short Taylor theorem

If f(x) is a polynomialMathworldPlanetmathPlanetmath with integer coefficients and x0 and h integers, then the congruenceMathworldPlanetmathPlanetmathPlanetmathPlanetmath

f(x0+h)f(x0)+f(x0)h(modh2) (1)

is in force.

Proof.  Because of the linear properties of (1) we can confine us to the monomialsf(x):=xn.  Then  f(x)=nxn-1.  By the binomial theoremMathworldPlanetmath we have

(x0+h)n=x0n+nx0n-1h+h2P(x0) (2)

where P(x0) is a polynomial in x0 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