Question

Given polynomial `f(x)=x^3-2x^2-40x-64` and a factor `x-8` how do you find all other factors?

Answer 1

The other factors are `(x+2)` and `(x+4)`

Explanation:

We perform a synthetic division

`color(white)(aaaa)` `8` `color(white)(aaaa)` `|` `color(white)(aaaa)` `1` `color(white)(aaaa)` `-2` `color(white)(aaaaa)` `-40` `color(white)(aaaaaa)` `-64`

`color(white)(aaaaaa)` `color(white)(aaaa)` `|` `color(white)(aaa)` `color(white)(aaaaaaa)` `8` `color(white)(aaaaaa)` `48` `color(white)(aaaaaaaa)` `64`

`color(white)(aaaaaaaaaa)`------------------------------------------------------------

`color(white)(aaaa)` `color(white)(aaaaaa)` `color(white)(aaaaaa)` `1` `color(white)(aaaaa)` `6` `color(white)(aaaaaaa)` `8` `color(white)(aaaaaaaa)` `color(red)(0)`

The remainder is `=0`

The quotient is `=x^2+6x+8`

This is factorised as

`x^2+6x+8=(x+2)(x+4)`