Let q and f(x) be an odd characteristic and an irreducible polynomial of degree m over F_q, respectively. Then, suppose that F(x) = x^mf(x+x^-1) becomes irreducible over F_q. This paper shows that the conjugate zeros of F(x) with respect to F_q form a normal basis in F_q^2m if and only if those of f(x) form a normal basis in F_q^m and the part of conjugates given as follows are linearly independent over F_q,

{γ - γ^-1, (γ - γ^-1)^q, … , (γ - γ^-1)^qm-1},

where γ is a zero of F(x) and thus a proper element in F_q^2m. In addition, from the viewpoint of q-polynomial, this paper proposes an efficient method for checking whether or not the conjugate zeros of F(x) satisfy the condition.