  In recent years, several families of scattered polynomials have been
investigated in the literature. However, most of them only exist in odd
characteristic. In [B. Csajb\'ok, G. Marino and F. Zullo: New maximum scattered
linear sets of the projective line, Finite Fields Appl. 54 (2018), 133-150; G.
Marino, M. Montanucci and F. Zullo: MRD-codes arising from the trinomial
$x^q+x^{q^3}+cx^{q^5}\in\mathbb{F}_{q^6}[x]$, Linear Algebra Appl. 591 (2020),
99-114], the authors proved that the trinomial
$f_c(X)=X^{q}+X^{q^{3}}+cX^{q^{5}}$ of $\mathbb{F}_{q^6}[X]$ is scattered under
the assumptions that $q$ is odd and $c^2+c=1$. They also explicitly observed
that this is false when $q$ is even. In this paper, we provide a different set
of conditions on $c$ for which this trinomial is scattered in the case of even
$q$. Using tools of algebraic geometry in positive characteristic, we show that
when $q$ is even and sufficiently large, there are roughly $q^3$ elements $c
\in \mathbb{F}_{q^6}$ such that $f_{c}(X)$ is scattered. Also, we prove that
the corresponding MRD-codes and $\mathbb{F}_q$-linear sets of
$\mathrm{PG}(1,q^6)$ are not equivalent to the previously known ones.
