# 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

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## 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.

#### 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

#### BibTeX

@misc{reference.wolfram_2022_decompose, author="Wolfram Research", title="{Decompose}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Decompose.html}", note=[Accessed: 10-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_decompose, organization={Wolfram Research}, title={Decompose}, year={1988}, url={https://reference.wolfram.com/language/ref/Decompose.html}, note=[Accessed: 10-June-2023 ]}