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# 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.

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

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

$P\!-\!Q\;=:\;P\!+\!(-Q)$ |

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

Keywords:

coefficient

Related:

OppositeNumber, Unity, BasicPolynomial, MinimalPolynomialEndomorphism

Type of Math Object:

Definition

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Reference

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## Mathematics Subject Classification

11C08*no label found*12E05

*no label found*13P05

*no label found*

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new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

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new question: Prove a formula is part of the Gentzen System by LadyAnne

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new question: A problem about Euler's totient function by mbhatia