Advancing Solutions to Higher-Degree Polynomials: 
A Novel Recurrence Approach via EMS’s Theorem

Mourad Sultan Ezouidi; Taoufik Gassoumi

doi:doi:10.11648/j.ajaa.20261302.11

Research Article |

| Peer-Reviewed

Advancing Solutions to Higher-Degree Polynomials: A Novel Recurrence Approach via EMS’s Theorem

Mourad Sultan Ezouidi^*

, Taoufik Gassoumi

Published in American Journal of Astronomy and Astrophysics (Volume 13, Issue 2)

Received: 7 April 2026 Accepted: 16 April 2026 Published: 29 April 2026

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Solving higher-degree polynomial equations remains a fundamental challenge in both pure and applied mathematics. While quadratic, cubic, and quartic equations have known algebraic solutions, no general radical solution exists for degree five and higher (Abel–Ruffini theorem). This work introduces a novel recurrence-based methodology for deriving exact roots of polynomials of arbitrary degree, based on Ezouidi Mourad Sultan's Theorem (EMST). Unlike traditional algebraic techniques that are often restricted to degrees four or less or rely on numerical approximations, this framework allows for the explicit determination of roots, including irrational, complex, and multiple roots, across any polynomial degree. By systematically leveraging the structure of polynomial coefficients through recursive relationships, this approach extends the capabilities of classical methods and enhances their precision. The method is demonstrated through comprehensive examples involving irreducible and high-degree polynomials of degree 8, producing exact roots in closed form. Comparative analyses with established techniques such as Cardano's method, Newton's Method, and the Rational Root Theorem highlight the advantages of this recurrence formulation, including exactness, no reliance on initial guesses, and applicability to any degree. The EMST-based methodology offers a unified pathway toward exact solutions for longstanding algebraic problems.

Published in	American Journal of Astronomy and Astrophysics (Volume 13, Issue 2)
DOI	10.11648/j.ajaa.20261302.11
Page(s)	59-73
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Recurrence, Polynomial Roots, EMS’s Theorem, Higher-degree Polynomials, Algebraic Equations, Exact Solutions

1. Introduction

Polynomial equations serve as foundational elements across various fields in mathematics, engineering, and the sciences, underpinning models in control theory, optimization, physics, and combinatorics. While explicit solutions for quadratic, cubic, and quartic equations have been established since the Renaissance, the general solution for polynomials of degree five and higher remains one of the most significant unresolved problems in algebra. The groundbreaking work of Galois and Abel proved the impossibility of solving the general polynomial of degree five or greater through radicals, revealing intrinsic algebraic limitations

[1]

Classical solution methods, such as the Rational Root Theorem, Newton’s iterative method, Cardano’s method, and polynomial decomposition, often provide only partial or approximate solutions. The Rational Root Theorem is limited to rational roots and becomes computationally prohibitive for high-degree or irrational roots

[2, 3]

. Newton’s method, widely employed for numerical approximation, depends heavily on initial guesses and frequently fails to accurately identify multiple, complex, or closely spaced roots

[4-6]

. Moreover, these techniques do not offer explicit algebraic formulas, restricting their application to approximate solutions

[7-10]

Recent advances in algebraic and computational techniques—such as spectral theory, orthogonal polynomials, and symbolic computation—have contributed to understanding specific classes of polynomials, including Hermite, Legendre, Chebyshev, Jacobi, and q-orthogonal polynomials

[11-15]

. Polynomial solutions to differential equations, recurrence relations, and combinatorial interpretations have further expanded this framework

[16-20]

. Despite these developments, the derivation of a universal explicit formula for the roots of arbitrary-degree polynomials remains an open challenge.

Explorations involving discriminants, recursive relationships, and algebraic curves have provided partial insights, such as bounds on root locations and conditions for root localization

[21-26]

.Nonetheless, a comprehensive, explicit, and exact solution applicable to all polynomial degrees remains a fundamental open problem in algebra.

Recurrence-based methods for solving higher-degree polynomials are presented in

[27-32]

. Algebraic techniques for polynomial root finding appear in

[33-36]

. Advanced approaches for complex root analysis are discussed in

[37-40]

. Laguerre equation techniques, Gibbs phenomenon analysis, cubic equation decomposition, root bounding, and general polynomial solution frameworks are covered in

[41-45].

Multiple root computation schemes for high-degree equations are found in

[46-50]

. Numerical methods and iterative techniques for polynomial solutions are covered in

[51-55]

. Additional results on difference equations and singular solutions to polynomial problems are addressed in

[56-61]

This work presents a recurrence-based methodology grounded in Mourad Sultan Ezouidi’s Theorem (EMST). This approach exploits recursive relationships among polynomial coefficients to explicitly determine all roots—rational, irrational, complex, and multiple—for polynomials of any degree. Unlike classical algebraic or purely numerical methods, the EMST framework offers a unified, exact pathway to solutions, thereby extending the scope of traditional techniques while overcoming their inherent limitations. Through rigorous theoretical formulation and illustrative examples involving high-degree and irreducible polynomials, this recurrence approach establishes a new paradigm for the explicit and precise resolution of polynomial equations.

