Q-If-is-a-root-of-the-equation-x-3-3x-1-0-prove-that-the-other-roots-are-2-2-and-2-2-Please-help-

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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