New Six Formulas of Radical Roots Developed by Using an Engineering Methodology to Solve Sixth Degree Polynomial Equation in General Forms by Calculating All Solutions Nearly in Parallel

This paper proposes new six formulas allowing to calculate all roots of sixth degree polynomial equation nearly in parallel while including the use of radical expressions, which is extending a new engineering methodology to solve polynomial equations of nth degree where the value of n can exceed five. This methodology is based on developing the roots of nth degree polynomial equation according to a distributed structure of radical terms, where each term is built by multiplying two radicals presenting the roots of polynomial equations with inferior degrees. This distributed structure of terms is allowing them to neutralize each other during multiplications, which forward calculations toward eliminating radicalities, suppressing complex terms and reducing degrees. As a result, this paper is proposing new two theorems solving sixth degree polynomial equation in complete forms while relying on two different approaches built on the same engineering methodology of roots architecting, which allow calculating solutions nearly in parallel. This engineering methodology is scalable to solve higher degrees of polynomial equations while extending the same distributed architecture of terms whereas re-engineering the expressions of included sub-terms in order to manifest the same outcomes of reciprocal neutralization, radicality suppression and degrees reduction during calculations. Therefore, this paper is also presenting the engineered requirements and techniques along with details in order to scale the used methodology by projecting it on nth degree polynomial equations where the possibility of calculating the values of all roots nearly in parallel whereas the polynomial degrees can exceed the quantic form. The new proposed engineering methodology in this paper is listing all necessary logic, techniques and formulas to solve nth degree polynomial equations in general forms stage-by-stage while relying on the use of radical expressions, which will scale the results of this paper toward solving highly complex equations.