For odd q, f(x) = x(q+1)/2 + ax is a PP of Fq if and only if a2− 1 is a nonzero square in Fq. Moreover, the polynomial f(x) + cx is a PP of Fq for (q − 3)/2 values of c ∈ Fq [1939].

The polynomial xr(f(xd))(q−1)/d is a PP of Fq if (r, q − 1) = 1, d | q − 1, and f(xd) has no nonzero root in Fq.