# The Simplest Cubic Root

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

**and with high**

*Ip(x)=B/3A<1***coefficients leading to relatively steeper architecture between turning points.**

*x*This post assumes math at high school level.

## T**he Simplest Cubic Root**

Strikes a shrinking cord ** EJ** within the range of the Turning Points to intercept the x axis close to the target

**as shown in Diag 1 with example function**

*Root B***. Point**

*y=x³-1.5x²-5x+6***is 180 deg rotationally symmetrical with**

*G*

*Root B.*As the chord shrinks it rotates clockwise about the Inflection Point ** Ip** until Intercept

**is concentric with point**

*E***(undefined) as shown in Diag 2 where Intercept**

*G***and**

*J***both become concentric with**

*Approx Root B***. Hence:**

*Root B*** Approx Root B=Root B** exactly when

*E=G.*## A**pproximate Root Formula**

Because we don’t know the x coordinate of point ** G** we can use a given proximal value, Constant

**and use Similar Triangles to calculate**

*D=6***Referring to Diag 3 as follows:**

*Approx Root B*** Approx Root B=Ip(x)*D/(D-Ip(y) **hence;

** Approx Root B=0.5*6/(6–3.25)=1.09 **which compares with

**actual.**

*1.1*## R**emaining Roots**

To complete the task, the 2nd and 3rd roots are simply ‘downloaded’ from the ‘Extended Quadratic Equation’ I presented in a post article, *Cubic Polynomials-A Simpler Approach**.*

Where Factor ** L=-1.09**, which we have just found,

**and**

*A***are usual cubic coefficients and**

*B***the constant.**

*D*## S**ummary**

This post introduces Similar Triangles within the symmetrical architecture of the Cubic polynomial to find the ‘Simplest Root’. The method strikes a cord through the Inflection Point ** Ip**, requiring only basic math for a sound root approximation method.

More importantly I have tried to nurture intuitive learning, relating the math with the graph; making math work for you; not you for it!