Complex Numbers Question - None

Question

If 1, a_1, a_2, \ldots, a_{n-1} are n roots of unity, then the value of (1 - a_1)(1 - a_2)(1 - a_3) \cdots (1 - a_{n-1}) is equal to

Accepted Answer

Solution: If 1, a_1, a_2, \ldots, a_{n-1} are n roots of unity, they are roots of x^n - 1 = 0. We can write:

x^n - 1 = (x-1)(x-a_1)(x-a_2)\cdots(x-a_{n-1})

Dividing both sides by (x-1):

\frac{x^n-1}{x-1} = x^{n-1} + x^{n-2} + \cdots + x + 1 = (x-a_1)(x-a_2)\cdots(x-a_{n-1})

Setting x = 1:

n = (1-a_1)(1-a_2)\cdots(1-a_{n-1})

∴ Answer is (c) n.

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