# opposite polynomial

The opposite polynomial^{} of a polynomial $P$ in a polynomial ring $R\beta \x81\u2019[X]$ is a polynomial β$-P$β such that

$$P+(-P)=\text{\pi \x9d\x9f\x8e},$$ |

where 0 denotes the zero polynomial^{}. βIt is clear that β$-P$β is obtained by changing the signs of all of the coefficients of $P$, i.e. (http://planetmath.org/Ie)

$$-\underset{\mathrm{\Xi \xbd}=0}{\overset{n}{\beta \x88\x91}}{a}_{\mathrm{\Xi \xbd}}\beta \x81\u2019{X}^{\mathrm{\Xi \xbd}}=\underset{\mathrm{\Xi \xbd}=0}{\overset{n}{\beta \x88\x91}}(-{a}_{\mathrm{\Xi \xbd}})\beta \x81\u2019{X}^{\mathrm{\Xi \xbd}}.$$ |

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

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

Forming the opposite polynomial is a linear mapping ββ$R\beta \x81\u2019[X]\beta \x86\x92R\beta \x81\u2019[X]$.

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 |