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Quartic Polynomials — Calculate Roots Using Twin Function Nodes

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. I will include the method in this post for completeness.

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

Consider the following function:

4x⁴–8x³-3x²+5x+1, shown in blue in the graph below and its twin 4x⁴+8x³-3x²-5x+1 derived from SOSO shown in dotted red.

We can see that Nodes K and L are created by the simultaneous SOSO equations, which after cancellation of like terms; D0=0, Lx and Kx=+-√D/B=+-√5/8=+-0.79.

Calculate Root B

Root B can be calculated without SOSO of course as the principal function intercepts the y axis at D0 where y=1 and the gradient dy/dx =D=+5. Hence by Newton, -Δx=Do/(dy/dx)= -1/5=-0.2 which compares with the actual root B=-0.19.

Calculate Root A

Node K: (-0.79, 0.69), hence using Newton’s approximation:

Δx=0.69/(dy/dx)=0.69/16x³-24x²-6x+5=0.69/-13.13=-0.053 hence: