as an inverse of ApartSquareFree:

Wolfram Language code: ApartSquareFree[(x - 1) / (x(x + 1) ^ 2)]
Wolfram Language code: Together[%]
Wolfram Research (2007), ApartSquareFree, Wolfram Language function, https://reference.wolfram.com/language/ref/ApartSquareFree.html.

Text

Wolfram Research (2007), ApartSquareFree, Wolfram Language function, https://reference.wolfram.com/language/ref/ApartSquareFree.html.

CMS

Wolfram Language. 2007. "ApartSquareFree." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ApartSquareFree.html.

APA

Wolfram Language. (2007). ApartSquareFree. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ApartSquareFree.html

BibTeX

@misc{reference.wolfram_2026_apartsquarefree, author="Wolfram Research", title="{ApartSquareFree}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/ApartSquareFree.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_apartsquarefree, organization={Wolfram Research}, title={ApartSquareFree}, year={2007}, url={https://reference.wolfram.com/language/ref/ApartSquareFree.html}, note=[Accessed: 13-June-2026]}