Question 28
At the beginning of work, the memory of computer contained a single polynomial $x^2 - 1$. Each minute, the computer can do one of two things: choose any polynomial $f(x)$ from its memory and memorize also polynomials $f^2(x) - 1$ and $f(x^2 - 1)$, or chooses any two different polynomials $g(x)$ and $h(x)$ from its memory and memorizes the polynomial $\frac{1}{2}(g(x) + h(x))$. Is it possible that after some time the memory of computer contains polynomial $P(x) = \frac{1}{1024}(x^2 - 1)^{2048} - 1$?