# zero of polynomial

Let $R$ be a subring of a commutative ring $S$. If $f$ is a polynomial^{} in $R[X]$, it defines an evaluation homomorphism from $S$ to $S$. Any element $\alpha $ of $S$ satisfying

$$f(\alpha )=\mathrm{\hspace{0.33em}0}$$ |

is a zero of the polynomial $f$.

If $R$ also is equipped with a non-zero unity, then the polynomial $f$ is in $S[X]$ divisible by the binomial
$X-\alpha $ (cf. the factor theorem). In this case, if $f$ is divisible by ${(X-\alpha )}^{n}$ but not by
${(X-\alpha )}^{n+1}$, then $\alpha $ is a zero of the *order* $n$ of the polynomial $f$. If this order is 1, then $\alpha $ is a *simple zero* of $f$.

For example, the real number $\sqrt{2}$ ($\in \mathbb{R}$) is a zero of the polynomial ${X}^{2}-2$ of the polynomial ring $\mathbb{Q}[X]$.

Title | zero of polynomial |

Canonical name | ZeroOfPolynomial |

Date of creation | 2013-03-22 18:19:50 |

Last modified on | 2013-03-22 18:19:50 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 13P05 |

Classification | msc 11C08 |

Classification | msc 12E05 |

Related topic | PolynomialFunction |

Related topic | ZerosAndPolesOfRationalFunction |

Defines | zero of polynomial |

Defines | order of zero |

Defines | order |

Defines | simple zero |

Defines | simple |