Question

How do you long divide `(2x^4+x^3-5x^2+13x-6)/(x^2+3x-2)`?

Answer 1

`color(blue)(2x^2-5x+14`and remainder `color(blue)(-39x+22`

Explanation:

`(2x^4+x^3-5x^2+13x-6) / (x^2+3x-2) = 2x^2-5x+14` and remainder of
`-39x+22`

`color(white)(..........)color(white)(............)2x^2-5x+14`
`x^2+3x-2|overline(2 x^4+x^3-3x^2+13x-6)`
`color(white)(....................)ul(2x^4+6x^3-4x^2)`
`color(white)(..........................)-5x^3-x^2+13x`
`color(white)(............................)ul(-5x^3-15x^2+10x)`
`color(white)(..................................)14x^2+3x-6`
`color(white)(..................................)ul(14x^2+12x-28)`
`color(white)(..........................................)-39x+22`

`color(blue)((2x^4+x^3-5x^2+13x-6) / (x^2+3x-2) = 2x^2-5x+4` and remainder `color(blue)(-39x+22`

Answer 2

`2x^2-5x+14 -(39x-22)/(x^2+3x-2)`

Explanation:

This is a format that I decided upon specifically for Socratic

`" "2x^4+color(white)(6)x^3-5x^2+13x-6`
`color(magenta)(+2x^2)(x^2+3x-2)-> ul(2x^4+6x^3-4x^2 larr" Subtract"`
`" "0-5x^3-color(white)(15)x^2+13x-6`
`color(magenta)(-5x)(x^2+3x-2)-> " "ul(-5x^3-15x^2+10x larr" Subtract"`
`" "0+14x^2+color(white)(.)3x-6`
`color(magenta)(+14)(x^2+3x-2)->" "ul(14x^2+42x-28larr" Sub."`
`" "0" "color(magenta)(-39x+22)`

Where `-39x+22` is the remainder

`color(magenta)( 2x^2-5x+14+ (-39x+22)/(x^2+3x-2) )`

note that `+(-39x+22)/(x^2+3x-2)` is the same as `-(39x-22)/(x^2+3x-2)` giving:

`2x^2-5x+14 -(39x-22)/(x^2+3x-2)`