Spivak, Ch, 25, 2(v): Is there some specific technique to factorize $x^3-x^2-x-2$ or must one guess that 2 is root?

Question

The following problem appears in Ch. 25, "Complex Numbers" of Spivak's Calculus

2 (v) Solve the equation $x^3-x^2-x-2=0$.

Is there some specific technique to factorize this? Must one identify simply by guessing that $2$ is a root?

Once we know that $2$ is a root then finding the other two roots becomes a simple question of solving a quadratic for which there are formulas.

I know there are complicated formulas for the solution of a cubic equation, but it's not clear to me what standard procedure is for solving such equations. Does one look up the complicated formulas, or is there some technique for factorization?

Answer 1

Using the rational root theorem on the above cubic, you only need to check whether $\pm 1,\pm2$ are roots. Otherwise there's no rational root and the problem is a little harder.

In those cases you'd usually have to use the cubic formula or some simplified version thereof. But thankfully they don't usually come up in exams. This Mathologer video on the cubic equation may dispel some fears that it is "super complicated".