Given this
$\dfrac{x+1}{x-i}$
What can I do with that?
If i do polynomial division as below, I get:
$$x-i)\overline{x+1} = 1 + \frac{1+i}{x-1}$$
however, I'm not even sure if this is legal maths. The other thing I think I could to is remove the complex demoninator with the complex conjugate to get:
$$\dfrac{x^2-x(i-1)-i}{x^2+1}$$
but I do not know what to do with this from this point on. I tried polynomial division to get:
$$1- \frac{-x(i-1)-1}{x^2+1}$$
Once again, i'm not sure if this is the correct path. Some help would be appreciated. Thanks
Polynomial division (quotient and remainder) in ${\Bbb C}[x]$ gives $$x+1 = 1\cdot (x-i) + (1+i).$$