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

1. ## 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 a0, a1, a2,..., an are real numbers that satisfy (a0/1) + (a1/2) + (a2/3) +...+ (an/n+1) = 0
show that the equation a0 + a1x + a2x2 +...+ anxn = 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.

2. ## 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 $\displaystyle \int_0^1 f(x)\;dx =f(w)\iff f(w)=0$