# opposite polynomial

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. (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