Question

Given polynomial `f(x)=x^3+2x^2-51x+108` and a factor `x+9` how do you find all other factors?

Answer 1

The answer is `=(x+9)(x-4)(x-3)`

Explanation:

`f(x)=x^3+2x^2-51x+108`

`(x+9)` is a factor

We do a long division

`color(white)(aaaa)` `x^3+2x^2-51x+108` `color(white)(aaaa)` `∣` `x+9`

`color(white)(aaaa)` `x^3+9x^2` `color(white)(aaaaaaaaaaaaaaa)` `∣` `x^2-7x+12`

`color(white)(aaaa)` `0-7x^2-51x`

`color(white)(aaaaaa)` `-7x^2-63x`

`color(white)(aaaaaaaa)` `-0+12x+108`

`color(white)(aaaaaaaaaaaa)` `+12x+108`

`color(white)(aaaaaaaaaaaaaa)` `+0+0`

Therefore,

`(x^3+2x^2-51x+108)/(x+9)=x^2-7x+12`

We can factorise the quotient

`x^2-7x+12=(x-4)(x-3)`