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Let $a, b, c$ be real numbers with $a \neq 0$ and let $α, β$ be the roots of the equation $a x^{2} + b x + c = 0$ . Then $a^{3} x^{2} + a b c x + c^{3} = 0$ has roots is

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$α^{2} β, β^{2} α$

$α, β^{2}$

$α^{2} β, β α$

$α^{3} β, β^{3} α$

answer is A.

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Detailed Solution

$(α + β) α β = \frac{- b}{a} \times \frac{c}{a} = - \frac{b c}{a^{2}}$

$\Rightarrow α^{2} β + β^{2} α = \frac{- b c}{a^{2}}$

$α^{2} β \times β^{2} α = \frac{c^{3}}{a^{3}}$

$\Rightarrow α^{3} β^{3} = \frac{c^{3}}{a^{3}}$

$\Rightarrow$ The Q.E. is $x^{2} - x (α^{2} β + β^{2} α) + α^{3} β^{3} = 0$

$\Rightarrow a^{3} x^{2} + a b c x + c^{3} = 0$

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