<s>[INST] How to factorize $x^3 - 7x + 6$? 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?
 [/INST] 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*}
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