This is a strange question, but consider the following type of equation, which falls out of a problem I'm working on in a different field:

$(1-n)^3 x^2 + c\_{1}nx + c\_{2}n = 0$

where $n,c\_{1}, c\_{2} \geq 0$ and $\in \mathbb{Z}$. By the rational root theorem, I believe that any integer solution of $x$ must be a factor of $c\_{2}n$ - but because $n$ is common to every co-efficient bar the leading one, can I reason as follows?

Reasoning: $(1-n)^3x^2$ must be divisible by $n$, which means that unless $n=1$, the co-efficient cannot be divided by $n$ and thus either $n=1$, or $x = wn$.

I'm not sure ifI'm on the right track here - would I instead have to conclude that $x^2 = wn$ or can I make the inference I have above? I have a whole class of polynomials were all terms bar the leading one have a repeated factor, and I'd be grateful for any advice!