Question Number 146083 by mathmax by abdo last updated on 10/Jul/21
![F(x)=x^n −e^(inα) 1) roots of F(x)? 2) factorize F(x) inside C[x]](https://www.tinkutara.com/question/Q146083.png)
$$\mathrm{F}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{n}} \:−\mathrm{e}^{\mathrm{in}\alpha} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{roots}\:\mathrm{of}\:\mathrm{F}\left(\mathrm{x}\right)? \\ $$$$\left.\mathrm{2}\right)\:\mathrm{factorize}\:\mathrm{F}\left(\mathrm{x}\right)\:\mathrm{inside}\:\mathrm{C}\left[\mathrm{x}\right] \\ $$
Answered by Olaf_Thorendsen last updated on 10/Jul/21
$$\left.\mathrm{1}\right)\:{x}^{{n}} \:=\:{e}^{{in}\alpha} \\ $$$${x}\:=\:{e}^{{i}\left(\alpha+\frac{\mathrm{2}{k}\pi}{{n}}\right)} ,\:{k}\:=\:\mathrm{0},\:\mathrm{1},…,\:{k}−\mathrm{1} \\ $$$$\left.\mathrm{2}\right) \\ $$$$\mathrm{F}\left({x}\right)\:=\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({x}−{e}^{{i}\left(\alpha+\frac{\mathrm{2}{k}\pi}{{n}}\right)} \right) \\ $$