Relationship between roots of a polynomial and coefficients

Question

Let $p(x)=a_0+a_1x+\ldots+a_nx^n$ be a polynomial with real coefficients, which of the following assumptions guarantee that $p(x)$ has a zero in $[0,1]$

1) $a_0 < 0$ and $a_0+a_1+\ldots +a_n >0$

2) $a_0+\frac{a_1}{2}+\ldots + \frac{a_n}{n+1} =0$

3) $\frac{a_0}{1 \times 2} + \frac{a_1}{2 \times 3} + \ldots +\frac{a_n}{(n+1) \times (n+2)} = 0$.

I tried to analyze this problem based on the relationship between roots and coefficients that I studied in elementary analysis class but with no success. Any hints?

Answer 1

Hints.

(1) Intermediate Value Theorem.

(2) Let $$q(x)=a_0x+\frac{a_1}{2}x^2+\ldots + \frac{a_n}{n+1}x^{n+1}\ .$$ Then $q(0)=0$ and $q(1)=0$. Does this guarantee that $q'(x)=0$ somewhere in $[0,1]$?

(3) Let $$r(x)=\frac{a_0}{1 \times 2}x^2 + \frac{a_1}{2 \times 3}x^3 + \ldots +\frac{a_n}{(n+1) \times (n+2)}x^{n+2}\ .$$ Then $r(0)=0$ and $r(1)=0$ and $r'(0)=0$. Does this guarantee that $r''(x)=0$ somewhere in $[0,1]$?