Question

If the sum of two of the roots of $x^4-5x^3-3x^2+15x=0$ is zero, then the roots are

• A. $0,5,\pm \sqrt{3}$
• B. $0,5,\pm \sqrt{2}$
• C. $0,5,\pm \sqrt{5}$
• D. $0,5,\pm \sqrt{6}$

Answer 1

The correct option is A $0,5,\pm \sqrt{3}$

Let $\alpha ,\beta ,\gamma ,\delta$ be the roots of $x^4-5x^3-3x^2+15=0$
Such that $\alpha +\beta =0$
$S_1=\alpha +\beta +\gamma +\delta =5\Rightarrow \gamma +\delta =5S_3=\alpha \beta +\alpha \gamma +\alpha \delta +\beta \gamma +\beta +\gamma \delta =-15\Rightarrow (\alpha +\beta )\gamma +(\alpha +\beta )\delta +\alpha \beta +\gamma \delta =-15\Rightarrow \alpha \beta +\gamma \delta =-15$
Now from option only option A satisfies all conditions