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I'm trying to deepen my understanding of the Fundamental Theorem of Algebra, which asserts that every non-constant single-variable polynomial with complex coefficients has as many roots as its degree, counting multiplicities. Specifically, for a polynomial equation of degree n, there should be exactly n roots in the complex number system.

For example, consider the function $y=x^3+5x^2+x+3$. When plotting this function on a 2D y-x plane, it appears there is only one root, which contradicts the theorem's assertion that there should be three roots for a cubic polynomial.

Question 1: How do we find the other two imaginary roots for this equation? I understand that complex roots come in conjugate pairs when the polynomial has real coefficients, but I'm unsure how to mathematically derive these roots.

Question 2: Is there a way to visualize this function and its roots in a 3D environment, allowing us to see the imaginary roots? I'm interested in understanding how these complex roots are considered roots of the function, as they must cause the function's output to be zero. Any suggestions on tools or techniques for plotting this in 3D would be greatly appreciated.

I've tried plotting the function in 2D and understand the basics of finding real roots, but the concept of visualizing and finding complex roots is where I'm seeking more help.

J. W. Tanner
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user97662
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  • Q1 doesn't have an answer for higher-degree polynomials. The FTA doesn't give you a method to find these roots; it only tells you they exist. In general, there is no formula for the roots of polynomials of degree 5 and higher. (There is a cubic equation that can be applied to your particular example, however.) – Sambo Feb 26 '24 at 21:26
  • That there is only one real root does not contradict the theorem that there are three roots in the complex numbers – J. W. Tanner Feb 26 '24 at 21:38
  • See https://math.stackexchange.com/a/1101169/589 and https://math.stackexchange.com/a/3236061/589 – lhf Feb 26 '24 at 22:17

1 Answers1

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While FTOA tells us that every polynomial has a certain number of roots, it doesn't give us a method for finding them. In general, while the roots of any linear or quadratic polynomial can be found through basic algebra, finding the roots of a cubic or quartic is quite difficult and for quintics and higher there is no guarantee that the roots can be exactly calculated.

Beyond that, many of the methods used for working with real roots can be extended to complex numbers, for example:

  1. Vieta's formulas still apply.

  2. A version of the rational root theorem can be written in terms of Gaussian rationals.

  3. If you can find some of the roots exactly, you can perform polynomial long division to reduce the degree of the polynomial.

  4. You can use a method like Newton-Raphson to find approximate values of the roots.

As for visualisation, it can be difficult to efficiently display four dimensions of information (two each for the real and imaginary parts of the inputs and outputs). As linked in the comments, domain colouring is one such method. A few other methods are discussed in this video, including vector fields and Riemann spheres.

ConMan
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