I want to describe the polynomials with integer coefficients and the property that $f'(x) \mid f(x)$ (the derivative divides the polynomial).
So I know that $f(x)$ divides $g(x)$ if all of $f(x)$'s roots are roots of $g(x)$. I also know that $f'(x)$ has degree one less than $f(x)$. Finally, $f'(x)$ detects multiple roots. If $a$ is a root of $f(x)$ and $f'(x)$, it is at least a double root.
Using these facts I've conjectured that $f(x)$, if it has degree $n$, has 1 root with multiplicity $n$ or 2 roots, one with multiplicity $n-1$. But I can't prove it and I fear that I am making a mistake thinking just about linear terms.