Menu Close

F-x-x-n-e-in-1-roots-of-F-x-2-factorize-F-x-inside-C-x-




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]
$$\mathrm{F}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{n}} \:−\mathrm{e}^{\mathrm{in}\alpha} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{roots}\:\mathrm{of}\:\mathrm{F}\left(\mathrm{x}\right)? \\ $$$$\left.\mathrm{2}\right)\:\mathrm{factorize}\:\mathrm{F}\left(\mathrm{x}\right)\:\mathrm{inside}\:\mathrm{C}\left[\mathrm{x}\right] \\ $$
Answered by Olaf_Thorendsen last updated on 10/Jul/21
1) x^n  = e^(inα)   x = e^(i(α+((2kπ)/n))) , k = 0, 1,..., k−1  2)  F(x) = Π_(k=0) ^(n−1) (x−e^(i(α+((2kπ)/n))) )
$$\left.\mathrm{1}\right)\:{x}^{{n}} \:=\:{e}^{{in}\alpha} \\ $$$${x}\:=\:{e}^{{i}\left(\alpha+\frac{\mathrm{2}{k}\pi}{{n}}\right)} ,\:{k}\:=\:\mathrm{0},\:\mathrm{1},…,\:{k}−\mathrm{1} \\ $$$$\left.\mathrm{2}\right) \\ $$$$\mathrm{F}\left({x}\right)\:=\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({x}−{e}^{{i}\left(\alpha+\frac{\mathrm{2}{k}\pi}{{n}}\right)} \right) \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *