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

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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) .
$$\mathrm{If}\:\alpha,\:\beta,\:\gamma,\:\delta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$${x}^{\mathrm{4}} \:+\:{x}^{\mathrm{3}} \:+\:{x}^{\mathrm{2}} \:+\:{x}\:+\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{then}\:\mathrm{find} \\ $$$$\alpha^{\mathrm{2021}} \:+\:\beta^{\mathrm{2021}} \:+\:\gamma^{\mathrm{2021}} \:+\:\delta^{\mathrm{2021}} \:. \\ $$
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
$$\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)=\mathrm{x}^{\mathrm{5}} −\mathrm{1}=\mathrm{0} \\ $$$$\Rightarrow\mathrm{x}^{\mathrm{5}} =\mathrm{1}\Rightarrow\alpha^{\mathrm{5}} =\beta^{\mathrm{5}} =\gamma^{\mathrm{5}} =\delta^{\mathrm{5}} =\mathrm{1} \\ $$$$\Rightarrow\alpha^{\mathrm{2020}} =\beta^{\mathrm{2020}} =\gamma^{\mathrm{2020}} =\delta^{\mathrm{2020}} =\mathrm{1} \\ $$$$\Rightarrow\alpha^{\mathrm{2021}} +\beta^{\mathrm{2021}} +\gamma^{\mathrm{2021}} +\delta^{\mathrm{2021}} =\alpha+\beta+\gamma+\delta=−\mathrm{1} \\ $$
Answered by AntonCWX last updated on 18/Jan/25

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