A polynomial  $p\left(x\right)=x^4+a{x}^3+b{x}^2+cx+d$ has roots $\sqrt{2}$, e and $\pi$ and no other roots. Let $I={\int }_{\sqrt{2}}^{e}p\left(x\right)dx$ and $J={\int }_e^{\pi }p\left(x\right)dx$, then which of following option(s) is/are INCORRECT?

$I and J$ must have opposite signs

 $I and J$ can be both positive but not both negative

$I and J$ can be both negative but not both positive 

 Not enough information is provided to compare the signs of  $I and J$