Question

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

Answer 1

Using the Long Division method ->

Explanation:

First step I would do is remove the negative sign from the fraction.
`-(3x^3+2x^2-13x-4)/-(x-3)` `=(3x^3+2x^2-13x-4)/(x-3)`
Proceed with Long Division.

..........................`(+3x^2+11x+20)`
`color(white)((x-3)/color(black)(x-3)(3x^3+2x^2-x^2-7x-7)/color(black)(")"bar(3x^3+2x^2-13x-4)))`
.............`-(3x^3-9x^2)`
..................------------------------------------
.................`(0x^3+11x^2-13x)`
..........................`-(11x^2-33x)`
..................------------------------------------
..........................`(-0X^2+20x-4)`
...........................................`-(20x-60)`
.................--------------------------------------
................................................`(0x+56)`

Note: Ignore the full stops, I'm new at this :/

You may want to copy it on paper to make it clearer.

So, to simplify the fraction from the above:

`-(3x^3+2x^2-13x-4)/-(x-3)` = `3x^2+11x+20+(56/(x-3))`

Answer 2

This really is long division. It just looks different!

`3x^2+11x+20+56/(x-3)`

Explanation:

For convenience I have written `3-x` as `-x+3`

`" "-3x^3-2x^2+13x+4`
`color(magenta)(3x^2)(-x+3)->ul( -3x^3+9x^2) larr" Subtract"`
`" "0-11x^2+13x+4`
`color(magenta)(11x)(-x+3)->" "ul(-11x^2+33x )larr" Subtract"`
`" "0" " -20x+4`
`color(magenta)(20)(-x+3)->" "ul(-20x+60) larr" Subtract"`
`" "color(magenta)(0-56 larr "Remainder")`

`(-3x^3-2x^2+13x+4)/(3-x) = color(magenta)(3x^2+11x+20-56/(3-x))`

To match Ching's answer consider `-56/(3-x)`

Not that `3-x` is the same as `-(+x-3)`

So we have `" "-(56/(-(x-3))) ->+56/(x-3)`

Giving: `" "3x^2+11x+20+56/(x-3)`