A tool-kit for finding Quintic Polynomial roots starting with ‘OSOS’ Quadratics — This post presents viable Quintic roots solutions, using a confected Y-Axis 180 deg rotation mirror image ‘OSOS’ twin function, with 2 opposing Quadratics to replicate 2 roots then a Cubic concoction for 1 and the last 2 gifted. For any 5-root Quintic polynomial, it is only required to find 3…

Solving Quadratics using gradients at the roots — An important characteristic of the Quadratic Polynomial y=ax²+bx+c is that it retains its parabolic ax² shape and size wherever it is located in the x-y grid. The other terms, bx +c are just the, “Sit where we say”, ushers! This architectural symmetry and stability means that the gradients at the…

Exploiting Polynomial ‘Perfect Cube’ architecture to build your own function — without the usual approach of specifying all 3 roots — This post explores the ‘components of perfect and imperfect cubics’ and shows how you can use them to build a cubic that meets particular characteristics. While math books generally provide instruction on building polynomials up from their roots, another way to build them is by modifying a ‘perfect’ polynomial and…

An intuitive way to find the 2nd and 3rd roots — While we are all familiar with finding the roots of a quadratic using the Quadratic Equation, it can be complex to find the roots of a higher order polynomial. We typically need to find one factor by brute force, then divide through to create a quadratic. I’ve been working on…

Approximate Cubic Roots Using Generic Quadratic Relationships — This post presents a novel method of calculating approximate roots of Cubic polynomials y=x³+Cx+D by exploiting underlying Quadratic relationships. It presents a very simple one-liner for a root while more importantly for students it graphically relates root dependency on function shape and the constant D.

Another Shot at Cubic Roots By Upgrading to Quartic — This Time Using Quadratics — In a previous post, Finding Cubic Roots Using Quartic Hybrids, I showed how to find Cubic roots by increasing the order to Quartic and then utilizing the amazing architectural symmetries of Quartics in what I call hybrid Big W genetics. In this post, I adopt an even deeper genetic posture…

Build your own Quartic Polynomials to order at any address in the Grid! — Students can be tasked with finding the roots of polynomials without requiring a detailed understanding of the underlying functional architecture or the position of key metrics in the X-Y Grid. …

Increasing the order of Cubic polynomials to Quartic to find Roots to very close approximations — It might seem counterproductive to increase the order of a polynomial to find the roots but this post does it using the architecture of Quartics, in particular the symmetry of what I call hybrid Big W genetics. While upgrading avoids using a Cubic equation altogether and gets it right the…

Designing Quartics with specified Turning Points Using The Genetic Architecture ~ Without Cubic Equations — While polynomials are frequently derived from the roots up and the focus of math education seems to be to find them again, this doesn’t address the significance of Tps in some robotics and AI applications. …