  In this note, we employ the techniques of Swan (Pacific J. Math. 12(3):
1099-1106, 1962) with the purpose of studying the parity of the number of the
irreducible factors of the penatomial
$X^n+X^{3s}+X^{2s}+X^{s}+1\in\mathbb{F}_2[X]$, where $s$ is even and $n>3s$.
Our results imply that if $n \not\equiv \pm 1 \pmod{8}$, then the polynomial in
question is reducible.
