Alright, these can't be the only ways...

I use the theorem and get all these possible rational zeros. And I could go through them one by one seeing if one of them works out and yeah, there's like a list of 20 of them (but really 40) I have to sift through hoping I get lucky. And so the ones I try out I use upper and lower bounds to try and enhance my search by restricting what I search for to narrow in on it. Even this though is time consuming. And I'm not allowed to use a graphing calculator. I have to do this at least within 5 minutes because my quizzes are timed.

Did anyone invent a better method that's not in the textbook that I can use to get these done quickly? Time is of the essence.