Question

How do you factor `x^3-3x^2+6x-18`?

Answer 1

`x^3-3x^2+6x-18 = (x^2+6)(x-3)`

`color(white)(x^3-3x^2+6x-18) = (x-sqrt(6)i)(x+sqrt(6)i)(x-3)`

Explanation:

Notice that the ratio between the first and second terms is the same as that between the third and fourth terms, so this will factor by grouping:

`x^3-3x^2+6x-18 = (x^3-3x^2)+(6x-18)`

`color(white)(x^3-3x^2+6x-18) = x^2(x-3)+6(x-3)`

`color(white)(x^3-3x^2+6x-18) = (x^2+6)(x-3)`

That's as far as you can go using Real coefficients since `x^2+6 >= 6 > 0` for any Real value of `x`. If you allow Complex coefficients then this factors further as:

`color(white)(x^3-3x^2+6x-18) = (x^2-(sqrt(6)i)^2)(x-3)`

`color(white)(x^3-3x^2+6x-18) = (x-sqrt(6)i)(x+sqrt(6)i)(x-3)`