Question

How do you long divide `(x^4+x^3-13x^2-25x-12) / (x^2+2x+1)`?

Answer 1

The quotient is `=x^2-x-12` and the remainder is `=0`

Explanation:

Let's do the long division

`color(white)(aaaa)` `x^4+x^3-13x^2-25x-12` `color(white)(aaaa)` `∣` `x^2+2x+1`

`color(white)(aaaa)` `x^4+2x^3+x^2` `color(white)(aaaaaaaaaaaaaaa)` `∣` `x^2-x-12`

`color(white)(aaaa)` `0-x^3-14x^2-25x`

`color(white)(aaaaaa)` `-x^3-2x^2-x`

`color(white)(aaaaaaaaa)` `0-12x^2-24x-12`

`color(white)(aaaaaaaaaaa)` `-12x^2-24x-12`

`color(white)(aaaaaaaaaaaaaaa)` `-0-0-0`

The quotient is `=x^2-x-12` and the remainder is `=0`

Answer 2

`x^2-x-12`

Explanation:

`" "x^4+x^3-13x^2-25x-12`
`color(red)(x^2)(x^2+2x+1) -> ul(x^4 +2x^3+x^2" " larr" subtract"`
`" "0 -x^3-14x^2-25x-12`
`color(red)(-x)(x^2+2x+1)->color(white)(.)ul(-x^3-2x^2color(white)(.)-x larr" subtract")`
`" "0-12x^2-24x-12`
`color(red)(-12)(x^2+2x+1)->" "color(white)(.)ul(-12x^2 -24x-12larr" subtract")`
`" "0" " +0" "+0`

There is no remainder

`(x^4+x^3-13x^2-25x-12)/(x^2+2x+1)" " =" " color(red)(x^2-x-12`