[Summation of numerical sequences and differences in a series]

## [Integer and integer-valued polynomials]

For the polynomial $f(x)=x^{3}-x$, find $\Delta^{2} f(x)$.

Explain, without using divisibility arguments, why $f(x)$ is divisible by 6 for all integer $x$.

#
$\Delta f(x)=(x+1)^{3}-(x+1)-\left(x^{3}-x\right)=3 x^{2}+3 x, \Delta^{2} f(x)=\Delta(\Delta f(x))=6 x+6, \Delta^{3} f(x)=6$. According to problem $\underline{61448}$

$f(x)=0+0+6 C_{x}^{2}+6 C_{x}^{3}$.

The numbers $C_{x}^{2}$ and $C_{x}^{3}$ are integers for any integer $x$.

## Answer

$\Delta^{2} f(x)=6 x+6$.