to do this problem you just set z= x+yi then just factor away right? i just need to know because homework is worth 40% of my grade in this class....ty very much!
It might be hard to find the four roots. Here is what it would look like if I do what FernandoRevilla said.
Solution:
$\displaystyle z^4+1$
$\displaystyle =z^4-i^2$
$\displaystyle =(z^2+i)(z^2-i)$
$\displaystyle =(z^2+i)(z^2-(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)^2)$
$\displaystyle =(z^2+i)(z+\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i) (z-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i)$
$\displaystyle =(z^2-(-i))(z+\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)(z-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i)$
$\displaystyle =(z^2-(\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i)^2)(z+\frac{\sqrt{2}}{2}+\frac {\sqrt{2}}{2}i)(z-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i)$
$\displaystyle =(z+\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i)(z-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)(z+\frac{\s qrt{2}}{2}+\frac{\sqrt{2}}{2}i)(z-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i)$
Why is Latex so confusing.....
Why not at this step write $\displaystyle z^2+i = 0 \Rightarrow z^2 = -i \Rightarrow z = \pm i\sqrt{i} $ or
$\displaystyle z^2-i \Rightarrow z^2 = i \Rightarrow z = \pm \sqrt{i}$ (or it's just the difference of two squares).
(So the four roots of the given equation are $\displaystyle \sqrt{i}$, $\displaystyle -\sqrt{i}$, $\displaystyle i\sqrt{i}$, and $\displaystyle -i\sqrt{i} )$ Thus:
$\displaystyle z^4+1 = (z^2+i)(z^2-i) = (z-i\sqrt{i})(z+i\sqrt{i})(z-\sqrt{i})(z+\sqrt{i})$.
An alternative:
$\displaystyle \sqrt[4]{-1}=\sqrt[4]{\cos \pi/4+i\sin \pi/4}=$
$\displaystyle \left\{{z_k=\cos (\pi/4+k\pi/2)+i\sin (\pi/4+k\pi/2):k=0,1,2,3}\right\}=$
$\displaystyle \left\{{\sqrt{2}/2(1+i),\sqrt{2}/2(-1+i),\sqrt{2}/2(-1-i),\sqrt{2}/2(1-i)}\right\}$
So,
$\displaystyle z^4+1=(z-z_0)(z-z_1)(z-z_2)(z-z_3)$
in the form lanierms provided i.e. separated real and imaginary parts of the roots.
No, it's not okay! It is very easy to look at a solution given by someone else and tell yourself that you understand it now, but even if you do understand every step, you have not had the experience of finding the steps for yourself- and that is where learning occurs.