Part (i) I can solve and understand that the solutions are $Z=e^\frac{2ki\pi}{5}$ for $k = 0,1,2,3,4$
Its the part (ii) I cannot understand. Could someone kindly give me a detailed simple step by step explanation of your logic if you are able to solve it.
I can see how the quartic is obtained from expanding $(Z+1)^5 = Z^5 +10Z^3 +10Z^2 + 5Z +1$ And since this is equal to $Z^5$ the $Z^5$'s cancel and you are left with the quartic that they gave you, but where do i go from here?
With kind regards,
Thank you.
As André Nicolas wrote in his comments, using the $\theta$ values you already discovered: $(z+1)^5=z^5\Rightarrow(\frac{z}{z+1})^5=1\Rightarrow \frac {z}{z+1}=e^{i\theta}\Rightarrow z=e^{i\theta}(z+1)\Rightarrow e^{i\theta}=z(1-e^{i\theta})\Rightarrow z=\frac{1}{1-e^{i\theta}}$