$\displaystyle x^4-x^2+1$ Factorize.
I've set $\displaystyle x^2=t \rightarrow t^2-t+1=0$. Using the formula, I get $\displaystyle x_{1,2}=\pm\sqrt\frac{1+i\sqrt3}{2} x_{3,4}=\pm\sqrt\frac{1-i\sqrt3}{2}$ What now?!
$\displaystyle x^4-x^2+1$ Factorize.
I've set $\displaystyle x^2=t \rightarrow t^2-t+1=0$. Using the formula, I get $\displaystyle x_{1,2}=\pm\sqrt\frac{1+i\sqrt3}{2} x_{3,4}=\pm\sqrt\frac{1-i\sqrt3}{2}$ What now?!
$\displaystyle \pm\sqrt\frac{1+i\sqrt3}{2}$ can be rewritten as $\displaystyle \pm\sqrt{e^{i\frac{\pi}{3}}}$ and then you can determine the roots of that to be $\displaystyle e^{i\frac{\pi}{6}}$ and $\displaystyle e^{i\frac{7\pi}{6}}$. For the other roots, the approach is similar.
Indeed, but what are $\displaystyle x_{1}, x_{2}, x_{3}, x_{4}$, and most importantly, how do you get them?
icemanfan, you are beyond me, care to explain? Why is it, that $\displaystyle \frac{1+i\sqrt3}{2} = e^{i\frac{\pi}{3}}$
The solution should be $\displaystyle (x^2+\sqrt3x+1)(x^2-\sqrt3x+1)$. How can you get it?
$\displaystyle x_{i}(i=1\ldots4)$ are the roots you have posted in your first message ( see Algebric proofs Section at Fundamental theorem of algebra - Wikipedia, the free encyclopedia )
$\displaystyle x+iy=r\mathrm{e}^{i\theta}$, where $\displaystyle r=\sqrt{x^{2}+y^{2}}$ and $\displaystyle \theta=\mathrm{atan}(y/x)$ ( see Euler's formula - Wikipedia, the free encyclopedia ).
But what if you do not "see" what the factor is (in this case, $\displaystyle (x^2+\sqrt3x+1)(x^2-\sqrt3x+1)$; some tips to see such beast?), or if introducing "new" variable gets you nowhere (in this case, $\displaystyle t^2-t+1=0$)?
What are considered best practices (in this case)?