Question

If 1, alpha 1, alpha 2 alpha 3 are the fourth roots of unity then the value of 1 + alpha 1 into 1 + alpha 2 into 1 + alpha 3 is equal to

Answer 1

Answer:

Step-by-step explanation:

1, α¹ , α² & α³ are fourth roots of unity

x⁴ - 1 = (x -1)(x - α¹)(x - α²)(x - α³)

Let say value of (1 + α¹)(1 + α²)(1 + α³) = P

P = (1 + α¹)(1 + α²)(1 + α³)

lets multiply both sides by (1- α¹)

P(1- α¹) = (1- α¹)(1 + α¹)(1 + α²)(1 + α³)

=> P(1- α¹) = (1- α²)(1 + α²)(1 + α³)

=> P(1- α¹) = (1- α⁴)(1 + α³)

α⁴ = 1

=> P(1- α¹) = (1- 1)(1 + α³)

=> P(1- α¹) = 0(1 + α³)

=> P(1- α¹) = 0

=> P = 0