Question: If 1, a1, a2,……an–1 are the n roots of unity, then (1 – a1</su...

Question

If 1, a₁, a₂,……a_n–1 are the n roots of unity, then (1 – a₁) (1 – a₂) …. (1 – a_n–1) is equal to –

Answer 1

Sol. Being the roots of unity, 1, a₁, a₂, …, a_{n –1} are the roots of xⁿ – 1 = 0, so that

xⁿ – 1 = (x – 1) (x – a₁) (x – a₂) …. (x – a_n–1)

Ž (x – a₁) (x – a₂) …. (x – a_n–1) = $\frac{x^{n} - 1}{x - 1}$

= x^n–1 + x^n–2 + ….. + x + 1

Ž (x – a₁) (x – a₂) ….. (x – a_n–1) = x^n–1 + x^n–2 + …. + x + 1.

Putting x =1 in the above equation, we now get

(1 – a₁) (1 – a₂) …. (1 – a_n–1)

= $\underset{ntimes}{\overset{1 + 1 + .... + 1 + 1}{︸}}$ = n.

Question: If 1, a<sub>1</sub>, a<sub>2</sub>,……a<sub>n–1</sub> are the n roots of unity, then (1 – a<sub>1</su...