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  (1)

Represent a polynomial as a composition of polynomials:

Scope  (3)

A composition of more than two polynomials:

No decomposition:

A polynomial with symbolic coefficients:

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:

Introduced in 1988
 (1.0)