Roots of the Derivative of a Polynomial

Just took a qualifying exam (and did pretty well on it I think), but got stumped by one of the questions.

Show that if a polynomial has all real roots then it's derivative has only real roots as well.

I couldn't construct a counterexample, so it seems true to me.

If the roots are all distinct you can use Rolle's Theorem on each interval between the roots. But what would you do if you had multiple real roots? Why are you guaranteed real roots of the derivative in this case? Thanks in advance for any help.