Factoring Cubic Equations I’ve been trying to figure out how to factor cubic equations by studying a few worksheets online such as the one here and was wondering is there any generalized way of factoring these types of equations or do we just need to remember a bunch of different cases. 
For example, how would you factor the following: 

$x^3 – 7x^2 +7x + 15$

I'm have trouble factoring equations such as these without obvious factors. 

The general method involves three important rules for polynomials with integer coefficients:


*

*If $a+\sqrt b$ is a root, so is $a-\sqrt b$.

*If $a+ib$ is a root, so is $a-ib$.

*If a root is of the form $p/q$, then p is an integer factor of the constant term, and q is an integer factor of the leading coefficient.


Since this is cubic, there is at least one real rational solution (using rules 1 & 2). Let's find it using rule 3: 
$$q = 1,\ p \in \{\pm1,\pm3,\pm5,\pm15\}$$
Brute force to find that $-1$ is a solution. 
So we can now write the polynomial as:
$$(x+1)(x-a)(x-b)=x^3 - 7x^2 + 7x +15$$
Now use:
$$a+b=8$$
$$ab-a-b=7$$
Solve to get:
$$a(8-a)-a-(8-a)=7 \to a^2-8a+15 = 0$$
And get the two remaining solutions.
