Question

If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3-p{x}^2+qx-r=0$, then the value of $\sum {\alpha }^2\beta$ is equal to

• A. $pq+3r$
• B. $pq+r$
• C. $pq-3r$
• D. $q^2/r$

Answer 1

The correct option is C $pq-3r$

$\alpha +\beta +\gamma =p;\alpha \beta +\alpha \gamma +\beta \gamma =q;\alpha \beta \gamma =r$

$\sum {\alpha }^2\beta ={\alpha }^2\beta +{\beta }^2\alpha +{\beta }^2\gamma +\beta {\gamma }^2+{\alpha }^2\gamma +\alpha {\gamma }^2=(\sum \alpha )(\sum \alpha \beta )-3\alpha \beta \gamma =pq-3r$