Let $p$ be a polynomial of degree $n$. Prove that it has at most n zeros.
Use induction and mean value theorem.
I don't understand how to do the induction. I used $n=0$ for the base case which is obvious and then I assumed that for degree $n$ that the polynomial had at most n roots. Then I was trying to prove that for $n+1$ it had at most $n+1$ roots.
Hint: (I'll let you formalize it)
Let $g(x) \in P_{n+1}$ be a polynomial from the set of all polynomials of degree $n+1$ ($P_{n+1}$).
Assume that $g$ has $k > n+1$ distinct roots.
Clearly, $g'(x)\in P_n$, and since $g$ has $k$ distinct roots, there are $(k-1)$ points where $g'(x) = 0$. However, $k - 1 > (n+1) - 1 > n$, and we come to a conclusion the $g'(x)$ has $k-1 > n$ roots...