If C are the distinct roots of the equation x 2 - x + 1 = 0 , then 101 + 107 is equal to

Question

If $α, β \in C$ are the distinct roots of the equation $x^{2} - x + 1 = 0,$ then $α^{101} + β^{107}$ is equal to

Smart Achievers · Accepted Answer

(1)

$We have,$ $x^{2} - x + 1 = 0$

$\Rightarrow x = \frac{- 1 (- 1) \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{3} i}{2}$

$∴$ $Let α = \frac{1 + \sqrt{3} i}{2} and β = \frac{1 - \sqrt{3} i}{2}$

$\Rightarrow α = - ω^{2} and β = - ω, where ω = \frac{- 1 + \sqrt{3} i}{2}$

$and ω^{2} = \frac{- 1 - \sqrt{3} i}{2}$

$Now, α^{101} + β^{107} = {(- ω^{2})}^{101} + {(- ω)}^{107}$

$= - ω^{202} + {(- ω)}^{107}$

$= - ω - ω^{2} = - (ω + ω^{2}) = - (- 1)$ $(∵ 1 + ω + ω^{2} = 0)$

$= 1$

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