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(C) Let the roots of the equation $x^3+b x^2+c x+d=0$ be $\alpha, \beta, 	ext{ and } \gamma$.From the relation between roots and coefficients,we have

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$8 d+b^3=4 b c$

Vedclass · Answer

(C) Let the roots of the equation $x^3+b x^2+c x+d=0$ be $\alpha, \beta, 	ext{ and } \gamma$.From the relation between roots and coefficients,we have $\alpha+\beta+\gamma=-b$.According to the given condition,one root is the sum of the other two,so let $\alpha=\beta+\gamma$.Substituting this into the sum of roots equation: $\alpha+\alpha=-b$,which implies $2\alpha=-b$ or $\alpha=-\frac{b}{2}$.Since $\alpha$ is a root of the equation,it must satisfy $x^3+b x^2+c x+d=0$.Substituting $x=-\frac{b}{2}$: $\left(-\frac{b}{2}ight)^3+b\left(-\frac{b}{2}ight)^2+c\left(-\frac{b}{2}ight)+d=0$.$-\frac{b^3}{8}+\frac{b^3}{4}-\frac{b c}{2}+d=0$.Multiplying the entire equation by $8$: $-b^3+2b^3-4bc+8d=0$.$b^3-4bc+8d=0$,which simplifies to $b^3+8d=4bc$.Thus,option $C$ is correct.

The condition to be satisfied in order that one root of $x^3+b x^2+c x+d=0$ is the sum of the other two roots,is

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