opposite polynomial

The opposite polynomial of a polynomial $P$ in a polynomial ring $R[X]$ is a polynomial  $-P$  such that

$$P+(-P)=\text{𝟎},$$

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. (http://planetmath.org/Ie)

$$-\sum _{\nu =0}^n{a}_{\nu }{X}^{\nu }=\sum _{\nu =0}^n(-a_{\nu })X^{\nu }.$$

The opposite polynomial may be used to define subtraction of polynomials:

$$P-Q=:P+(-Q)$$

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

Title	opposite polynomial
Canonical name	OppositePolynomial
Date of creation	2013-03-22 14:47:41
Last modified on	2013-03-22 14:47:41
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	9
Author	pahio (2872)
Entry type	Definition
Classification	msc 11C08
Classification	msc 12E05
Classification	msc 13P05
Related topic	OppositeNumber
Related topic	Unity
Related topic	BasicPolynomial
Related topic	MinimalPolynomialEndomorphism