If `1,alpha_(1),alpha_(2),alpha_(3),...,alpha_(n-1)` are n, nth roots of unity, then `(1-alpha_(1))(1-alpha_(2))(1-alpha_(3))...(1-alpha_(n-1))` equals to

Options

A. `sqrt(3)`
B. `1//2`
C. n
D. 0

Answer

C

Explanation

It is given that
`,alpha_(1),alpha_(2),…….,alpha_(n-1)` are the roots of `x^(n)-1=0`.
`therefore x^(n)-1=(x-1)(x-alpha_(1))(x-alpha_(2))………..(x-alpha)(n-1)`
`rArr (x^(n)-1)/(x-1)=(x-alpha_(1))(x-alpha_(2))……….(x-alpha_(n-1))`
`rArr (x-alpha_(1))(x-alpha_(2))(x-alpha_(3)).....(x-alpha_(n-1))=x^(n-1)+x^(n-2)+.......+x^(2)+x+1`....................(i)
Putting `x=-1` in the above equation, we get
`rArr (1-alpha_(1))(-1-alpha_(2))(-1-alpha_(3)).....(x-alpha_(n-1))`
`rArr (x^(n-1))/(x-1) = (x-alpha_(1))(x-alpha_(2))(1-alpha_(3))......(1+alpha_(n-1))=(-1)^(n-1)+(-1)^(n-2)+(-1)^(n-3)+...............+(-1)+(-1)^(0)`...................(ii)
If n is odd, then RHS of the above equality contains an odd number of terms with alternately positive negative signs.
`therefore` RHS =1, Also`(-1)^(n-1)=1`.
`therefore (1-alpha_(1))(1-alpha_(2))(1-alpha_(3)).......(1+alpha_(n-1))=1`.