Question

If 1, α₁, α₂, ...... α_n–1 are the roots of unity, then (1 + α₁)(1 + α₂) ...... (1 + α_n–1) is equal to (when n is even) ______.

पर्याय

• n – 1

• n

• 0

• 2

Answer 1

If 1, α₁, α₂, ...... α_n–1 are the roots of unity, then (1 + α₁)(1 + α₂) ...... (1 + α_n–1) is equal to (when n is even) 0.

Explanation:

xⁿ – 1 = 0

⇒ x = `(1)^(1/n)`

Given 1, α₁, α₂, ...... α_n–1 are the n^th roots of unity, then (xⁿ – 1) = (x – 1)(x – α₁)(x – α₂)(x – α₃)(x – α₄) ................ (x – α_n–1)

 Put x = –1, we get

[(–1)ⁿ – 1] = (–1 – 1)(–1 – α₁)(–1 – α₂)(–1 – α₃)(–1 – α₄) …….. (–1 – α_n–1)

Since, n is even

⇒ (–2)(–1 – α₁)(–1 – α₂)(–1 – α₃)(–1 – α₄) ......... (–1 – α_n–1) = 0

(1 + α₁)(1 + α₂)(1 + α₃)(1 + α₄) …. (1 + α_n–1) = 0