Question 30
Initially, the blackboard contains two polynomials $x^3-3x^2+5$ and $x^2-4x$. If polynomials $f(x)$ and $g(x)$ are written on the blackboard, it is permitted to write any polynomial of the form $f(x)\pm g(x)$, $f(x)g(x)$, $f(g(x))$, or $cf(x)$, where $c$ may be any (not necessarily integer) constant. Is it possible after several such operations to write a nonzero polynomial of the form $x^n-1$? (K. Tyschuk)