Q. If α is a root of the equation x^3 −3x−1=0, prove that the other roots are 2−α^2 and α^2 −α−2. Please help.


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Question Number 33193 by SammyKT last updated on 12/Apr/18

$Q . If α is a root of the equation$ $x^{3} - 3 x - 1 = 0,$ $prove that the other roots are$ $2 - α^{2} and α^{2} - α - 2 .$ $Please help .$
	Answered by MJS last updated on 12/Apr/18

	$x^{3} + 0 x^{2} - 3 x - 1 = 0$ $(x - α) (x - β) (x - γ) = 0$ $x^{3} - (α + β + γ) x^{2} + (α β + α γ + β γ) x - α β γ = 0$ $\Rightarrow - (α + β + γ) = 0$ $now we set β = 2 - α^{2} and γ = α^{2} - α - 2$ $α + 2 - α^{2} + α^{2} - α - 2 = 0$ $true$
	Commented by SammyKT last updated on 13/Apr/18

	$Thank you$
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