Let $f(x,y) := (x+1)\cdots (x+j)-(y+1)\cdots (y+k) \in \mathbb{F}_{q}$, where $j>k$. Do you know if there is a **quick** way to show that no polynomial of the form $ax+by+c$, with $(a,b)\in \mathbb{F}_{q} \times \mathbb{F}_{q}\setminus\{(0,0)\}$, can divide $f(x,y)$?

I originally posted this question here: https://math.stackexchange.com/questions/1975446/non-divisible-by-a-linear-factor However, I consider that MO is a more suitable venue for it.

Thanks for your replies, comments, web-links, bibliographical recommendations, etc.