How do I factor polynomials with power 3 and above? All of the videos I've found do things like "let's try -1. oh, it works! okay, now let's divide it out and find the next root".  Is there really no other strategy other than guess and check?  A couple of times I've come across these high power polynomials in contests and I just have no clue what to do with them.  I'm asking this question specifically now because there's a question in my textbook that asks me to find the two imaginary roots of a polynomial of 4th power.  Clearly guess and check is not going to work as well with imaginary numbers, and even with real ones it seems like a pain.  What strategies do you guys use to factor?
 A: There is in fact a Cubic Formula, which can give you the roots, and make factoring trivial...although it is a little more complicated than the famed Quadratic Formula; see here https://math.vanderbilt.edu/schectex/courses/cubic/. And there is also a Quartic Formula, but that is much more complicated https://en.wikipedia.org/wiki/Quartic_function#Solving_a_quartic_equation . And in fact there is $\textbf{not}$ a general equation for 5th and higher order polynomials; a famous theorem known as Abel–Ruffini theorem.
But that is besides the point.
For factoring, particularly in contests; there is normally a trick to see, which is what makes the problems so rich. There are often various symmetries that can be used, or a quantity you can add and subtract which allows you to complete squares, polynomial division, etc.
If we are told roots are rational, then there is a lot that can be accomplished by guess and check, which is often what is taught in high school. However in the case where you have a Quartic Polynomial, I would assume the first step would be to split it into a product of two Quadratic Polynomials and go from there, since if there are 2 imaginary roots, they are probably conjugates of one another which can be expressed in one quadratic polynomial.
There is no general strategy of factoring, just various techniques.
