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Question
If 1, α1, α2, ...... αn–1 are the roots of unity, then (1 + α1)(1 + α2) ...... (1 + αn–1) is equal to (when n is even) ______.
Options
n – 1
n
0
2
MCQ
Fill in the Blanks
Solution
If 1, α1, α2, ...... αn–1 are the roots of unity, then (1 + α1)(1 + α2) ...... (1 + αn–1) is equal to (when n is even) 0.
Explanation:
xn – 1 = 0
⇒ x = `(1)^(1/n)`
Given 1, α1, α2, ...... αn–1 are the nth roots of unity, then (xn – 1) = (x – 1)(x – α1)(x – α2)(x – α3)(x – α4) ................ (x – αn–1)
Put x = –1, we get
[(–1)n – 1] = (–1 – 1)(–1 – α1)(–1 – α2)(–1 – α3)(–1 – α4) …….. (–1 – αn–1)
Since, n is even
⇒ (–2)(–1 – α1)(–1 – α2)(–1 – α3)(–1 – α4) ......... (–1 – αn–1) = 0
(1 + α1)(1 + α2)(1 + α3)(1 + α4) …. (1 + αn–1) = 0
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Cube Root of Unity
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