] decomposes a polynomial, if possible, into a composition of simpler polynomials.
Decompose gives a list of the polynomials which can be composed as to give the original polynomial.
The set of polynomials is not necessarily unique.
Decomposition is an operation which is independent of polynomial factorization.
See the Mathematica book: Section 3.3.4.
See also: FactorList, Solve.
This polynomial can be rewritten in a variable that is a polynomial in the original variable.
Here is a more complicated example. We first set up a triple composition.
The functions in the decomposition are different from the three we started with, although in this case the degrees are the same.
Recomposing the three new polynomials brings us back to the original composition.