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Rotating cubic roots into managed quadratic segments where the math is easier!

This post presents a novel method for approximating 1st roots of cubic polynomials that avoids the lengthy gradient and height calculation iterations associated with Newton’s and some of my own previously posted methods.

It particularly addresses the task of approximating cubic polynomial roots in ‘difficult to get to’ locations - namely where root gradients are low and close to turning points. Such ‘architecture’ would normally require two or more iterations to solve.

This method exploits the amazing symmetry of cubic polynomials and their ‘component architecture’ by ‘rotating’ roots into segments where the underlying quadratic architecture approximates the cubic curvature.

This means we can use the simpler quadratic math to calculate a root!

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 a simpler approach that I’m sharing in this post.

My goal is to make the process a lot simpler by eliminating polynomial long division.

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

I hope it helps you to think in a fresh way about cubic polynomials, and how they can be simpler than first meets the eye! …

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Photo by Susan Kuriakose on Unsplash

A very simple Cubic polynomial root approximation using architectural symmetry and similar triangles

The architectural symmetry of Cubic polynomials supports many root approximation methods, helping intuitive understanding and with a little work, can be transposed to higher orders to minimise more complex math. This post expands on an earlier post, Cubic Polynomials — Using Similar Triangles to Approximate Roots, which promoted the use of Similar Triangles derived from Turning and Inflection Points.

Both methods are very simple and it is difficult to quantify pros and cons in a short post. The latter, being very easily formularised and not requiring calculus beyond finding the Inflection Point Ip is the quicker and simpler of the 2, is best suited when Ip(x)=B/3A<1 and with high x coefficients leading to relatively steeper architecture between turning points. …

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Photo by Martin Sanchez on Unsplash

Applying Similar Triangles to approximate roots aided by ‘natural’ Cubic architecture

While similar triangles can be used with polynomials of any order, this introductory post is presented in cubics to exploit the compliant architecture while highlighting the practicality and simplicity of similar triangle math.

The method has particular advantages in higher order polynomials when supported by various changeable architectures, as it can alleviate onerous iterations of gradient and height calculations. This will be the subject of another post.

This post assumes math at high school level.

Similar Triangles

Similar triangles strike a chord across the x axis between points or nodes being line intercepts on the function.

The closer the nodes are together, the more accurate the cord intercept will approximate the root as shown in Diagram 1 below, where similar triangles DEF and GEH result in: approximate root G=E-Δx=E-(EH)*DF/EF) which compares with Root A, shown in red. …

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Melbourne Shot Tower Photo by Alex Knight on Unsplash

Using twin function proximal nodes for application of Newton’s Approximation to find roots

While Newton’s approximation is universal, its accuracy largely depends on having application nodes proximal to roots to eliminate the need for iterations. It is also impacted by the steepness of the gradient at the root and the variability of the function’s root gradient with the node tangent. To improve options for node selection I recently posted, Quartic Polynomials-Calculate Roots Using Twin Function Nodes, which uses the architecture and symmetry of polynomials to find roots using ‘SOSO’; a confected twin function with Same Opposite Same Opposite +/- coefficients.

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Courtesy Icepoint Unsplash

Exploiting architectural symmetry of Quartic Polynomials to calculate roots using Twin Function Nodes created by SOSO

This post demonstrates the application to Quartic polynomials of SOSO which is a confected twin function with Same Opposite Same Opposite +/- coefficients. It creates strategic node intercepts, for the application of Newton’s or other approximation methods where applicable.

I recently posted Polynomial Roots — ‘SOSO’ Easy with Curved Segment transplants, using the ‘architecture’ and symmetry of Cubic Polynomials to find roots by curved Segment Transplants Incorporating Node gradients or ‘STING’ as an alternative to Newton’s straight line approximation.

My focus here is on 1st and 2nd roots by SOSO, as I have previously presented a modified Quadratic Equation for finding the 3rd and 4th roots given that the 1st and 2nd. You can read about this in Cubic Polynomials — A Simpler Approach.

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

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. …

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Photo by Mae Mu on Unsplash

Exploiting the amazing symmetry of cubic polynomials to quickly find the first root using Newton’s method.

Using Newton’s Approximation to find the roots of polynomials is commonly taught. However, this can involve repeated iterations to achieve the required level of accuracy.

This post shows how to achieve high accuracy in a single application. The method exploits the amazing symmetry of cubic polynomials around their Inflection Point, (Xip,Yip), to find an optimal starting point for Newton’s method. I do this by exploiting polynomials’ rotational symmetry to ‘visually’ transpose a selected ‘satellite’ point on the graph to its counterpart. …

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A further look at polynomial ‘division by vision’, covering non-unity lead coefficients and quadratic remainders.

This post expands on methods I’ve explored in Polynomials—Division by Vision with Remainders (Part A), and Polynomials — Division by Vision, which avoid long division by ‘downloading’ coefficients from the original polynomials into the solution.

My earlier posts were limited to simpler divisions resulting in linear and quadratic quotients.

With a little extra effort, I’ll now show how the ‘division by vision’ methodology applies to non-unity leading coefficients, and higher order quotients.

Admittedly, big quotients do stretch the ‘division by vision’ claim a little, but it will be handy for programmers and those who seek to understand and marvel at the architecture of polynomials. …

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A fresh way to apply ‘division by vision’ in higher order polynomials with quadratic quotients with remainders, while saving time.

In my post Polynomials — Division by Vision, I presented a further method for dividing higher order polynomials, without long division.

I call the approach ‘division by vision’ as it helps you to ‘visualise’ the outcome of the equation. The main advantage of ‘division by vision’ is its speed, and less chance of errors (in absence of a calculator).

While useful in many applications, the method, derived from my original polynomial factoring post, Polynomial Division — by Formula, was limited to division resulting in quadratic quotients without remainders. …


Greg Oliver

I’m passionate about maths and making it simpler to enjoy. Retired Engineer and Accountant currently writing maths book at a high school level.

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