I just wanna prove this statement- if a polynomial has at least one real zero then it will be equal to its derivative function somewhere for at least one x. I have proved the same for even degree polynomials using Rolle's theorem.
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As kind of an outline of a proof: for an odd polynomial where the coefficient of the highest power term is $1$, the polynomial is strictly positive when $x$ is larger than the largest root ($R_L$), and $R_L$ isn't a double root, the derivative function will be larger than the polynomial for $x = R_L$, but will grow more slowly than the polynomial, so eventually the two will cross. From this use a similar argument for other cases (first coefficient is $-1$, function is strictly negative when $x$ is less than least root, largest or smallest root is a double root). – Χpẘ Sep 10 '17 at 20:45