Question

How do you use polynomial long division to divide `(2x^3-x+1)div(x^2+x+1)` and write the polynomial in the form `p(x)=d(x)q(x)+r(x)`?

Answer 1

`2(x-1) -(x-2)/(x^2+x-1)`
`color(white)()`

`d(x)=2`
`q(x)=(x-1)`
`r(x)=-(x-2)/(x^2+x-1)`

Explanation:

Note that `r(x)` will be the remainder and that `d(x)q(x)` will be a factorisation.

Note that I am using a place keeper for convenience of formatting:
`0x^2` which has no value.

`" "2x^3+0x^2-x+1`
`color(magenta)(2x)(x^2+x+1)-> ul(2x^3+2x^2+2xlarr" Subtract")`
`" "0-2x^2-3x+1`
`color(magenta)(-2)(x^2+x+1)->" "ul(-2x^2-2x-2larr" Subtract")`
`" "0-x+2larr" Remainder"`

`color(magenta)((2x-2)+(-x+2)/(x^2+x+1))`

Write as: `2(x-1) -(x-2)/(x^2+x-1)`

Note that I have changed the remainder's sin (for the whole) from positive to negative to change `-x` to `+x`

Thus we have:

`d(x)=2`
`q(x)=(x-1)`
`r(x)=-(x-2)/(x^2+x-1)`