# short Taylor theorem

If $f(x)$ is a polynomial^{} with integer coefficients and ${x}_{0}$ and $h$ integers, then the congruence^{}

$f({x}_{0}+h)\equiv f({x}_{0})+{f}^{\prime}({x}_{0})h\phantom{\rule{veryverythickmathspace}{0ex}}(mod{h}^{2})$ | (1) |

is in force.

Proof. Because of the linear properties of (1) we can confine us to the monomials $f(x):={x}^{n}$. Then ${f}^{\prime}(x)=n{x}^{n-1}$. By the binomial theorem^{} we have

${({x}_{0}+h)}^{n}={x}_{0}^{n}+n{x}_{0}^{n-1}h+{h}^{2}P({x}_{0})$ | (2) |

where $P({x}_{0})$ is a polynomial in ${x}_{0}$ with integer coefficients. The equality (2) may be written as the asserted congruence (1).

Title | short Taylor theorem |
---|---|

Canonical name | ShortTaylorTheorem |

Date of creation | 2013-04-01 13:19:12 |

Last modified on | 2013-04-01 13:19:12 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 1 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 11A07 |