A good place to start would be
Wikipedia - Newton's Method. I'd also recommend reading
Wolfram's Mathworld entry on Newton's Method.
The main obstruction with using it for user-input functions is that you must know both f(x) and f'(x) beforehand. If, in your case, the polynomial is spelled out for you, it should be rather easy to implement.
Give us some code to glean through and we can provide a bit more help.