Sydney Opera House-SOSO curves with a million fitted tiles! Photo by Arvin Wiyono on Unsplash

Polynomial Roots — Applying Curved Segment transplants from ‘SOSO’ Nodes

Using the ‘architecture’ to find roots from confected ‘SOSO’ nodes with transplanted segments having the same initial gradients as the recipient root segments.

Since I presented Cubic Polynomials — A Simpler Approach to find the last two roots of any polynomial given the other roots, my focus has been on improving node coordinates for application of Newton’s Approximation in finding first roots.

While exact lengthy numerical methods and calculators are of course available, I prefer to use ‘visual architecture’ as it aids learning, while also finding good approximate solutions. My objective in this post is twofold:

• to simplify and improve initial node coordinates by confecting twin Same Opposite Same Opposite, or ‘SOSO’ functions with intercepts, and;
• to replace Newton’s straight line hypotenuse approximations using Segment Transplants Incorporataing Node Gradients, or ‘STING’. It is a transplanted curved segment with the same node gradient as a root segment.

This post assumes knowledge of algebra and introductory calculus (differentiating polynomials) at the high school level.

The ideas

SOSO is a confected twin function with Same Opposite Same Opposite +/- coefficients.

It creates strategic node point intercepts for the application of STING transplants.

Consider the following function x³+4x²-5x-15, shown in blue in the graph below, and its twin x³-4x²-5x+15, derived from SOSO shown in red.

SOSO showing how it’s done

SOSO twin functions showing intercepts and roots

The two functions are 180-degree rotationally symmetrical about the x-y axis origin (0,0). Root C rotates to root J on the twin function. The two triangular shapes, JBM and CKN, rotate in concert and it can be seen that these triangles ‘house’ the roots B and C of one function, and J and K of its twin. A remarkable feature of SOSO is that the…