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[parent] opposite polynomial (Definition)

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

$\displaystyle P+(-P) = \textbf{0},$
where $ \textbf{0}$ means the zero polynomial. It is clear that $ -P$ is gotten by changing the signs of all coefficients of $ P$, i.e.
$\displaystyle -\sum_{\nu = 0}^n a_\nu X^\nu = \sum_{\nu = 0}^n (-a_\nu)X^\nu.$

The opposite polynomial may be used in the 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.
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See Also: opposite number, unity, minimal polynomial (endomorphism)

Keywords:  coefficient

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Cross-references: linear mapping, subtraction, clear, zero polynomial, polynomial ring, polynomial
There are 2 references to this entry.

This is version 3 of opposite polynomial, born on 2004-11-04, modified 2004-11-05.
Object id is 6447, canonical name is OppositePolynomial.
Accessed 1067 times total.

AMS MSC12E05 (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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