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Question Number 215811 by MATHEMATICSAM last updated on 18/Jan/25

If α, β, γ, δ are the roots of x^4 + x^3 + x^2 + x + 1 = 0 then find α^(2021) + β^(2021) + γ^(2021) + δ^(2021) .

Ifα,β,γ,δaretherootsofx4+x3+x2+x+1=0thenfindα2021+β2021+γ2021+δ2021.

Answered by A5T last updated on 18/Jan/25

(x−1)(x^4 +x^3 +x^2 +x+1)=x^5 −1=0 ⇒x^5 =1⇒α^5 =β^5 =γ^5 =δ^5 =1 ⇒α^(2020) =β^(2020) =γ^(2020) =δ^(2020) =1 ⇒α^(2021) +β^(2021) +γ^(2021) +δ^(2021) =α+β+γ+δ=−1

(x−1)(x4+x3+x2+x+1)=x5−1=0⇒x5=1⇒α5=β5=γ5=δ5=1⇒α2020=β2020=γ2020=δ2020=1⇒α2021+β2021+γ2021+δ2021=α+β+γ+δ=−1

Answered by AntonCWX last updated on 18/Jan/25

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