# short Taylor theorem

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

 $\displaystyle f(x_{0}\!+\!h)\;\equiv\;f(x_{0})+f^{\prime}(x_{0})h\;\;\pmod{h^{% 2}}$ (1)

is in force.

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

 $\displaystyle(x_{0}\!+\!h)^{n}\;=\;x_{0}^{n}\!+\!nx_{0}^{n-1}h+h^{2}P(x_{0})$ (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 ShortTaylorTheorem 2013-04-01 13:19:12 2013-04-01 13:19:12 pahio (2872) pahio (2872) 1 pahio (2872) Definition msc 11A07