short Taylor theorem

If $f(x)$ is a polynomial with integer coefficients and $x_0$ and $h$ integers, then the congruence

$f(x_0+h)\equiv f(x_0)+f^{\prime }(x_0)h(mod{h}^2)$ (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