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
xq+xq3+cxq5βFq6β[x], Linear Algebra Appl. 591 (2020),
99-114], the authors proved that the trinomial
fcβ(X)=Xq+Xq3+cXq5 of Fq6β[X] is scattered under
the assumptions that q is odd and c2+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 q3 elements cβFq6β such that fcβ(X) is scattered. Also, we prove that
the corresponding MRD-codes and Fqβ-linear sets of
PG(1,q6) are not equivalent to the previously known ones