  For every integer $k$ there exists a bound $B=B(k)$ such that if the
characteristic polynomial of $g\in \operatorname{SL}_n(q)$ is the product of
$\le k$ pairwise distinct monic irreducible polynomials over $\mathbb{F}_q$,
then every element $x$ of $\operatorname{SL}_n(q)$ of support at least $B$ is
the product of two conjugates of $g$. We prove this and analogous results for
the other classical groups over finite fields; in the orthogonal and symplectic
cases, the result is slightly weaker. With finitely many exceptions $(p,q)$, in
the special case that $n=p$ is prime, if $g$ has order $\frac{q^p-1}{q-1}$,
then every non-scalar element $x \in \operatorname{SL}_p(q)$ is the product of
two conjugates of $g$. The proofs use the Frobenius formula together with upper
bounds for values of unipotent and quadratic unipotent characters in finite
classical groups.
