A theorem to rule them all

Calculus Level 2

True or False?

Let a0,a1,,ana_0, a_1 , \ldots , a_n be reals such that a01+a12++ann+1=0 \dfrac {a_0}1 + \dfrac{a_1}2 + \cdots + \dfrac{a_n}{n+1} = 0 , then there exists a real z[0,1]z\in [0,1] such that a0+a1z++anzn=0a_0 + a_1 z + \cdots + a_n z^n = 0 .

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