5. - 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 iff its reciprocal polynomial is irreducible, and f(x) is primitive iff its reciprocal polynomial is primitive.
Palindromic and Reciprocal Polynomials are investigated.