If the sum of two of the roots of x^3+p x^2-q | vedclass

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(A) Let the roots of the equation $x^3+p x^2-q x+r=0$ be $\alpha, \beta, \gamma$.Given that the sum of two roots is zero,let $\alpha+\beta=0$,which im

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$-r$

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(A) Let the roots of the equation $x^3+p x^2-q x+r=0$ be $\alpha, \beta, \gamma$.Given that the sum of two roots is zero,let $\alpha+\beta=0$,which implies $\beta=-\alpha$.From the relation between roots and coefficients:$1) \alpha+\beta+\gamma = -p \implies 0+\gamma = -p \implies \gamma = -p$.$2) \alpha \beta + \beta \gamma + \gamma \alpha = -q \implies \alpha(-\alpha) + \gamma(\alpha+\beta) = -q$.Since $\alpha+\beta=0$,we have $-\alpha^2 + 0 = -q$,so $\alpha^2 = q$.$3) \alpha \beta \gamma = -r \implies \alpha(-\alpha)\gamma = -r \implies -\alpha^2 \gamma = -r$.Substituting $\alpha^2=q$ and $\gamma=-p$ into the equation $-\alpha^2 \gamma = -r$:$-q(-p) = -r \implies pq = -r$.

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

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