Decompose

Decompose[poly,x]

decomposes a polynomial, if possible, into a composition of simpler polynomials.

Details and Options

  • Decompose gives a list of the polynomials Pi which can be composed as to give the original polynomial.
  • The set of polynomials Pi is not necessarily unique.
  • Decomposition is an operation which is independent of polynomial factorization.

Examples

open allclose all

Basic Examples  (3)

Represent a polynomial as a composition of polynomials:

Compositions of the same polynomials in different orders:

A decomposition with three polynomials:

Scope  (5)

A composition of more than two polynomials:

No decomposition:

A polynomial with symbolic coefficients:

A polynomial with complex coefficients:

Decompose a polynomial over the integers modulo 3:

Options  (1)

Modulus  (1)

Decompose a polynomial over integers modulo 2:

Applications  (1)

Solve some polynomial equations of degrees higher than 4 in terms of radicals:

Solve by solving and then etc:

Check that these indeed are the roots of f:

Wolfram Language solvers use Decompose automatically:

Properties & Relations  (2)

Composition of polynomials given by Decompose gives the original polynomial:

Use Fold to compose the polynomials:

Use Expand to show that the result is equal to f:

Use Factor to represent a polynomial as a product of irreducible factors:

f can be factored but not decomposed; g can be decomposed but not factored:

Possible Issues  (1)

Decompose ignores possible decompositions with inner polynomials that are linear:

Wolfram Research (1988), Decompose, Wolfram Language function, https://reference.wolfram.com/language/ref/Decompose.html.

Text

Wolfram Research (1988), Decompose, Wolfram Language function, https://reference.wolfram.com/language/ref/Decompose.html.

BibTeX

@misc{reference.wolfram_2020_decompose, author="Wolfram Research", title="{Decompose}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Decompose.html}", note=[Accessed: 19-January-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_decompose, organization={Wolfram Research}, title={Decompose}, year={1988}, url={https://reference.wolfram.com/language/ref/Decompose.html}, note=[Accessed: 19-January-2021 ]}

CMS

Wolfram Language. 1988. "Decompose." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Decompose.html.

APA

Wolfram Language. (1988). Decompose. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Decompose.html