If α , β , γ , δ are the roots of the equation x 4 + x 3 + x 2 + x + 1 = 0 , then α 2021 + β 2021 + γ 2021 + δ 2021 is equal to

Step-by-step solution

We know that x 4 + x 3 + x 2 + x + 1 = 0 can be written as x 5 - 1 x - 1 = 0 ⇒     x 5 = 1 i.e. x = 1 1 5 So α ,   β ,   γ and δ are the 5 th  roots of unity except 1 . Now, from the concept of roots of unity, we know α n + β n + γ n + δ n + 1 n = 0 if   n   is not a multiple of   5 5 if   n   is a multiple of   5 So, α 2021 + β 2021 + γ 2021 + δ 2021 = - 1 .