The opposite polynomial of a polynomial $P$ in a polynomial ring $R[X]$ is a polynomial $-P$ such that $$P+(-P) = \textbf{0},$$ where ${0}$ denotes the zero polynomial. It is clear that $-P$ is obtained by changing the signs of all of the coefficients of $P$ , i.e. $$-\sum_{\nu = 0}^n a_\nu X^\nu = \sum_{\nu = 0}^n (-a_\nu)X^\nu.$$

The opposite polynomial may be used to define subtraction of polynomials: $$P-Q := P+(-Q)$$

Forming the opposite polynomial is a linear mapping $R[X]\to R[X]$ .