Roots of a power series in an interval Let $ a_0 + \frac{a_1}{2} + \frac{a_2}{3} + \cdots + \frac{a_n}{n+1} = 0 $
Prove that $ a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n = 0 $ has real roots into the interval $ (0,1) $
I found this problem in a real analysis course notes, but I don't even know how to attack the problem. I tried to affirm that all coefficients are zero, but that is cleary not true, we have many cases when the result is 0 but $ a_i \ne 0$ for some $i$. I have tried derive/integrate, isolate and substitute some coefficients ($ a_0 $ and $a_n $ where my favorite candidates). Work with factorials (and derivatives and factorials) but could not find a way to prove. I have many pages of useless scratches.
Any tips are welcome.

Consider the function $f(x)=a_0x+ \frac{a_1}{2}x^2+\cdots + \frac{a_n}{n+1}x^{n+1}$. Then $f(1)=0$ is given, and $f(0)=0$ is clear. Rolle's theorem now shows there is $x \in (0,1)$ such that $f'(x)=0$.
