Question

How do you use synthetic division to divide `X^3 + 2x^2 - 5x - 6 div (x + 3)`?

Answer 1

The remainder is `0` and the quotient is `=x^2-x-2`

Explanation:

Let's perform the synthetic division

`color(white)(aaaa)` `-3` `|` `color(white)(aaaa)` `1` `color(white)(aaaaa)` `2` `color(white)(aaaaaa)` `-5` `color(white)(aaaaa)` `-6`

`color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `color(white)(aaaa)` `-3` `color(white)(aaaaaaa)` `3` `color(white)(aaaaaaa)` `6`

`color(white)(aaaaaaaaa)` #`_________________________________________________________`#

`color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `1` `color(white)(aaa)` `-1` `color(white)(aaaaaa)` `-2` `color(white)(aaaaaa)` `color(red)(0)`

The remainder is `0` and the quotient is `=x^2-x-2`

Therefore,

`(x^3+2x^2-5x-6)/(x+3)=(x^2-x-2)`

Also,

If `f(x)=x^3+2x^2-5x-6`

`f(-3)=(-3)^3+2(-3)^2-5(-3)-6`

`=-27+18+15-6`

`=0`