Finding real root of polynomial degree n using MVT for integrals

**voyance** · December 11th 2012, 07:15 PM

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

**LordoftheFlies** · January 1st 2013, 05:25 PM

Let $f(x)=\displaystyle\sum_0^n a_i x^i$ and we have $\displaystyle\int_0^1 f(x)\;dx=\sum_0^n \dfrac{a_i}{i+1}=0$
By the MVT there exists $w\in (0,1)$ such that $\int_0^1 f(x)\;dx =f(w)\iff f(w)=0$

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