How does one find complex roots of a polynomial? Can you please keep the explanation simple, because I am still a high school student? Please try to keep the answer at the level of a high school pre-calculus student. Also, can you please explain why your method works; don't just give me a method to find imaginary/complex roots of a polynomial. Thank you!


closed as too broad by José Carlos Santos, Yves Daoust, Somos, Jyrki Lahtonen, Xander Henderson Jul 27 '18 at 12:54

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  • $\begingroup$ (1 of 3) If you are asking for an explanation which only uses high school math, you almost certainly aren't going to find one. The very fact that a polynomial of degree $n$ has $n$ roots (counting multiplicity) requires a proof that is pretty far beyond the scope of high school math (my favorite proof of the Fundamental Theorem of Algebra is essentially a corollary of Liouville's Theorem, a theorem in complex analysis; you need to know a fair bit about complex differentiation before you get there). $\endgroup$ – Xander Henderson Jul 27 '18 at 12:52
  • $\begingroup$ (2 of 3) The next hurdle is showing that for most polynomials, you have literally no hope of finding the roots analytically. This follows from Galois theory, which is usually taught as part of a course in abstract algebra, either to senior undergraduate math majors, or to junior graduate students. The basic result says that if you have a polynomial of degree 5 or greater, then there is no general formula for writing down the roots in terms of radicals (i.e. there is no analog of the quadratic formula once you get beyond degree 4). $\endgroup$ – Xander Henderson Jul 27 '18 at 12:53
  • $\begingroup$ (3 of 3) This means that the only thing that you are going to be able to do most of the time is approximate roots numerically. In terms of numerical approximation, there are a lot of techniques. The usual introductory technique is probably Newton's method, which relies on results from calculus. One could probably explain to you how Newton's method works, but it would require an introduction to derivatives, at the very least. Again, this is beyond the scope of most high school precalculus classes. I don't see how anyone can reasonably answer you question, and am therefore voting to close it. $\endgroup$ – Xander Henderson Jul 27 '18 at 12:54

It’s a very complicated question actually. There are formulas to find roots for polynomials of degree 3&4, (in this case it’s very similar to polynomials of degree 2), but no formulas for polynomials of degree higher than 5. Yet maybe someone who is much better at mathematics can explain it in a simple way!

  • $\begingroup$ Well can you give me some methods to do it then? Are there any methods that exist for a polynomial higher than degree 5 that exist to do so? They need not be formulas. And can they be explained at the level of a high school pre-calc student? $\endgroup$ – Ethan Chan Jul 17 '18 at 12:42
  • $\begingroup$ @Moo Thanks for the comment! $\endgroup$ – Charlie Chang Jul 17 '18 at 13:02

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