How to factorize $x^3 - 7x + 6$?

Question

How do you factorize this polynomial:

$${x^3 - 7x + 6}$$

Some online solver doesn't even work saying: using GCF method doesn't work, but sites like Mathway.com gave me the answer, is there a pre-step you need to do before factorizing?

the answer is $(x-1)(x-2)(x+3)$.

This is actually part (b) of a question, it said use the answer for part (a) i.e $x^3-8$ and factorize. I don't get the relationship, what does this hint actually shows?

Answer 1

The method below will not always work, but it is another way to arrive at the same answer for this cubic.

This is difficult because it is a cubic, but if you consider the quadratic $x^2 - 7x + 6$, that factors nicely as $(x-1)(x-6)$. So we rewrite our given cubic as $$x^3 - 7x + 6 = (x^3 - x^2) + (x^2 - 7x + 6).$$ Factorising each bracket, we have \begin{align*} x^3 - 7x + 6 &= (x^3 - x^2) + (x^2 - 7x + 6)\\ &= x^2(x - 1) + (x - 1)(x - 6)\\ &= (x - 1)(x^2 + x - 6)\\ &= (x - 1)(x - 2)(x + 3). \end{align*}

Answer 2

As the degree is three, one of the factors must be linear. A linear factor with rational coefficients means there is a rational root and by the rational root theorem, it must be a divisor of $6$ ($\pm1,\pm2,\pm3,\pm6$). Can you see that one of these is indeed a root?