α^2021 + β^2021 + γ^2021 + δ^2021 for x^4 + x^3 + x^2 + x + 1 = 0

If are the roots of the equation ,  then is equal to

(A) –4
(B) –1
(C) 1
(D) 4

Solution:

First, simplify the given equation.

Use property of nth root of unity i.e.

Step 1 of 2:

Given, ….(1)

Multiply by x on both sides

…..(2)

From (1), we get

So, equation (2) becomes

Therefore roots are

i.e.

Step 2 of 2:

Now it is given that \text{are roots of equation (1)

And for , 5th root is 1

So, from the property of the nth root of unity. Suppose the positive value of the integer is the nth root of unity.

All the n roots of the nth roots of unity are in GP. If we add all the n roots of the nth root, the resulting value equals zero.

Therefore,

Final Answer:

Hence, Option (B) is correct.