We say $t \in A$ is algebraic over $Z(A)$ if there exists $z_0, z_1, \ldots, z_n \in Z(A)$ such that $$z_0 + z_1t+ \cdots + z_nt^n = 0 \ \textrm{ and } \ \ z_n \neq 0.$$
Let $A = \mathbb{F}<X>$ be the free algebra over a field $\mathbb{F}$ on a set $X$ containing at least two elements. Prove that to $t \in A \setminus \mathbb{F}$, i.e., $t$ is not a constant polynomial, then $t$ is not algebraic.