Reciprocal polynomial

In algebra, the reciprocal polynomial p^∗ of a polynomial p of degree n with coefficients from an arbitrary field, such as

is the polynomial

Essentially, the coefficients are written in reverse order. They arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix.

In the special case that the polynomial p has complex coefficients, that is,

the conjugate reciprocal polynomial, p^† given by,

where ${\displaystyle {\overline {a_{i}}}}$ denotes the complex conjugate of ${\displaystyle a_{i}\,\!}$, is also called the reciprocal polynomial when no confusion can arise.

A polynomial p is called self-reciprocal if p(x) = p^∗(x).

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