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Show that the product of a polynomial and its reciprocal polynomial is a palindromic polynomial

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Show that the product of a polynomial and its reciprocal polynomial is a palindromic polynomial.
Hint Consider the zeros.

Definition of reciprocal polynomial of f(x) for the book Introduction to the Theory of Error-Correcting Codes, by Vera Pless, 3rd edition Page 58 and 59.

If f(x) is a polynomial of degree m, the reciprocal polynomial of f(x) is defined to be .
If , its reciprocal polynomial equals ; that is, the coefficients are written in reverse order.

Theorem: If is a root of is a root of g(x), reciprocal polynomial of f(x). Also f(x) is irreducible if its reciprocal polynomial is irreducible, and f(x) is primitive iff its reciprocal polynomial is primitive.