Question

(x3 + 1) is divided by (x2 – x + 1)?

Answer 1

$x^3=x\cdot x^2$, and $x(x^2-x+1)=x^3-x^2+x$. Subtracting this from the dividend gives a remainder of

$(x^3+1)-(x^3-x^2+x)=x^2-x+1$

$x^2=1\cdot x^2$, and $1(x^2-x+1)=x^2-x+1$. Subtracting this from the previous remainder gives a new one of

$(x^2+x-1)-(x^2-x+1)=2x-2$

$x^2$ does not divide $2x$, so we're done, and we've found that

$\dfrac{x^3+1}{x^2-x+1}=x+1+\dfrac{2x-2}{x^2-x+1}$

i.e. you get a quotient of $x+1$ and a remainder of $2x-2$.