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

`(x^3+1)/(x^2-x+1)=x+1+(2x-2)/(x^2-x+1)`

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