Polynomials in the Complex Plane

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When you plot a polynomial


in the complex plane with ComplexPlot, its roots appear as black dots. The color function goes from to counterclockwise around each of the zeros, passing through the continuous sequence that might be described as red, orange, yellow, green, cyan, blue, magenta and back to red. For a double root, the color sequence wraps around the zero twice, and analogously for higher orders. For a polynomial of degree , a peripheral curve enclosing all the roots goes through color cycles. For simplicity, the coefficients are restricted to integers between and 3.

You can choose to see the values of the numerically computed roots.


Contributed by: S. M. Blinder (January 2020)
Open content licensed under CC BY-NC-SA



[1] D. J. Velleman, "The Fundamental Theorem of Algebra: A Visual Approach," The Mathematical Intelligencer, 37(4), 2015 pp. 12–21. doi:10.1007/s00283-015-9572-7.

[2] S. Wolfram. "Phase of a Complex Polynomial" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/PhaseOfAComplexPolynomial.


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