Question 30
Find all functions $f : \mathbb{Q}[X] \to \mathbb{R}$ such that for all polynomials $P(x)$ and $Q(x)$ in $\mathbb{Q}[X]$:
i. $f(P(Q(x))) = f(Q(P(x)))$;
ii. $f(P(x)Q(x)) = f(P(x)) + f(Q(x))$, where $P(x)Q(x) \neq 0$.
i. $f(P(Q(x))) = f(Q(P(x)))$;
ii. $f(P(x)Q(x)) = f(P(x)) + f(Q(x))$, where $P(x)Q(x) \neq 0$.