Factoring a complex polynomial

[TheCanadian]

I've attached two equivalent complex equations, where one is written as a polynomial with 7 terms and the other is the factored form. I was just wondering how one can immediately write down the factored form based on the equation with 7 terms? Is there anything obvious (e.g. coefficient 1) or any particular method (possibly a general one) a person can use to find the factored form of the polynomial?

 

[fresh_42]

I've attached two equivalent complex equations, where one is written as a polynomial with 7 terms and the other is the factored form. I was just wondering how one can immediately write down the factored form based on the equation with 7 terms? Is there anything obvious (e.g. coefficient 1) or any particular method (possibly a general one) a person can use to find the factored form of the polynomial?

$1+z+z^2+\dots +z^{n-1} = \frac{z^n - 1}{z-1}$. I.e. the roots of $z^n-1$ are $1$ plus the roots of $1+z+z^2+\dots +z^{n-1}$. They are called roots of unity ($z^n = 1$) and are equally distributed along the unit circle, starting at $(1,0)$. Therefore they are multiples of the angle $\frac{2 \pi}{n}$, i.e. $\exp (i{\frac{k}{n} 2 \pi})$ with $k=0,\dots ,n-1$.