PlanetMath: opposite polynomial

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opposite polynomial

(Definition)

The opposite polynomial of a polynomial in a polynomial ring is a polynomial such that

$\displaystyle P+(-P) = \textbf{0},$

where $\textbf{0}$ denotes the zero polynomial. It is clear that

is obtained by changing the signs of all of the coefficients of

$\displaystyle -\sum_{\nu = 0}^n a_\nu X^\nu = \sum_{\nu = 0}^n (-a_\nu)X^\nu.$

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

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

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

"opposite polynomial" is owned by pahio. [ full author list (2) ]

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Keywords:

coefficient

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Cross-references: linear mapping, subtraction, coefficients, clear, zero polynomial, polynomial ring, polynomial
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This is version 5 of opposite polynomial, born on 2004-11-04, modified 2008-03-11.
Object id is 6447, canonical name is OppositePolynomial.
Accessed 2106 times total.

Classification:

AMS MSC:	12E05 (Field theory and polynomials :: General field theory :: Polynomials )
	11C08 (Number theory :: Polynomials and matrices :: Polynomials)
	13P05 (Commutative rings and algebras :: Computational aspects of commutative algebra :: Polynomials, factorization)

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