Let p be a polynomial. Show that the roots of p' are real if the roots of p are real.
The Attempt at a Solution
So we start with a root of p', call it r. We want to show that r is real. Judging by the condition given, I am assuming that we have to relate this root of p' to a root of p so that we can be guaranteed r is real. I am pretty stuck on this one. I've tried to see how Taylor's theorem could help here (since this is an exercise from the section where Taylor's theorem is introduced), but no such luck. Maybe some application of l'Hopital's, since this relates a function to it's derivative? I've tried this but have come up blank. Any words of wisdom would be seriously appreciated!