Let $a_0, ... , a_n$ be real numbers satisfying $$\frac{a_n}{n+1}+\frac{a_{n-1}}{n}+ ... + \frac {a_1}{2}+a_0 = 0.$$ Show that the polynomial $$p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ has at least one zero in the interval $(0,1)$.
The solution is that let $$f(x) = \frac{a_n}{n+1}x^n + \frac {a_{n-1}}{n}x^{n-1}+ ...+\frac {a_1}{2}x + a_0$$ ,and then apply the Rolle's theorem.
But, I am wondering how the existence of root of $f(x)$ implies the existence of root of $p(x). $
Hint. Instead, consider the function $$F(x)=xf(x)= \frac{a_n}{n+1}x^{n+1} + \frac {a_{n-1}}{n}x^{n}+ \dots+\frac {a_1}{2}x^2 + a_0x. $$ Then $F(1)=F(0)=0$ and $F'(x)=p(x)$.