The problem is that, for the function $f$:
\begin{equation*}
f(x)=c_{0} + c_{1}x + c_{2}x^{2} + \ldots + c_{n}x^{n}, c_i \in \mathbb Z
\end{equation*}
then for $f(x)=0$, how many distinct solutions for $x$ at most ?
What I have known:
- there are $1$ solution at most when $n=1$.
- and $2$ at most for $n=2$, since I know $f(x)$ can be rewritten as $(x-a)(x-b)$.
But about the subsequent parts, $n=3,4,5,...$
Is there any guru could offer advices ? THANKS !