Finding real root of polynomial degree n using MVT for integrals

I really need help for this problem. I don't even know how to start this problem. Anyway, here's the problem:

if a_{0, }a_{1, }a_{2},..., a_{n} are real numbers that satisfy (a_{0}/1) + (a_{1}/2) + (a_{2}/3) +...+ (a_{n}/n+1) = 0

show that the equation a_{0 }+ a_{1}x + a_{2}x^{2} +...+ a_{n}x^{n} = 0 has at least one real root.

I am forced to use mean value theorem for integrals. In fact, I can't even see a straight answer for this even without the MVT for integrals. I just know that both equations are continuous and differentiable all over because they're polynomials.

Re: Finding real root of polynomial degree n using MVT for integrals

Let $\displaystyle f(x)=\displaystyle\sum_0^n a_i x^i$ and we have $\displaystyle \displaystyle\int_0^1 f(x)\;dx=\sum_0^n \dfrac{a_i}{i+1}=0$

By the MVT there exists $\displaystyle w\in (0,1)$ such that $\displaystyle \int_0^1 f(x)\;dx =f(w)\iff f(w)=0$