How to factorise a cubic?

Question

What are the steps in factorising a cubic?

For example $x^{3} - 2x^{2} - 7x - 4$

I know that $-1$ is a root so it will start off as $(x+1)$(...something?)

Answer 1

Since you already know that $x+1$ is a factor, try to extract it i.e

$$x^{3} - 2x^{2} - 7x - 4 = (x^3+x^2) -(3x^2+3x) -(4x+4) $$ $$(x^3+x^2) -(3x^2+3x) -(4x+4) = (x+1)(x^2-3x-4)=(x+1)(x+1)(x-4)$$

Hence you have $$x^{3} - 2x^{2} - 7x - 4= (x+1)(x+1)(x-4) $$

Answer 2

Since you know $-1$ is a root you can use Ruffini's rule:

Then $x^3-2x^2-7x-4=(x+1)(x^2-3x-4)$

And you can easily see that $-1$ and $4$ are roots of $(x^2-3x-4)$

Then $x^3-2x^2-7x-4=(x+1)(x^2-3x-4)=(x+1)(x+1)(x-4)$

Note: Before using Ruffini's rule you can also see that $4$ is a root using Rational root theorem

Answer 3

Yes, then $\, f = (x\!+\!1)(x^2\!+bx-4)$ $\,\overset{\large x\,=\,1}\Longrightarrow\,-12 = f(1) = 2(b-3)\,\Rightarrow\, b = -3$