Assume that $R$ is a unital algebra (maybe non-commutative) over an infinite field $F$ and $f$ is a polynomial from $R[x]$ with all coefficients from the center $Z(R)$.
Moreover, the coefficient of the highest term of $f$ is not a zero-divisor.
Is it true that $f(a)$ is not a zero-divisor for some $a\in F$?
I think the answer is "not true", but I can't find any counterexample.
Some remarks:
(1) Since all coefficients of $f$ belong to $Z(R)$, we also have $f(a)\in Z(R)$ for all $a\in F$. Therefore, we do not have here to specify whether we consider left or right zero-divisors.
(2) The condition that $F$ is infinite is important, since otherwise the answer is obviously "not true".
(3) Assume $f(a)$ is a zero-divisor for all $a\in F$. We denote $f(x)=r_n x^n + \cdots + r_1 x + r_0$ for some $r_0,\ldots, r_n$ from $Z(R)$, where $r_n$ is not zero-divisor by the condition.
Considering $f(0)$ we obtain that $r_0$ is a zero-divisor. In particular, $n\geq 1$.