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Sextic Polynomial Square Roots — Division Tool kit
Finding Sextic Square Root Functions With a ‘Division by Vision’ Tool kit
Most students are familiar with Square and Cube roots of real numbers and many encounter ‘Perfect Square’ Quadratics at some time during their courses, but less so for the square roots of higher order Polynomials.
This post finds Square roots of Perfect Sextics by creating a simple tool kit derived from my earlier posts, Polynomials — Division by Vision and Polynomials-Division by Vision with Remainders Parts A and B where x^n coefficients and constants of the numerators and denominators constitute a simple equation delivering the quotient.
This post assumes math at high school level.
Firstly ‘Division by Vision’ revision
DIVISION by VISION
The ‘Division by Vision’ formulas (without remainders) present as:
Sextic/Cubic=Cubic
Numerator: y=x⁶+Bx⁵+Cx⁴+Dx³+Ex²+Fx+G
Denominator: y=x³+bx²+cx+d
Quotient: y=x³+(B-b)x²+[C-c-(B-b)b]x+G/d.
Example
Divide y= x⁶+x⁵-13x⁴-0.92x³+34.17x²-14.8x-4.96
By: y=x³-0.36x²-11.19x+14.17
By Vision: y=x³+(1+0.36)x²+[-1.81+0.49]x-4.96/14.17 hence:
Quotient y=x³+1.36x²-1.32x-0.35
Note: the [C-c-(B-b)b] coefficient of x requires both b and c of the denominator to be known but when working with Square roots this is replaced with the DIVISION TOOL KIT formulation, (4C-B²)/8 requiring only Numerator coefficients B an C.
DIVISION TOOL KIT
Sextic-Square Root
The square root function of a Sextic, y=Ax⁶+Bx⁵+Cx⁴+Dx³+Ex²+Fx+G is a Cubic polynomial where, with no remainder and assuming a monic Ax⁶ coefficient, has the form: y=x³+(B-b)x²+[(C-c)-(B-b)b]x+d where the coefficient [(C-c)-(B-b)b] of x can be transposed into this post’s headline form:
[(C-c)-(B-b)b]=(4C-B²)/8 as follows:
Divide: y=Ax⁶+Bx⁵+Cx⁴+Dx³+Ex²+Fx+G
