Question

If one root of $x^2+px+q=0$ may be the square of the othe, then $p^3+q^2+q=$

• A. $4pq$
• B. $3pq$
• C. $2pq$
• D. $pq$

Answer 1

The correct option is B $3pq$

We have,

$\begin{array}{c}{x}^2+px+q=0\alpha ,{\alpha }^2\\ {\alpha }^3=q\\ \alpha +{\alpha }^2=-p\\ {\alpha }^3{\left(1+\alpha \right)}^3=-p^3\\ q\left[1+{\alpha }^3+3\alpha \left(1+\alpha \right)\right]=-p^3\\ q\left[1+q+3\left(-p\right)\right]=-p^3\\ q+q^2+p^3=3pq\end{array}$

Hence, Option $B$ is correct answer.