Therefore, $$G_{m}^{*}(x)=G_{dq}^{*}(x)=(a^{dq}+b^{dq})/\alpha=Q(x)(a^{d}+b^{d})/\alpha.$$ This and the Binet formula (\[bineformulados\]) imply that $G_{m}^{*}(x)=G_{d}^{*}(x) Q(x)$.

Characterization of the strong divisibility property {#gcd:characterization}
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In this section we prove the main results of this paper. Thus, we prove the necessary and sufficient condition for the strong divisibility property for generalized Fibonacci polynomial of the Fibonacci type. We also prove that the strong divisibility property holds partially for generalized Fibonacci polynomial of the Lucas type. The other important result in this section is that the strong divisibility property holds partially for a generalized Fibonacci polynomial and its equivalent. The results here therefore provide a complete characterization of the strong divisibility property satisfied by the GFP of Fibonacci type.

We note that if $G_{m}^{*}(x)$ and $G_{n}^{\prime}(x)$ are two equivalent polynomial from Table \[equivalent\], then $\gcd(G_{m}^{*}(x), G_{n}^{'}(x))$ is either one or $G_{\gcd(m,n)}^{*}(x)$. However, it is not true in general. Here we give an example of a couple GFP polynomials that do not satisfies this property. Firstly we define a Fibonacci type polynomial $$G_0^{\prime}(x)=0, \; G_1^{\prime}(x)= 1,\;  \text{and} \;  G_{n}^{\prime}(x)= (2x+1) G_{n - 1}^{\prime}(x) + G_{n - 2}^{\prime}(x) \text{ for } n\ge 2.$$ We now define the equivalent polynomial of the Lucas type $$G_0^{*}(x)=2, \; G_1^{*}(x)= 2x+1,\;  \text{and} 