Question

How do you find f(3) given `f(x)=x^3-7x^2+5x-6` using synthetic division?

Answer 1

The remainder is `-27` and the quotient is `=x^2-4x-7`

Explanation:

Let's perform the synthetic division

`color(white)(aaaa)` `3` `|` `color(white)(aaaa)` `1` `color(white)(aaaa)` `-7` `color(white)(aaaaaa)` `5` `color(white)(aaaaaa)` `-6`

`color(white)(aaaaa)` ###|#`color(white)(aaaa)` ###color(white)(aaaaaa)#`3` `color(white)(aaaaa)` `-12` `color(white)(aaaa)` `-21`

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

`color(white)(aaaaa)` ###|#`color(white)(aaaa)` `1` `color(white)(aaaa)` `-4` `color(white)(aaaaaa)` `-7` `color(white)(aaaaaa)` `color(red)(-27)`

The remainder is `-27` and the quotient is `=x^2-4x-7`

Therefore,

`(x^3-7x^2+5x-6)/(x-3)=(x^2-4x-7)-27/(x-3)`