any concrete relationship between roots of polynomial and coeefficients+special pattern

Question

if $a_0$, $a_1$, ..,$a_n$ are real numbers such that $a_0+\frac{a_1}{2}+\frac{a_2}{3}+\frac{a_n}{n+1}=0$ prove the polynomial $a_0+a_1x+...+a_nx^n=0$ has a solution between 0,1.
This is a problem after Rolle theorem topic in real analysis.
let $f(x)=a_0+a_1x+...+a_nx^n$
I know polynomial above, is bounded and real valued and continuous on $[0,1]$. also the derivative of $f(x)$ exists for all point in $(0,1)$. so by Rolle theorem, if $f(0)=f(1)=0$ I can conclude there is a point $c\in(0,1)$ such that $f^{'}(c)=0$
My question is how can I then claim anything about the roots of f? ( I remember intermediate value theorem from calculus, but apparently this must not be the case here according to my text book )

Answer 1

Hint: Try to integrate!

(I still can't comment, so I wrote an answer)

Answer 2

Let $ n $ be a positive integer, and $ F $ the function defined as : $ F:x\mapsto\int\limits_{0}^{x}{f\left(t\right)\mathrm{d}t} $

Since $ F\left(1\right)-F\left(0\right)=\int\limits_{0}^{1}{f\left(t\right)\mathrm{d}t}=\sum\limits_{k=0}^{n}{\frac{a_{k}}{k+1}}=0 $, and $ F $ is $ \mathcal{C}^{1} $ on $ \left[0,1\right] \cdot $ Rolle's theorem allows us to say that : $$ \left(\exists c\in\left[0,1\right]\right),\ F'\left(c\right)=f\left(c\right)=0 $$