If α, β ∈ C are the distinct roots, of the equation x^2 – x + 1 = 0, then α^(101) + β^(107) is equal to
If α, β ∈ C are the distinct roots, of the equation x2 – x + 1 = 0, then α101 + β107 is equal to
(1) –1 (2) 0 (3) 1 (4) 2
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Note that:
x^3+1=(x+1)(x^2−x+1)
So since
α is a root of x^2−x+1=0
we have:
α^3+1=(α+1(α^2−α+1)
=(α+1)(0)
=0
So:α^3=−1
Similarly:
β^3=−1
Also:
x^2−x+1=(x−α)(x−β)
=x^2−(α+β)x+αβ
So:
α+β=1
αβ=1
Note that:
α^2+β^2=(α+β)^2−2αβ
=1-2
=−1
So:
α^101+β^107
=α^(3×33+2)+β^(3×35+2)
According to indices rule
(α^3)^33×α^2+(β^3)^35×β^2
Remember:
α^3=-1
β^3=-1
So substituting we have
(-1)^33×α^2+(-1)^35×β^2
=-α^2-β^2
=-(α^2+β^2)
Remember
α^2+β^2=-1
So from there we have
-(-1)
=1
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