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If-are-the-roots-of-x-4-x-3-x-2-x-1-0-then-find-2021-2021-2021-2021-




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
(x1)(x4+x3+x2+x+1)=x51=0x5=1α5=β5=γ5=δ5=1α2020=β2020=γ2020=δ2020=1α2021+β2021+γ2021+δ2021=α+β+γ+δ=1
Answered by AntonCWX last updated on 18/Jan/25

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