This is a toy version of a [problem I have recently posted][1].

Imagine playing the following game. You choose a polynomial $B$ over a finite field $\mathbb F_p$ of degree $\deg B\le p-1$ (where $p$ is a large prime). I can modify your polynomial any way I wish, except that I am not allowed to add or change the monomials of degree exceeding $0.9p$. My goal is to make the resulting polynomial split completely into linear factors; if I fail to do so, you win.

* What polynomial $B$ will you start the game with to win? 
* Is there a comprehensible classification of all winning (for the first player) polynomials?

[1]:https://mathoverflow.net/q/330636/9924