Question

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

• A. $–r$
• B. $r$
• C. $2r$
• D. $–2r$

Answer 1

The correct option is A

$–r$

Explanation for the correct option.

Step 1: Find the third root.

For the cubic equation $x^3+p{x}^2-qx+r=0$, the sum of the three roots is $-p$.

But it is given that the sum of the two roots is $0$. So the third root is $-p$.

Step 2. Find the relation between $p,q,r$

As $-p$ is a root of the equation $x^3+p{x}^2-qx+r=0$, so substitute $-p$ for $x$ and simplify.

$$\begin{gathered} {\left(-p\right)}^3+p{\left(-p\right)}^2-q\left(-p\right)+r=0 \\ \Rightarrow -p^3+p^3+pq+r=0 \\ \Rightarrow pq=-r \end{gathered}$$

Hence the correct answer is option A.