If 1, z₁, z₂, , z_ n-1 are the n th roots of unity, (1-z₁ ) (1-z₂ ) (1-z_ n-1 ) is equal to

Step-by-step solution

We have given that, 1, z₁, z₂, z₃, . z_ n-1 are the n th root of unity. aligned & z^n-1=(z-1) (z-z₁ ) (z-z₂ ), ., (z-z_ n-1 ) & (z-1) (z^ n-1 +z^ n-2 + . .+z^2+z+1 ) & =(z-1) (z-z₁ )+ . . (z-z_ n-1 ) & (z^ n-1 +z^ n-2 + . .+z^2+z+1 )= (z-z₁ ) & (z-z₁ )= (z-z_ n-1 ) aligned Put z=1 , we get aligned & (1+1+ .+n times ) & = (1-z₁ ) (1-z₂ )+ . (1-z_ n-1 ) & n= (1-z₁ ) (1-z₂ ) (1-z_ n-1 ) aligned