Question

Suppose p is a polynomial with n distinct real roots. Show that p' has
at least (n-1) distinct real roots.

Answer 1

Conditions

Suppose $p$ is a polynomial with $n$ distinct real roots. Show that $p'$ has at least $n-1$ distinct real roots.

Solution

Let's use Rolle's Theorem. It claims that for differentiable functions with two points with an equal value, their derivative has a root between these points.

The polynomial is a differentiable function, and as we have $n$ roots, so we have $n-1$ intervals, where at two distinct points we have equal values (zero, as they are roots).

We have the following n-1 intervals

$$(\alpha_1, \alpha_2), (\alpha_2, \alpha_3), \dots, (\alpha_{n-1}, \alpha_n)$$ $$P(\alpha_1) = P(\alpha_2) = \dots = P(\alpha_n) = 0$$

In each interval by Rolle's Theorem we have 1 root for derivative function. Totally $n-1$ roots. And the proof is done.

Q.E.D.