The condition for one root of $a x^{3} + b x^{2} + c x + d = 0$ is sum of the other two roots, is

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$8 a^{2} d - 4 a b c + b^{3} = 0$

$6 a^{2} d + 7 a b c + b^{2} = 0$

$- 8 a^{2} d + 13 a b c + b^{4} = 0$

$5 a^{2} d - 3 a b c + c^{3} = 0$

answer is A.

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Detailed Solution

The given equation is $a x^{3} + b x^{2} + c x + d = 0$

Let $α, β, γ$ are the roots of given equation

By the given condition, one root is sum of other two roots

i.e., $α = β + γ$

$∴ s_{1} = α + β + γ$

$\Rightarrow 2 α = \frac{- b}{a}$

$\Rightarrow α = \frac{- b}{2 a}$

$∴ α$ is one roots of the given equation

$∴ f (\frac{- b}{2 a}) = 0$

$\Rightarrow a {(\frac{- b}{2 a})}^{3} + b {(\frac{- b}{2 a})}^{2} + c (\frac{- b}{2 a}) + d = 0$

$\Rightarrow \frac{- b^{3}}{8 a^{2}} + \frac{b^{3}}{4 a^{2}} - \frac{b c}{2 a} + d = 0$

$\Rightarrow 8 a^{2} d - 4 a b c + b^{3} = 0$

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