How to Find Polynomial Roots Step-By-Step
Your Daily Dose of Math, Science, and Python
It’s a well-known fact that all polynomials
can be factorized as
where the 𝑥_i are the roots (zeros) of the polynomial. The roots can be real or complex and don’t all have to be different. This is called the fundamental theorem of algebra, but “fundamental” does not mean “easy”. In fact, it required one of the greatest mathematicians of all time, Carl Friedrich Gauß, to prove it. Incidentally, he proved it in 1799 in his Ph.D. thesis. Well, if that isn’t an example of summa cum laude, then what? While it is good to know that such a factorization is possible, it is all but clear how to obtain it. This is a hard problem indeed, and for quintic (𝑛=5n=5) or higher polynomials, no exact solution formula exists. However, there are a few tricks that allow us to obtain exact solutions for many cases, after all. And this is what this article is about. First I will show you the tricks, then we will see step-by-step examples implemented in Python, that you can play around with, yourself.
