*f* is a function of *x*; fix a number *h* > 0.

First difference function is \Delta *f* (*x*) = *f*(*x*
+ *h*) - *f*(*x*).

Second difference function is \Delta^2 *f* (*x*) =
\Delta *f*(*x* + *h* - \Delta *f*(*x*).

Third difference function is \Delta^3 *f* (*x*) =
\Delta^2 *f*(*x* + *h*) - \Delta^2 *f*(*x*).

Etc.

Now suppose *f* is a polynomial function; *f* has * degree*
*n*
means that it has the form
*f*(*x*) = *a*_*n* *x*^*n* +
*a*_{*n*-1}*x*^{*n*-1}+ ... + *a*_1 *x*
+ *a*_0
with *a*_*n* not 0.

0. If *f* is of degree 0, it is a constant; \Delta *f* (*x*)
= 0.

1. If *f* is of degree 1 (linear) then \Delta *f* (*x*)
is constant
and so \Delta^2 *f* is zero.

2. If f is of degree 2 (quadratic) then \Delta^2 *f* (*x*)
is constant
and so \Delta^3 *f* is zero. (Check this for *f*(*x*) =
*x*^2; why is this
enough?)

3. Fill this in yourselves, and figure out the easiest way to check it.

Etc. State and prove the corresponding fact for polynomials
of degree *n*.