Graphical solution of algebraic equations; d'pres Lill

Abstract Lill's graphical method with a pseudo-billiard to identify the real roots of real algebraic equations provides a fundamental basis for the mathematical theory of paper folding, as beautifully demonstrated in a YouTube video of Polster; https://www.youtube.com/watch?v=IUC-8P0zXe8} uploaded in 2019. In an article, "Polynomials as Polygons" in Math Intelligencer 2017, Tabachinikov rivisits an anonymous paper subsequent toLill's original paper, and explores a graphical approach for identifying the imaginary roots. Guided by the above articles and Horner's method, we naturally consider a polynomial with complex coefficients as a polygonal line in the Argand--Gauss plane and a root of the equation as a string of similar triangles ``reflecting" on the segments of the line, thereby unifying the seemingly disparate geometrical approaches to real and imaginary roots. We also examine the factorization of polynomials and the multiplicity of roots from a graphical viewpoint. For instance, a repeated root of a real cubic equation is a tri-tangent line to a certain three-parabola. This refines the well known classic result of Beloch. In the final section, we discuss a method for identifying real quadratic factors of real polynomicals.