2. EMS’s Theorem and Proof

Let P(x)

= \sum_{r = 0}^{n} {{{(- 1)}^{r} l}_{r - 1}^{q} x}^{n - k}

(1)

be a polynomial of degree n. The new coefficients of the polynomial, denoted as

S_{k - 1}^{q}

, is defined as a function.

l_{- 1}^{q}

l_{0}^{q}

l_{1}^{q}

l_{2}^{q}

l_{3}^{q}

l_{4}^{q}

l_{5}^{q}

,,,,,,

l_{n - 2}^{q}

l_{n - 1}^{q}

and can be computed using the formula:

s_{k - 1}^{r} = \sum_{r = 0}^{k} c_{n - r}^{k - r} ({n)}^{r} ({- l_{0}^{q})}^{k - r} l_{r - 1}^{q}

(2)

Equation (2) represents the EMS’s Theorem, where

l_{- 1}^{q}

l_{0}^{q}

l_{1}^{q}

l_{2}^{q}

l_{3}^{q}

l_{4}^{q}

l_{5}^{q}

,,,,,,

l_{n - 2}^{q}

l_{n - 1}^{q}

are the terms derived from the coefficients of the polynomial, and the formula allows for the calculation of exact roots, whether they are rational, irrational, real, or complex.

Graphical abstract

Proof: Recurrence formula

Download: Download full-size image

Figure 1. Graphical abstract of the EMST recurrence method for solving higher-degree polynomials.

The proof of the EMST involves understanding how the roots of a polynomial are related to its coefficients. By using recursive method we examine the polynomial’s structure and find a method to derive the exact values of the roots. This proof demonstrates that the formula provides an exact solution for all types of roots, including repeated and complex ones.

Let

\{P (n) \exists n_{0} \in (0, 1,,, n - 1) ∕ P (n_{0}) ⟹) P (n - 1\}

is true

For (k=0), application of EMST Equation (2) yields:

s_{- 1}^{r} = \sum_{r = 0}^{0} c_{n - r}^{0 - r} {(n)}^{r} ({- l_{0}^{q})}^{0 - r} l_{r - 1}^{q} = c_{n}^{- 1 + 1} = c_{n}^{0} = 1

(3)

For (k=1):

s_{0}^{r} = \sum_{r = 0}^{1} c_{n - r}^{1 - r} {(n)}^{r} ({- l_{0}^{q})}^{1 - r} l_{r - 1}^{q} = c_{n}^{1} {(n)}^{0} ({- l_{0}^{q})}^{1} l_{- 1}^{q} + c_{n}^{0} {(n)}^{1} ({- l_{0}^{q})}^{0} l_{0}^{q} = {- c}_{n}^{1} l_{0}^{q} l_{- 1}^{q} + n l_{0}^{q}

l_{- 1}^{q}

=1 then

s_{0}^{r} = \sum_{r = 0}^{1} c_{n - r}^{1 - r} {(n)}^{r} ({- l_{0}^{q})}^{1 - r} l_{r - 1}^{q} = {- c}_{n}^{1} l_{0}^{q} l_{- 1}^{q} + n l_{0}^{q}

(4)

For (k=2):

s_{1}^{r} = \sum_{r = 0}^{2} c_{n - r}^{2 - r} {(n)}^{r} ({- l_{0}^{q})}^{2 - r} l_{r - 1}^{q} = c_{n}^{2} {(n)}^{0} ({l_{0}^{q})}^{2} l_{- 1}^{q} - c_{n - 1}^{1} {(n)}^{1} ({l_{0}^{q})}^{1} l_{0}^{q} + n^{2} l_{1}^{q} = (c_{n}^{2} - n c_{n - 1}^{1}) ({l_{0}^{q})}^{2} + n^{2} l_{1}^{q}

= {- c_{1}^{1} c}_{n}^{2}) ({l_{0}^{q})}^{2} + n^{2} l_{1}^{q}

When

l_{- 1}^{q}

=1, it follows that

s_{1}^{r} = \sum_{r = 0}^{2} c_{n - r}^{2 - r} {(n)}^{r} ({- l_{0}^{q})}^{2 - r} l_{r - 1}^{q} = (c_{n}^{2} - n c_{n - 1}^{1}) ({l_{0}^{q})}^{2} + n^{2} l_{1}^{q} = {- c_{1}^{1} c}_{n}^{2}) ({l_{0}^{q})}^{2} + n^{2} l_{1}^{q}

i.e

s_{1}^{r} = {- c_{1}^{1} c}_{n}^{2}) ({l_{0}^{q})}^{2} + n^{2} l_{1}^{q}

(5)

