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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?

xiao
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    There are a variety of methods to find roots. You can use Rational Root Theorem to find what rational roots are possible for the cubic equation and check if each of the potential rational roots work. If your equation happens to be in the form $x^3-px+q$ you can use Cardano's Formula to find roots. If all else fails you can use the (rather unwieldy) General cubic formula. – Kyan Cheung Jan 24 '21 at 03:35
  • By Factor Theorem, if $r$ is a root of a polynomial then $x-r$ divides said polynomial. Given this you can now just divide your polynomial by $x-r$ and get $(x-r)P(x)$ where $P$ is a polynomial. You can use long polynomial division or the more efficient Synthetic division to divide your polynomial. – Kyan Cheung Jan 24 '21 at 03:39
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    For quartic equations, it becomes substantially harder to find roots. The general formula can be seen here. By the Abel-Ruffini Theorem it is in general impossible to find roots of quintic or higher order equations, hence your best bet in those cases would be to use numerical solutions like Newton's method to estimate roots. (You can still use the Rational Root Theorem to see if any rational roots exist). – Kyan Cheung Jan 24 '21 at 03:47
  • If the roots are pure imaginary, eg $\pm ai$ then you simply have a factor $(x^2+a^2)$. If they're complex, then they must be conjugates, assuming the coefficients of the polynomial are all real. – PM 2Ring Jan 24 '21 at 03:50

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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.

Bertrand Einstein IV
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