Question

(X^3+1)dividend (x-1)

Answer 1

$x^3=x\cdot x^2$ and $x^2(x-1)=x^3-x^2$. So we have a remainder of

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

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

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

$x=x\cdot1$ and $1(x-1)=x-1$. Subtracting this from the previous remainder gives a new one of

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

and we're done since 2 does not divide $x$. So we have

$\dfrac{x^3+1}{x-1}=x^2+x+1+\dfrac2{x-1}$