For r (k=3):

s_{2}^{r} = \sum_{r = 0}^{3} c_{n - r}^{3 - r} {(n)}^{r} ({- l_{0}^{q})}^{3 - r} l_{r - 1}^{q} = {- c}_{n}^{3} {(n)}^{0} ({l_{0}^{q})}^{3} l_{- 1}^{q} + c_{n - 1}^{2} {(n)}^{1} ({l_{0}^{q})}^{2} l_{0}^{q} - c_{n - 2}^{1} {(n)}^{2} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q}

{= - c}_{n}^{3} ({l_{0}^{q})}^{3} + {nc}_{n - 1}^{2} ({l_{0}^{q})}^{3} - n^{2} c_{n - 2}^{1} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q} = {(- c}_{n}^{3} + {nc}_{n - 1}^{2}) ({l_{0}^{q})}^{3} - n^{2} c_{n - 2}^{1} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q}

i.e

s_{2}^{r} = c_{n}^{2} c_{n}^{3} ({l_{0}^{q})}^{3} - n^{2} c_{n - 2}^{1} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q}

(6)

For (k=4):

s_{3}^{r} = \sum_{r = 0}^{4} c_{n - r}^{4 - r} {(n)}^{r} ({- l_{0}^{q})}^{4 - r} l_{r - 1}^{q} = c_{n}^{4} {(n)}^{0} ({l_{0}^{q})}^{4} l_{- 1}^{q} - c_{n - 1}^{3} {(n)}^{1} ({l_{0}^{q})}^{3} l_{0}^{q} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q}

- c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

= (c_{n}^{4} - n c_{n - 1}^{3}) ({l_{0}^{q})}^{4} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

= - c_{3}^{1} c_{n}^{3} ({l_{0}^{q})}^{4} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

s_{3}^{r} = - c_{3}^{1} c_{n}^{4} ({l_{0}^{q})}^{4} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

(7)

For (k=5)

s_{4}^{r} = \sum_{r = 0}^{5} c_{n - r}^{5 - r} {(n)}^{r} ({- l_{0}^{q})}^{5 - r} l_{r - 1}^{q} = - c_{n}^{5} {(n)}^{0} ({l_{0}^{q})}^{5} l_{- 1}^{q} + c_{n - 1}^{4} {(n)}^{1} ({l_{0}^{q})}^{4} l_{0}^{q} - c_{n - 2}^{3} {(n)}^{2} ({l_{0}^{q})}^{3} l_{1}^{q} + c_{n - 3}^{2} {(n)}^{3} ({l_{0}^{q})}^{2} l_{2}^{q}

- c_{n - 3}^{1} {(n)}^{4} ({l_{0}^{q})}^{1} l_{3}^{q} + n^{5} l_{4}^{q}

= (- c_{n}^{5} + n c_{n - 1}^{4}) ({l_{0}^{q})}^{5} - c_{n - 2}^{3} {(n)}^{2} ({l_{0}^{q})}^{3} l_{1}^{q} + c_{n - 3}^{2} {(n)}^{3} ({l_{0}^{q})}^{2} l_{2}^{q}

- c_{n - 3}^{1} {(n)}^{4} ({l_{0}^{q})}^{1} l_{3}^{q} + n^{5} l_{4}^{q}

s_{4}^{r} = c_{4}^{1} c_{n}^{5} ({l_{0}^{q})}^{5} - c_{n - 2}^{3} {(n)}^{2} ({l_{0}^{q})}^{3} l_{1}^{q} + c_{n - 3}^{2} {(n)}^{3} ({l_{0}^{q})}^{2} l_{2}^{q} - c_{n - 4}^{1} {(n)}^{4} ({l_{0}^{q})}^{1} l_{3}^{q} + n^{5} l_{4}^{q}

(8)

Assume that the property

\{P (n) \exists n_{0} \in (0, 1,,, n) ∕ P (n_{0}) ⟹ P (n) is true\}

\to

holds

The objective is to demonstrate that

\{P (n) \exists n_{0} \in (0, 1,,, n) ∕ P (n_{0}) ⟹ P (n) is true\}

remains valid at order n+1

\{P (n) \exists n_{0} \in (0, 1,,, n + 1) ∕ P (n_{0}) ⟹ P (n) is true\}

For (k=n+1):

s_{n}^{r} = \sum_{r = 0}^{n + 1} c_{n - r}^{n + 1 - r} {(n)}^{r} ({- l_{0}^{q})}^{n + 1 - r} l_{r - 1}^{q} = 0 \forall k \notin [0, 1,,,,, n]

