Question

If all the roots of the equation $x^3+px+q=0,p,q\in R,q\ne 0$ are real, then which the following is correct?

• A. $p<0$
• B. $p=0$
• C. $p>0$
• D. $p\le 0$

Answer 1

The correct option is A $p<0$

Let $\alpha ,\beta ,\gamma$ be the roots of the given equation. Then,
$\alpha +\beta +\gamma =0\alpha \beta +\beta \gamma +\gamma \alpha =p\alpha \beta \gamma =-q$
Now,
$p=\alpha \beta +\beta \gamma +\gamma \alpha$
$\Rightarrow p=\frac{1}{2}\left[\right(\alpha +\beta +\gamma {)}^2-({\alpha }^2+{\beta }^2+{\gamma }^2)]$
$\Rightarrow p=-\frac{1}{2}({\alpha }^2+{\beta }^2+{\gamma }^2)$
As $q\ne 0$, so
$\alpha \beta \gamma \ne 0\Rightarrow \alpha \ne 0,\beta \ne 0,\gamma \ne 0$
Therefore,
${\alpha }^2+{\beta }^2+{\gamma }^2>0\Rightarrow -\frac{1}{2}({\alpha }^2+{\beta }^2+{\gamma }^2)<0\Rightarrow p<0$