Question

How do you divide `(-x^5-4x^3+x-12)/(x^2-x+3)`?

Answer 1

The remainder is `=(8x-15)` and the quotient is `=(-x^3-x^2-2x+1)`

Explanation:

Perform a long division

`color(white)(aa)` `-x^5+0x^4-4x^3+0x^2+x-12` `color(white)(aa)` `|` `x^2-x+3`

`color(white)(aa)` `-x^5+x^4-3x^3` `color(white)(aaaaaaaaaaaaaaaa)` `|` `-x^3-x^2-2x+1`

`color(white)(aaaa)` `0-1x^4-1x^3+0x^2`

`color(white)(aaaaaa)` `-1x^4+1x^3-3x^2`

`color(white)(aaaaaaaaa)` `0-2x^3+3x^2+x`

`color(white)(aaaaaaaaaaa)` `-2x^3+2x^2-6x`

`color(white)(aaaaaaaaaaaaaa)` `0+x^2+7x-12`

`color(white)(aaaaaaaaaaaaaaaa)` `+x^2-1x+3`

`color(white)(aaaaaaaaaaaaaaaaa)` `-0+8x-15`

Therefore,

`(-x^5+0x^4-4x^3+0x^2+x-12)/(x^2-x+3)`

`=(-x^3-x^2-2x+1)+(8x-15)/(x^2-x+3)`

The remainder is `=(8x-15)` and the quotient is `=(-x^3-x^2-2x+1)`