MathNet 2014 Question 36

Initially the polynomials $x^3-3x^2+5$ and $x^2-4x$ are written on a board. If polynomials $f(x)$ and $g(x)$ are already on the board, one may add $f(x)\pm g(x)$, $f(x)g(x)$, $f(g(x))$, and $cf(x)$, where $c$ is an arbitrary constant, not necessarily an integer. Can a polynomial $x^n-1$, for a positive integer $n$, appear after finitely many operations?

(K. Tyshchuk)