Sextic Polynomials — A Practical Approach
At a young age a builder taught me how to mark out a right angle using the 3,4,5 triangle rule with a tape measure. That math practicality has stuck with me and recently I have been working to decipher it into its ‘LEGO block’ constituents to increase intuitive understanding.
This post adopts a ‘Tool Kit’ of methods of finding polynomial roots using basic math including Similar Triangles, ‘internal Quadratic architecture’ and formula adaptations and creation of twin function ‘SOSO’ nodes as application points for root approximation methods. Its purpose is to foster intuitive thinking over convenience solutions.
The intention isn’t to beat a calculator. It is to give students confidence to play the function as they see it, to be creative and in the process strengthen their math abilities.
This post assumes math at a high school level.
The Architecture
Consider the following Sextic function:
y= x⁶+x⁵-13x⁴-0.9x³+34x²-14.5x-5.
Graph 1-Scale 16:1
The 5 Steps to Finding 6 Roots:
Extended Quadratic-Sextic Equation
Where the Factors K, L, M and N represent the known roots.
A, B, are usual x^n coefficients and G the constant=-5.
Step 1 — Root B
Consider the following function y= x⁶+x⁵-13x⁴-0.9x³+34x²-14.5x-5, shown in blue and twin y= x⁶-x⁵-13x⁴+0.9x³+34x²+14.5x-5 derived from SOSO shown in red.
Graph 2-SOSO with Nodes
Three Nodes H, V and J are created by the intersection of the SOSO twin and the primary function as follows:
y= x⁶+x⁵-13x⁴-0.9x³+34x²-14.5x-5; subtract the twin.
y= x⁶-x⁵-13x⁴+0.9x³+34x²+14.5x-5 hence:
y= 2x⁵-1.83x³-29x=x(2x⁴-1.8x²-29); solve by standard quadratic equation letting z=x², z=+4.28, from which x=+-2.07. x=0, +-2.07. Hence; Node H=(-2.07, -19.32), Node J=(+2.07, -19.32) and Node V=(0, -5).
To apply Similar Triangles to approximate Root B. We first need to raise Node H above the x axis by see-saw action.
See-Saw Rotation to raise the Node
A) Change the coefficient of x to avoid introducing a constant term into the simultaneous SOSO Quartic function. This see-saws the twin function proportionally with x so it is necessary to divide the required height shift at Node H(x) and reduce the current twin x coefficient F=14.5 accordingly as follows:
B) Reduce coefficient of x by 19.31/2.07=9.33; hence Coefficient F=14.5–9.33=5.17 thereby transposing Node H to (-1.9, 4.07) and Node J to (1.9, -13.68) as shown in Graph 6.
Graph 3-Node Shifts
Node H coordinates are again easily calculated by simultaneous equations between the new function: y= x⁶-x⁵-13x⁴+0.9x³+34x²+5.17x-5, and the principal function: y= x⁶+x⁵-13x⁴-0.9x³+34x²-14.5x-5
Hence: y=x(2x⁴-1.8x²-19.67) which can readily be solved by letting z=x².
Hence: x=0, +-1.907 and y=4.07.
Note: there is no known rule to determine a shift that would transpose a node exactly onto the root! The overshoot to y=4.07 occurred because the twin gradient is positive at x=-2.07.
Similar Triangles
Referring to my earlier Post and Diag 1 below, Nodes H and H’ strike a chord across the segment containing the target Root B shown in red.
Diag 1-Similar Triangles
Node H has transposed from (–2.07, -19.31) to (-1.9, 4.07) hence:
Δx=k*J/(k+m)=4.07*(-2.07+1.9)/(4.07+19.31)=-0.03.
Approx Root B=-1.9+Δx=-1.93, which equals actual x=-1.93.
Step 2 — Root C
Referring to Graph 2 below; separating the function into (a) Sextic-cubic Ya=x⁶+x⁵-13x⁴-0.9x³ and (b) Quadratic Yb=34x²-14.5x-5 shown in dotted black which closely approximates the principal function from the turning point Xtp well through Root C.
The Quadratic equation content directly delivers quadratic Root C=0.23 and also provides Quadratic Root k=0.65 which is the starting point for creation of a Node for application Similar Triangles approximation to Root D.
That was the easy one!
Graph 4-Root C with Quadratics
Step 3 — Root D
Referring to Graph 3 below; construct a 1st node intercept on the function using a Sextic-Cubic and Quadratic Contents.
1st Node Intercept
Again consider the Quadratic content Yb=34x²-14.5x-5 with Root K(J)=0.65. Since y=Ya+Yb the Sextic-cubic content Ya=x⁶+x⁵-13x⁴-0.9x³ (shown in dotted green) must intersect the principal function at Node K, where x=0.65. Therefore Ya=-2.4.
Graph 5-Establish Node K
2nd Node Intercept
Refer to graph 6
Establish the 2nd node as close as possible to the root with y positive. A simple method is to construct a segment from TP=(0.21, -6.55) through Node K=(0.65, -2.4) and using similar triangles, intercept the x axis at I=(0.9, 0). Use that intercept as a root of the Quadratic content from which the y value of the Sextic-cubic content can be calculated.
Graph 6-Establish Quadratic Root I
Refer to Graph 7
Next calculate the constant shift H of the Quadratic required to create Root I.
Let Yb=34x²-14.5x+H.
Since roots are +0.9 and -0.9–2*0.21=-0.48 as shown at Root J, Constant C=-0.432*34=-14.69. Adding this to Ya=x⁶+x⁵-13x⁴-0.9x³ to maintainy=Ya+Yb we get Ya=x⁶+x⁵-13x⁴-0.9x³+14.69 from which intercept Node K2=(0.9, 1.43).
Graph 7-Establish Node K2
Using Similar Triangles
Node K has transposed from (0.65, -2.4) to (0.9, 1.43) hence:
Δx=1.43*(0.9–0.65)/(1.43+2.4)=0.09
Approx Root B=0.9-Δx=0.81, compared with actual 0.8.
Step 4 — Root E
Referring to Graph 8, even after the shift to find Root B, the remoteness of Node J=(1.9, -13.68) coupled with its negative twin gradients makes overshoot in 1 iteration for Similar Triangles as used in Steps 1 and 3 too complex so we will adopt a variation for Root E.
Graph 8-Similar Triangles with Non Straddling Nodes
In my post The Simplest Cubic Root, Similar Triangles are adopted using a Cubic polynomial Inflection point Ip and the Constant term to construct a triangle hypotenuse which intercepts the x axis close to the target root. Unlike the preceding examples, the nodes don’t strike a cord across the x axis.
In this Sextic example, Node J=(2.07, -19.3) coordinates prior to shift, are to represent the constant term and after shift Node J=(1.9, -13.68), assumed to be proximal to Ip (not defined).
Using Similar Triangles
Δx=0.17*13.68/5.62=0.414.
Approx Root E=1.9–0.414=1.486 versus 1.49 actual.
Note 1: The assumption to use a proximal Ip in this particular case is borne out when it is considered that the worst case scenario for Ip=B