Suppose I have a polynomial: $x^3-8x^2+17x-4$. How do I know it will always be $(x-4)(x^2-4x+1)$ by solving it?
I'm struggling to figure out what to look for in the polynomial to give me a hint or clue of where to start to get the answer of $(x-4)(x^2-4x+1)$
If this cubic factors over the rationals, by Gauss's lemma it factors over the integers, and it must factor as (linear)(quadratic), i.e. as $(x - r)(x^2 + a x + b)$ where $r,a,b$ are integers, and $r$ is a root of the polynomial. Since $rb = 4$, there aren't too many possibilities to try for $r$ ...