Let $f(X) = a_n X^n + \dots+a_0$. Then the Laplace transform of $f$ is $g(s) = \mathscr{L}\{f\}(s) = \frac{n! a_n}{s^{n+1}} + \dots + \frac{a_0}{s}$. If you now define the polynomial transform of $f$ to be the polynomial with precisely the same roots as $g(s)$, i.e.
$$ g(s) = \mathscr{L}\{f\}(s) = \frac{n! a_n}{s^{n+1}} + \dots + \frac{a_0}{s} = 0 \\ \iff \\ a_n n! + \dots + a_0 s^n = 0 $$
so let $h(s) = a_n n! + \dots +a_0 s^n$ be our transformed polynomial. What relationships does it have with $f$?
Also, don't we have immediately that $h(0) \neq 0$ since $g(0) = \infty$?
Thanks.