How to simplify the following polynomial?
$ \begin{align} (t - \sqrt{3} \; e^{ \frac{\pi}{3} i }) (t - \sqrt{3} \; e^{ -\frac{\pi}{3} i }) &= t^2 - \sqrt{3} \; e^{ \frac{\pi}{3} i } \; t - \sqrt{3} \; e^{ -\frac{\pi}{3} i } \; t + 3\\ &= t^2 - \sqrt{3} \; t \; ( e^{ \frac{\pi}{3} i } + e^{ -\frac{\pi}{3} i } ) + 3 \end{align} $
I know the result is
$ t^2 - \sqrt{3} \; t + 3 $
but I can't see how to simplify the two complex numbers.
Is it a general rule, that if I've got a polynomial with $(t-c)(t-\bar c)$ where $c,\bar c \in \mathbb{C}$ and $\bar c$ is the complex complement of $c$, the resulting polnomial is $(t-c)(t-\bar{c})=t^2-ct-\bar{c}t+c\bar{c}=t^2-(c+\bar{c})t+|c|^2,$
PS.
$\mathbb{C}[t]$ means that the coefficients are in $\mathbb{C}$? Why we usually write $\mathbb{K}[t]$?