When, x5=1 then x5−1=0 ⇒(x−1)(x4+x3+x2+x+1)=0 Given, x4+x3+x2+x+1=0 has roots α,β,γ and 8 . ∴ Roots of x5−1=0 are 1,α,β,γ and 8 We know, Sum of pth power of nth roots of unity =0. (If p is not multiple of n ) or n (If p is multiple of n ) ∴ Here, Sum of pth power of nth roots of unity Here, p=2021, which is not multiple of 5 . ∴12021+α2021+β2021+γ2021+82021=0 ⇒α2021+β2021+γ2021+82021=−1