If the sum of two of the roots of $x^{3} + px^{2} - qx + r = 0$ is zero, then $pq$ is equal to

Question

Dr. Abubacker · Accepted Answer

(a)
Let $\alpha,\ \beta,\ \gamma$ are the roots of the given equation. Then,
$\alpha + \beta + \gamma = - p$
$\alpha\beta + \beta\gamma + \gamma\alpha = - q$
And $\alpha\beta\gamma = - r$
Now, $pq = (\alpha + \beta + \gamma)(\alpha\beta + \beta\gamma + \gamma\alpha)$
$= (0 + \gamma)\left\lbrack \alpha\beta + \gamma(\alpha + \beta) \right\rbrack\ (\because\alpha + \beta = 0\ is\ given)$
$= \alpha\beta\gamma$
$= - r$

Q. If the sum of two of the roots of $x^{3} + px^{2} - qx + r = 0$ is zero, then $pq$ is equal to

Explanation

Reviewed by Dr. Abubacker

Want more practice?