IIT JEE , AIEEE Forum - Mathematics , Algebra - what is the easiest way to solve these kinds of problem

sourab_MCA

if w is nth root of unity then

(1-w)(1-w²)(1-w³).........(1-w^n-1)

thnx

danny_007

take any value of "n" and solve if asked in MCQ's

danny_007

but its better if u take n >= 2 for every qs in wich u have to check (not only in complex)

hsbhatt

Since $\omega$ , $\omega^2$ ..., $\omega^{n-1}$ are roots of z^n = 1 .

$(z^n - 1) = (z-1)(z-\omega)(z-\omega^2)(z-\omega^3)...(z-\omega^{n-1})$

or $1+z+z^2+z^3+...+z^{n-1} = (z-\omega)(z-\omega^2)(z-\omega^3)...(z-\omega^{n-1})$

Now substitute z =1.

Hence $(1-\omega)(1-\omega^2)(1-\omega^3)...(1-\omega^{n-1}) = n$