A theorem to rule them all

Calculus Level pending

True or False?

Let \(a_0, a_1 , \ldots , a_n\) be reals such that \( \dfrac {a_0}1 + \dfrac{a_1}2 + \cdots + \dfrac{a_n}{n+1} = 0 \), then there exists a real \(z\in [0,1] \) such that \(a_0 + a_1 z + \cdots + a_n z^n = 0 \).

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