I'm familiar with basic forms of polynomial factoring like completing the square or factoring (e.g. finding that $x^2+x-6=(x+3)(x-2)$, but I'm currently working on integration of rational functions by partial fractions and I find myself encountering the factorizations of polynomials unlike those I've worked with before.
Given: $x^3 - x^2 - x + 1$
When I first look at this, I'm unsure of where to start. It's very unlike the example above. It's degree $3$, for one thing, and it has $4$ terms.
When I look ahead to the solution, the book says $f(1) = 0$, so we know that $(x-1)$ is a factor. Okay, now that they mention that, I can see it fits, but what?! How did they come up with that? Is that a general rule? Does anyone have a link to where I can learn about it?
Further, once it's known that $(x-1)$ is a factor, how do I then factor an $(x-1)$ out of a longer polynomial like this?
In other words, how does $x^3 - x^2 - x + 1 = (x-1)(x^2-1)$?
Where can I learn more about this type of factorization?