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Let $\alpha, \beta$ be the roots of the equation $a x^{2}+b x+c=0$. A root of the equation $a^{3} x^{2}+a b c x+c^{3}=0$ is
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The correct answer is:
$\alpha^{2} \beta$
We have, $a^{3} x^{2}+a b c x+c^{3}=0$
$\begin{aligned}
&\Rightarrow x^{2}+\frac{b}{a} \cdot \frac{c}{a} x+\left(\frac{c}{a}\right)^{3}=0 \\
&\Rightarrow x^{2}-(\alpha+\beta) \alpha \beta x+\alpha^{3} \beta^{3}=0 \\
&\Rightarrow x=\alpha^{2} \beta, \alpha \beta^{2} \Rightarrow \alpha^{2} \beta \text { is one of the root. }
\end{aligned}$
$\begin{aligned}
&\Rightarrow x^{2}+\frac{b}{a} \cdot \frac{c}{a} x+\left(\frac{c}{a}\right)^{3}=0 \\
&\Rightarrow x^{2}-(\alpha+\beta) \alpha \beta x+\alpha^{3} \beta^{3}=0 \\
&\Rightarrow x=\alpha^{2} \beta, \alpha \beta^{2} \Rightarrow \alpha^{2} \beta \text { is one of the root. }
\end{aligned}$
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