Consider $P(x)=a_0+a_1x+a_2x^2+\cdot\cdot\cdot\cdot\cdot\cdot +a_nx^n,a_i's \in \mathbb R$.The question is whether the following conditions can guarantee existence of a root of $P(x)$ in $[0,1]$?
a.$a_0<0$ and $a_0+a_1+ \cdot\cdot\cdot\cdot\cdot+a_n>0 $
b.$a_0+a_1/2+a_2/3+\cdot\cdot\cdot\cdot\cdot\cdot +a_n/(n+1)=0$
c.$a_0/1.2+a_1/2.3+\cdot\cdot\cdot\cdot\cdot\cdot +a_n/(n+1)(n+2)=0$
(a) is true since $P(0)<0$ and $P(1)>0$.but how to do other two?