In algebra, given a polynomial with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, denoted by or , is the polynomial That is, the coefficients of are the coefficients of in reverse order. They arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix. In the special case where the field is the complex numbers, when the conjugate reciprocal polynomial, denoted , is defined by, where denotes the complex conjugate of , and is also called the reciprocal polynomial when no confusion can arise. A polynomial is called self-reciprocal or palindromic if . The coefficients of a self-reciprocal polynomial satisfy for all . Properties Reciprocal polynomials have several connections with their original polynomials, including: . is a root of a polynomial if and only if is a root of . If then is irreducible if and only if is irreducible. is primitive if and only if is primitive. Other properties of reciprocal polynomials may be obtained, for instance: A self-reciprocal polynomial of odd degree is divisible by x+1, hence is not irreducible if its degree is > 1. Palindromic and antipalindromic polynomials A self-reciprocal polynomial is also called palindromic because its coefficients, when the polynomial is written in the order of ascending or descending powers, form a palindrome. That is, if is a polynomial of degree , then is palindromic if for . Similarly, a polynomial of degree is called antipalindromic if for . That is, a polynomial is antipalindromic if . Examples From the properties of the binomial coefficients, it follows that the polynomials are palindromic for all positive integers , while the polynomials are palindromic when is even and antipalindromic when is odd. Other examples of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials. Properties If is a root of a polynomial that is either palindromic or antipalindromic, then is also a root and has the same multiplicity. The converse is true: If a polynomial is such that if is a root then is also a root of the same multiplicity, then the polynomial is either palindromic or antipalindromic. For any polynomial , the polynomial is palindromic and the polynomial is antipalindromic. It follows that any polynomial can be written as the sum of a palindromic and an antipalindromic polynomial, since . The product of two palindromic or antipalindromic polynomials is palindromic. The product of a palindromic polynomial and an