Solution of a polynomial in interval $(0,1)$

Question

Let $\displaystyle a_0 + \frac{a_1}{2} + \frac{a_2}{3} + ... + \frac{a_n}{n+1} = 0$, where $a_i$'s are some real constants.

How can we prove that the equation $a_0 + a_1x + a_2x^2 + ... +a_nx^n = 0$ has at least one solution in the interval $(0,1)$?

Answer 1

Note that $a_0 + a_1x + a_2 x^2 + \cdots + a_nx^n$ is continuous and $$\int_0^1 (a_0 + a_1x + a_2 x^2 + \cdots + a_nx^n) dx = 0$$