Question

If two roots of the equation $x^5-x^4+8x^2-9x-15=0$ are $-\sqrt{3},1-2i$ then number of positive real roots are

Answer 1

$\because -\sqrt{3},1+2i$ are the two roots of the given equation and coefficients are rational numbers.
$\Rightarrow \sqrt{3},1-2i$ are also roots of the given equation.
Let the remaining root be $\alpha .$
Since, sum of roots equal to $1.$
$\sqrt{3}+(-\sqrt{3})+(1+2i)+(1-2i)+\alpha =1\Rightarrow \alpha =-1$
$\therefore$ The roots are $-1,\sqrt{3},1-2i,-\sqrt{3},1+2i$.
So, only one positive root i.e., $\sqrt{3}$