If $1, α_{2}, α_{3} . . . . . . . α_{n - 1}$ are $n^{t h}$ roots of unity then evaluate the following :-
$(3 + α_{1}) (3 + α_{2}) . . . . . . (3 + α_{n - 1})$

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Solution

$1, α, α . . . . α_{n - 1}$ are $n^{t h}$ roots of unit.

then

$x^{n} - 1 = (x - 1) (x - α_{1}) (x - α_{2}) . . . . (x - α_{n_{1}})$

$\frac{x^{n} - 1}{x - 1} = (x - α_{1}) (x - α_{2}) . . . (x - α_{n - 1})$

Put $x = - 3$

$\frac{(- 3)^{n} - 1}{- 4} = (- 1)^{n} (3 + α_{1}) (3 + α_{2}) . . . (3 + α_{n - 1})$

$(3 + α_{1}) (3 + α_{2}) . . . . . . (3 + α_{n - 1}) = \frac{(- 1)^{n} (1 - (- 3)^{n})}{4}$

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If $1, α_{2}, α_{3} . . . . . . . α_{n - 1}$ are $n^{t h}$ roots of unity then evaluate the following :-
$(3 + α_{1}) (3 + α_{2}) . . . . . . (3 + α_{n - 1})$