Question

How do you divide `( x^5 - x^3 + 5x^2 - 10x - 75)/(x - 2 )`?

Answer 1

The remainder is `color(red)(-51)` and the quotient is `=x^4+2x^3+3x^2+11x+12`

Explanation:

Let's perform a synthetic division

`color(white)(aaaa)` `2` `color(white)(aaaaaa)` `|` `color(white)(aa)` `1` `color(white)(aaaaaa)` `0` `color(white)(aaaa)` `-1` `color(white)(aaaa)` `5` `color(white)(aaa)` `-10` `color(white)(aaa)` `-75`
`color(white)(aaaaaaaaaaaa)`________________

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `color(white)(aaaaa)` `2` `color(white)(aaaaa)` `4` `color(white)(aaaaa)` `6` `color(white)(aaaa)` `22` `color(white)(aaaaa)` `24`
`color(white)(aaaaaaaaaaaa)`______________

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaa)` `1` `color(white)(aaaaa)` `2` `color(white)(aaaaa)` `3` `color(white)(aaaaa)` `11` `color(white)(aaa)` `12` `color(white)(aaaa)` `color(red)(-41)`

The remainder is `color(red)(-51)` and the quotient is `=x^4+2x^3+3x^2+11x+12`

Therefore,

`(x^5-x^3+5x^2-10x-75)/(x-2)`

`=x^4+2x^3+3x^2+11x+12-11/(x-2)`