Why can’t a non-constant polynomial have a constant interval? Given a polynomial that is not constant, of course, it doesn’t contain a constant interval. But how can we prove it?
Since the polynomial is differentiable over R, I came up with a solution that uses Lagrange mean value theorem n times and reduces the nth derivative to a constant. Since the leading term is not zero, there can not be a zero in the nth derivative, and that contradicts the mean value theorem. Therefore, the polynomial must not contain a constant interval.
However, I do realize that this solution is a bit complicated,  so is there a simpler solution(possibly elementary) that can prove this?

A polynomial of degree $n \gt 0$ has at most $n$ roots. A constant polynomial is of degree $0$.
A non constant polynomial $p$… can’t be constant. Therefore it is of degree $n\gt 0$. If it would be constant and equal to $a$ on an interval of strictly positive lenght, $p-a$ would have an infinite number of roots. A contradiction.
