Baltic Way 1996 · Problem 12

Solution:

Consider the polynomial $W(x) = 1996x + 1996$. As $W(-1) = 0$ we conclude that $-1 \in S$.

Now consider the polynomial $U(x) = -x^{1996} - x^{1995} - \cdots - x^{2} - x + 1996$. As $U(1) = 0$ we have $1 \in S$.

Finally, let $T(x) = -x^{10} + x^{9} - x^{8} + x^{7} - x^{6} + x^{3} - x^{2} + 1996$. Then $-2 \in S$ since $T(-2) = 0$.