Question

How do you simplify `(3x^{3}+4x+11)\div (x^{2}-3x+2)`?

Answer 1

`18/((x -1)(x-2))+ 25/(x - 2)+3x+9`

Explanation:

`cancel(3x^3) + 4x +11` `| ul(x^2 - 3x +2)`
`ul(cancel(3x^3) - 9x^2 +6x)` `(3x+9)`

`0+ cancel(9x^2)-2x+11`
`ul( cancel(9x^2)-27x+18)`
`0+25x-7`

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`(3x^3 + 4x +11) /(x^2 - 3x +2) =3x+9+(25x-7)/(x^2 - 3x +2)`

`=3x+9+(25x-7)/((x - 2)(x -1))=`

`18/((x -1)(x-2))+ 25/(x - 2)+3x+9`

Answer 2

`3x+9+(25x+29)/(x^2-3x+2)`

Explanation:

Note that I use place keepers of no value to help with formatting alignment.

`color(white)("dddddddddddddd.dddd")3x^3+0x^2+color(white)("d")4x+11`
`color(magenta)(+3x)(x^2-3x+2) ->color(white)("d")ul(3x^3-9x^2+color(white)("d")6xlarr" Subtract" )`
`color(white)("dddddddddddddddddddd")0+9x^2-color(white)("d")2x+11`
`color(magenta)(+9)(x^2-3x+2)->color(white)("d..ddddd")ul(9x^2-27x-18larr" Subtract"`
`color(white)("ddddddddddddddddddddddddd")color(magenta)(0+25x+29larr" Remainder"`

`color(magenta)( 3x+9+(25x+29)/(x^2-3x+2) )`