Why don't you first try to decompose the polynomial as much as you can? It's obvious that are roots , so you can divide it by , and again
the quotient has as a root so...etc.
In short, (iv) is nilpotent (why?), (i) is a zero divisor (you'll see this when you decomposed as explained above) and (ii)-(iii) are units (if since there's commutativity here.)
BTW, to ask for "smallest b s.t....etc." is nonsense here. Perhaps you meant "polynomial b of smallest degree..."?