Question

How do you divide `(-x^4-4x^3-9x^2-7x-7)/(x-2)`?

Answer 1

`-x^3-6x^2-21x-49` with a remainder of `-105`

Explanation:

We can do long division on this:

`color(white)((x-2)/color(black)(x-2)(-x^4-4x^3-9x^2-7x-7)/color(black)(")"bar(-x^4-4x^3-9x^2-7x-7)))`

We multiply `x` by `-x^3` to get to `-x^4`, and so we have:

`color(white)((x-2)/color(black)(x-2)(-x^4color(black)(color(white)(00)-x^3)-9x^2-7x-7)/color(black)(")"bar(-x^4-4x^3-9x^2-7x-7)))`
`color(white)((x-2)/(x-2)((color(black)(-x^4+2x^3)-9x^2-7x-7)/(color(black)bar(0x^4-6x^3-9x^2)-7x-7)))`

We multiply `x` by `-6x^2` to get to `-6x^3` and so we have:

`color(white)((x-2)/color(black)(x-2)(-x^4color(black)(color(white)(00)-x^3-6x^2)-7x-7)/color(black)(")"bar(-x^4-4x^3-9x^2-7x-7)))`
`color(white)((x-2)/(x-2)((color(black)(-x^4+2x^3)-9x^2-7x-7)/(color(black)bar(0x^4-6x^3-9x^2)-7x-7)))`
`color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4)-6x^3+12x^2)-7x-7)/(color(black)bar(color(white)(0x^4-)0x^3-21x^2-7x)-7)))`

We multiply `x` by `-21x` to get to `-21x^2` and so we have:

`color(white)((x-2)/color(black)(x-2)(-x^4color(black)(color(white)(00)-x^3-6x^2-21x)-7)/color(black)(")"bar(-x^4-4x^3-9x^2-7x-7)))`
`color(white)((x-2)/(x-2)((color(black)(-x^4+2x^3)-9x^2-7x-7)/(color(black)bar(0x^4-6x^3-9x^2)-7x-7)))`
`color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4)-6x^3+12x^2)-7x-7)/(color(black)bar(color(white)(0x^4-)0x^3-21x^2-7x)-7)))`
`color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4-2x^3)-21x^2+42x)-7)/(color(black)bar(color(white)(0x^4-0x^3)-0x^2-49x-7))))`

We multiply `x` by `-49` to get to `-49x` and so we have:

`color(white)((x-2)/color(black)(x-2)(-x^4color(black)(color(white)(00)-x^3-6x^2-21x-49))/color(black)(")"bar(-x^4-4x^3-9x^2-7x-7)))`
`color(white)((x-2)/(x-2)((color(black)(-x^4+2x^3)-9x^2-7x-7)/(color(black)bar(0x^4-6x^3-9x^2)-7x-7)))`
`color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4)-6x^3+12x^2)-7x-7)/(color(black)bar(color(white)(0x^4-)0x^3-21x^2-7x)-7)))`
`color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4-2x^3)-21x^2+42x)-7)/(color(black)bar(color(white)(0x^4-0x^3)-0x^2-49x-7))))`
`color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4-2x^3-1x^2)-49x+98))/(color(black)bar(color(white)(0x^4-0x^3-0x^2)-0x-105))))`

And so `(-x^4-4^3-9x^2-7x-7)/(x-2)=-x^3-6x^2-21x-49` with a remainder of `-105`