Factorization of composition of reciprocal polynomials with monomials.

A polynomial f (x) is referred to as reciprocal if it satisfies f (x) = ± x deg ⁡ f f (1 / x). It is established that if f (x) ∈ Q [ x ] is reciprocal and irreducible over Q , and r > 0 is an odd integer, then f (x r) has an irreducible reciprocal factor over Q. Furthermore, if f (x) has a root on the unit circle, and r > 0 is an arbitrary integer, then every irreducible factor of f (x r) is reciprocal. This generalizes a related result of Filaseta and Meade for reciprocal 0 , 1 -polynomials. As an outcome of our methods, we obtain some information on the irreducible factors over Q of f (x r) for an arbitrary f (x) ∈ Q [ x ]. [ABSTRACT FROM AUTHOR]