<s>[INST] finding real roots by way of complex I was given 
$$x^4 + 1$$
and was told to find its real factors. I found the $((x^2 + i)((x^2 - i))$ complex factors but am lost as to how the problem should be approached.
My teacher first found 4 complex roots ( different than mine)
$$( x - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i)( x - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i)$$
$$( x + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i)( x + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i)$$
He then mltiplied both pairs,resulting in:
$$(x^2 - \sqrt{2}x + 1) and (x^2 + \sqrt{2}x + 1)$$
How do I get the first four imaginary points, or the two distinct because I realize that each has a conjugate.
 [/INST] From $$x^4 + 1 = (x^2+i)(x^2-i)$$ we need to find the roots of each factor. Remember that $i = e^{i\pi/2}$, so $x=e^{i\pi/4}$ is a root of $x^2-i$. Hence, so is its conjugate $e^{-i\pi/4}$. By Euler's formula,
$$
e^{i\pi/4} = \cos(\pi/4) + i\sin(\pi/4) = \frac{\sqrt{2}}{2} + i\frac{\sqrt 2}{2}.
$$
I will leave you to do the others.
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