how can I divide for example $\frac{x^2+1}{2x+1}$ in $\frac{\mathbb{Z}_3[x]}{x^3+1}$? It's like normal polynomial dividing but here I got in first step $\frac{x}{2} (\frac{x^2}{2x}=\frac{x}{2}$).What is the modulo value of this statement? I need it for solve a linear equation solve, it's part of the Euclidan algorithm..but here I am stuck and I don't know what to do..Right answer is 2x+2 with remainder 2..But I don't know WHY:D
Please help..And please be patience for my poor English and absence of right formatting, today I've an exam so I am a bit hurry:D