First slide

Theory of expressions

Question

If $α, β$ are the roots of $x^{2} - x + 1 = 0,$ then $α^{5} + β^{5} =$

Easy

A

2
B

1
C

12
D

4

Solution

$x^{2} - x + 1 = 0 \Rightarrow x = \frac{1 \pm \sqrt{3}}{2}$
$\Rightarrow x = - ω, - ω^{2}$
$α^{5} + β^{5} = {(- ω)}^{5} + {(- ω^{2})}^{5}$
$= - ω^{2} - ω = 1$
(Cube roots of unity $1 + ω + ω^{2} = 0$ )

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