Consider $p$ and $q$ real polynomials in one variable of same degree
$d$, having $d$ real simple roots.
If $pq$ has $d$ real simple roots, which are the possible
configurations of its roots with respect to the roots of $p$ and $q$?
We defined a morsification on a Hirzebruch surface $\Sigma_n$ as a
curve with three components having $3n$ nodal points and intersecting
every fiber of the ruling generically in three real points.
We use trigonal curves and dessins in order to give an answer on how
to construct all possible configurations for a degree $d$.
Additionally, we show unicity for the configuration with a maximal
number of adjacent roots of one of the polynomials $p$, $q$ or $pq$.
