Roots of Polynomials
Rene Descartes determined that if there existed rational roots for a polynomial with integer coefficients, then these roots would be related to the leading coefiicient and constant term in the manner stated in the Rational Roots Theorem. When I was in high school in the early 80s, we used this theorem to determine all possible rational roots and then used synthetic division to determine which ones actually were roots. From there, you can decompose the polynomial into factors and try to determine any complex roots.
I still teach the Rational Roots Theorem, but make allowances for today’s technology. Instead of asking the students to determine all possible rational roots and test them out, we graph the polynomial and use the calculator to determine the rational roots. Then, we use these roots to break the polynomial down into prime factors, which can allow the students to determine complex roots that did not appear on the graph (I sometimes refer to the complex roots as Sir Not Appearing in This Film).
What if the roots aren’t rational? and other complications
The polynomials I use for exercises, quizzes and tests are set up so that each breaks down to a series of linear factors and one quadratic factor with complex roots, with all the factors having integer coefficients.
But – what if this isn’t the case, or what if there are irrational real roots? How can we find the complex roots of a polynomial in these situations?
Answer: Newton’s Method. Newton’s Method is fairly easy to understand in application to finding real roots on the Cartesian Plane, but it also works just as well to find complex roots. Choosing various real and complex seed values for Newton’s Method leads to finding both the real and the complex roots.
Something very interesting about this process is that if we color the seed values from the complex plane based on which root they eventually lead to, the colored depiction of the complex plane that is produced appears fractal in nature. Each collection of seed values that leads to the same root is said to lie in a particular Newton Basin.
Here is a great website on Newton Basins.
And another one here.
Here is a look at how they approach this topic at MIT.
(Answer: with technology – Mathematica)