TITLE: Rolle's theorem question?
QUESTION [0 upvotes]: Let $q(x) = x^n + ax + b$ where $a,b$ are elements of $\mathbb R$.  Prove that $q$ has at the most:
i) two distinct real roots of $n$ is even
ii)  three distinct real roots if $n $is odd.
Other than knowing this has to do with Rolle's theorem, I'm completely stuck.  Any help is appreciated.  Thanks.

REPLY [1 votes]: Hint:  How many real roots can $q^\prime(x)$ have?  If there are more than the given number of distinct roots for $q(x)$, what does Rolle's theorem say about the number of roots of the derivative (by considering consecutive pairs of roots of $q(x)$)?
{"set_name": "stack_exchange", "score": 0, "question_id": 596846}