- 1}^{\prime}(x) + Q_{n - 2}^{\prime}(x) \text{ for } n\ge 2.$$ It easy to see that $2Q_{n}^{'}(x)=Q_{n}(x)$ and that $\alpha=2$. Flórez, Junes and Higuita [@florezHiguitaJunes] worked similar problems for numerical sequences.

**Note.** The definition of Generalized Fibonacci Polynomial in [@florezHiguitaMuk] differs with definition on this papers due to the initial conditions on the Fibonacci type polynomials. Thus, the initial conditions for the Fibonacci type polynomials in [@florezHiguitaMuk] is $G_{0}(x)=p_{0}(x)=1$ and so implicitly $G_{-1}(x)=0$. However, our definition for the Lucas type polynomials are the same in both papers.

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Divisibility properties of Generalized Fibonacci Polynomials
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In this section we prove a few divisibility and $\gcd$ properties which are true for all GFP. These results will be used in a section later on to prove the main results of this paper.

Proposition \[prop2;1\] parts (1) and (2) is a generalization of Proposition 2.2 in [@florezjunes]. The proof here is similar the proof in [@florezjunes] since both use properties of integral domains. The reader can therefore update the proof in the afore-mentioned paper to obtain the proof of this proposition.

\[prop2;1\] Let $p(x), q(x), r(x),$ and $s(x)$ be polynomials.

(1) If $\gcd(p(x),q(x))=\gcd(r(x),s(x))=1$, then $$\gcd(p(x)q(x),r(x)s(x))=\gcd(p(x),r(x))\gcd(p(x),s(x))\gcd(q(x),r(x))\gcd(q(x),s(x)).$$

(2) If $\gcd(p(x),r(x))=1\gcd(q(x),s(x))=1$, then $$\gcd(p(x)q(x),r(x)s(x))=\gcd(p(x),s(x))\gcd(q(x),r(x)).$$

(3) If $z_1(x)=\gcd(p(x),r(x))$ 