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

`(x^3+1)/(x-1)=x^2+x+1+\frac2{x-1}`