Question

If \( \alpha, \beta, \gamma, \delta \) are the roots of the equation \( x ^{4}+ x ^{3}+ x ^{2}+ x +1=0 \), then \( \alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021} \) is equal to

(1) -4

(2) -1 

(3) 1

(4) 4

Answer 1

The correct option is (2) -1.

Given, x⁴ + x³ + x² + x + 1 = 0

⇒ (x - 1)(x⁴ + x³ + x² + x + 1) = 0

⇒ x⁵ - 1 = 0

⇒ x = (1)^1/5 = 1, α, β, γ, δ

∴ 1, α, β, γ, δ are the fifth roots of unity

Since, 2021 is not a multiple of 5.

⇒ 1²⁰²¹ + α²⁰²¹ + β²⁰²¹ + γ²⁰²¹ + δ²⁰²¹ = 0

⇒ α²⁰²¹ + β²⁰²¹ + γ²⁰²¹ + δ²⁰²¹ = - 1

∴ The value of α²⁰²¹ + β²⁰²¹ + γ²⁰²¹ + δ²⁰²¹ is - 1.