above several authors have been interested in the divisibility properties for Fibonacci type polynomials. However, the Lucas type polynomials have been less studied. Here we complete three cases of the SDP. Indeed, we give a characterization for the SDP for Fibonacci type polynomials and study both the SDP for Lucas type polynomials and the SDP for the combinations of Lucas type polynomials and Fibonacci type polynomials. Finally we provide an open question for the most general case of combination of two polynomials.

Generalized Fibonacci polynomials GFP {#General:Fibonacci:Polynomial}
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In the literature there are several definitions of generalized Fibonacci polynomials. However, the definition that we introduce here is simpler and covers other definitions. The background given in this section is a summary of the background given in [@florezHiguitaMuk]. However, the definition of generalized Fibonacci polynomial here is not exactly the same as in [@florezHiguitaMuk]. The *generalized Fibonacci polynomial* sequence, denoted by GFP, is defined by the recurrence relation

$$\label{Fibonacci;general}
G_0(x)=p_0(x), \; G_1(x)= p_1(x),\;  \text{and} \;  G_{n}(x)= d(x) G_{n - 1}(x) + g(x) G_{n - 2}(x) \text{ for } n\ge 2$$

where $p_0(x)$ is a constant and $p_1(x)$, $d(x)$ and $g(x)$ are non-zero polynomials in $\mathbb{Z}[x]$ with $\gcd(d(x), g(x))=1$.

For example, if 