Question

use De Moivre's theorem to find, in polar form, the five roots of the equation. Z root 5=1

Answer 1

Do you mean z⁵=1, which has 5 zeroes or roots.

z⁵-1=(z-1)(z⁴+z³+z²+z+1)=0, so z=1 is the only real root.

De Moivre: e^inθ=cos(nθ)+isin(nθ)=(cosθ+isinθ)⁵.

z⁵=e^iθ=1=cos(2πm)+isin(2θm), where m is an integer, 0≤m<5

z=e^iθ/5=cos(⅖πm)+isin(⅖πm). The 5 roots correspond to the vertices of a regular pentagon of radius 1.

The roots are found by plugging in the values of m.

When m=0 z=1 (the real root). When m=1, z=cos(⅖π)+isin(⅖π).

The roots correspond to the polar coordinates (1,0), (1,⅖π), (1,⅘π), (1,1⅕π), (1,1⅗π), which are the vertices of the pentagon.