I would think about it a little more directly. First,Jis an ideal because any polynomial multiplied by an element ofJwill give you an element ofJ. It is principal because it is generated byx.

For the second part, I can't seem to give a rigourous explanation that I am happy with. Try to write a few of the cosets.

They are all different and there is an infinite number of them because you can just keep raisingxto a higher power. I hope I didn't confuse you with that. I'll think about it some more and try to come up with a better explanation.