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).