The answer in the given form can be obtained directly by observing that it is a cubic in x^2x2.
x^6-1=(x^2)^3-1^3x6−1=(x2)3−13
=(x^2-1)((x^2)^2+x^2+!)=(x2−1)((x2)2+x2+!)
=(x-1)(x+1)(x^4+x^2+1)=(x−1)(x+1)(x4+x2+1)
=(x-1)(x+1)(x^2-x+1)(x^2-x+1)=(x−1)(x+1)(x2−x+1)(x2−x+1)
=(x-1)(x+1)(x-1/2-i sqrt3/2)(x-1/2+i sqrt3/2)=(x−1)(x+1)(x−12−i√32)(x−12+i√32)
(x+1/2-i sqrt3/2)(x+1/2+ isqrt3/2)(x+12−i√32)(x+12+i√32)
The expanded form can be directly obtained from
(x^6-1)=prod(x-e^(i(2kpi)/6))(x6−1)=∏(x−ei2kπ6), k=0, 1, 2,..,5.,
using the six 6th roots of 1, {e^(i(2kpi)/6)}{ei2kπ6}, k=0, 1, 2,..,5.,
