  Using subresultants, we modify a recent real-algebraic proof due to Eisermann
of the Fundamental Theorem of Algebra ([FTA]) to obtain the following
quantitative information: in order to prove the [FTA] for polynomials of degree
$d$, the Intermediate Value Theorem ([IVT]) is requested to hold for real
polynomials of degree at most $d^2$. We also explain that the classical proof
due to Laplace requires [IVT] for real polynomials of exponential degree. These
quantitative results highlight the difference in nature of these two proofs.
