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?
Best Answer
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*}