Polynomial decomposition

In mathematics, a polynomial decomposition expresses a polynomial f as the functional composition ${\displaystyle g\circ h}$ of polynomials g and h, where g and h have degree greater than 1.Algorithms are known for decomposing polynomials in polynomial time.

Polynomials which are decomposable in this way are composite polynomials; those which are not are prime or indecomposable polynomials (not to be confused with irreducible polynomials, which cannot be factored into products of polynomials).

In the simplest case, one of the polynomials is a monomial. For example,

decomposes into

since

Less trivially,

A polynomial may have distinct decompositions into indecomposable polynomials where ${\displaystyle f=g_{1}\circ g_{2}\circ \cdots \circ g_{m}=h_{1}\circ h_{2}\circ \cdots \circ h_{n}}$ where ${\displaystyle g_{i}\neq h_{i}}$ for some ${\displaystyle i}$. The restriction in the definition to polynomials of degree greater than one excludes the infinitely many decompositions possible with linear polynomials.

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