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Show that if $z = e^{i\theta}$ is a solution to $0 = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0$ [1] where all $a_i$ are real, then $0 = a_{n-1}\sin\theta + a_{n-2}\sin2\theta + \cdots + a_0 \sin ... 3answers 152 views ### simplification of an complex exponential equation There are these steps in a solutions manual I do not follow. I struggle to find any good and problem specific information about this kind of math wizardry on my own. I don't really know what to google ... 4answers 91 views ### rewrite$2ie^{i\pi}+i^3$i am asked to rewrite$2ie^{i\pi}+i^3$into$x+iy$form. i just tried all what i know so far, but couldnot come to solution. i said:$2ie^{i\pi}+i^3=2ie^{i\pi}-i$but further i am stuck really. i am ... 4answers 763 views ### Find all roots of$\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ... 2answers 72 views ### complex numbers - proof of this statement i am trying to prove this statement, i dont but how to start. $$\forall z,w \in \mathbb{C}\quad |z|^2+|w|^2=\frac{1}{2}(|z+w|^2+|z-w|^2)$$ can someone please show me how start? 1answer 39 views ### what is the difference - sorry for over-simplicity i am asking too simple question, sorry for that. what is the difference between these two imaginÃ¤r numbers?$\operatorname{Im}(| \sqrt2+3i|^2)$vs.$\operatorname{Im}((\sqrt2+3i)^2)| ...
$x^8 = \frac{1+i}{\sqrt{3} - i} = \frac{\sqrt[8]{\frac{2}{\sqrt{2}}}(\cos \frac{\pi}{4} + i \sin{\frac{\pi}{4}})}{2 \cos \frac{\pi}{6} + i \sin \frac{3\pi}{2}}$ What's the way to solve this kind of ...