  In this paper, the polynomials $P_a(x)=x^{2^l+1}+x+a$ with
$a\in\mathrm{GF}(2^k)$ are studied. New criteria for the number of zeros of
$P_a(x)$ in $\mathrm{GF}(2^k)$ are proved. In particular, a criterion for
$P_a(x)$ to have exactly one zero in $\mathrm{GF}(2^k)$ when $\gcd(l,k)=1$ is
formulated in terms of the values of permutation polynomials introduced by
Dobbertin. We also study the affine polynomial $a^{2^l}x^{2^{2l}}+x^{2^l}+ax+1$
which is closely related to $P_a(x)$. In many cases, explicit expressions for
calculating zeros of these polynomials are provided.
