Let $\displaystyle \mathcal{Z}[x]$ be the set of all polynomials in the variable x, with coefficients from $\displaystyle \mathcal{Z}$ and the usual operations of addition and multiplication.

Let $\displaystyle S$ be the set of polynomials in $\displaystyle \mathcal{Z}[x]$ with roots at $\displaystyle x=0$ and $\displaystyle x=1$. Prove that $\displaystyle S$ is a subring of $\displaystyle \mathcal{Z}[x]$.