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In this paper, we find a lower bound for the order of the group $\langle\theta+\alpha\rangle \subset \overline{\mathbb F_{q}}^*$, where $\alpha\in\mathbb F_{q}$, $\theta$ is a generic root of the polynomial$F_{A,r}(X)=bX^{q^r+1}-aX^{q^r}+dX-c\in \mathbb F_{q}[X]$ and $ad-bc\ne0$.

On the multiplicative order of the roots of $bX^{q^r+1}-aX^{q^r}+dX-c$