(9)

\{P (n) \exists n_{0} \in (0, 1,,, n + 1) ∕ P (n_{0}) ⟹ P (n + 1) is true\}

that is to say that

\{P (n) \exists n_{0} \in (0, 1,,, n + 1) ∕ P (n_{0}) ⟹ P (n + 1) is true\}

⟹ \{P (n) \exists n_{0} \in (0, 1,,, n + 1) ∕ P (n_{0}) ⟹ P (n + 1) is true\}

\{P (n) \exists n_{0} \in (0, 1,,, n + 1) ∕ P (n_{0}) ⟹ P (n) is true\}

is strongly verified.

3. Methods Comparison and Advantages of the Ezouidi Mourad Sultan’s Theorem (EMST)

Traditional methods for finding polynomial roots, such as Newton's Method and the Rational Root Theorem, each have significant limitations. Newton's Method, while widely used for numerical approximation, relies heavily on the choice of initial guess and may fail to converge or yield inaccurate results, especially for high-degree polynomials or when roots are closely spaced, complex, or repeated. Moreover, it provides only approximate values, making it unsuitable for cases where exact solutions are required. On the other hand, the Rational Root Theorem can only identify possible rational roots, offering no means to determine irrational or complex roots, and becomes inefficient for polynomials of higher degree due to the explosion in the number of candidate roots. Additionally, it cannot detect or quantify repeated roots, further limiting its applicability in comprehensive root-finding for general polynomials.

Unlike Newton's Method and the Rational Root Theorem, the Ezouidi Mourad Sultan’s Theorem (EMST) presents a powerful and unified approach for solving polynomial equations. The EMST discriminant formula enables the determination of exact roots for all types of polynomials, including those with irrational, complex, or repeated roots. Its general applicability extends to polynomials of any degree, even those well beyond the reach of classical methods, such as equations of the eighteenth degree and higher. Unlike iterative methods, EMST does not depend on initial guesses or approximations; instead, it derives all roots directly from the structure and coefficients of the polynomial. Moreover, it systematically identifies both distinct and repeated roots, as well as complex solutions, making it a comprehensive and robust solution method for the full spectrum of polynomial equations.

4. Application and Verification of the EMST on Polynomials of Degree 8

The effectiveness of the EMST is demonstrated by analyzing an eighth-degree polynomial equation, with exact solutions obtained via EMST systematically compared to the approximate results produced by Newton's Method and the Rational Root Theorem.

4.1. Recurrence Approach via EMS’s Theorem: First Example Consider the Following Polynomial

P(x)=

x^{8} - 16 x^{7} + {\frac{7204}{8^{2}} x}^{6} - \frac{232832}{8^{3}} x^{5} + \frac{4726206}{8^{4}} x^{4} - \frac{61697920}{8^{5}} x^{3} + \frac{505838740}{8^{6}} x^{2} - \frac{2381353600}{8^{7}} x + \frac{4928741769}{8^{8}} = 0

(10)

P(x)=

x^{8} - l_{0}^{q} x^{7} + {l_{1}^{q} x}^{6} - l_{2}^{q} x^{5} + l_{3}^{q} x^{4} - l_{4}^{q} x^{3} {+ l_{5}^{q} x^{2} - l}_{6}^{q} x + l_{7}^{q} = 0

(11)

Let’s focus on this polynomial to the power eight. To find out the roots of this polynomial we will determine the values of

s_{r}^{q}

relative respectively to

l_{r}^{q} where r belongs to the set of numbers

(-1, 0, 1, 2, 3, 4, 5, 6, 7)

l_{0}^{q} = 16, l_{1}^{q} = \frac{7204}{8^{2}} l_{2}^{q} = \frac{232832}{8^{3}}, l_{3}^{q} = \frac{4726206}{8^{4}}, l_{4}^{q} = \frac{61697920}{8^{5}} l_{5}^{q} = \frac{505838740}{8^{6}} l_{6}^{q}

\frac{2381353600}{8^{7}} l_{7}^{q}

\frac{4928741769}{8^{8}}

(12)

Using Ezouidi Mourad Sultan (EMST)

s_{k - 1}^{r} = \sum_{r = 0}^{k} c_{n - r}^{k - r} {(n)}^{r} ({- l_{0}^{q})}^{k - r} l_{r - 1}^{q}

(13)

For k=0 we write

s_{- 1}^{r} = \sum_{r = 0}^{0} c_{n - r}^{- r} {(n)}^{r} ({- l_{0}^{q})}^{- r} l_{r - 1}^{q} = c_{n}^{- 1 + 1} = c_{n}^{0} = 1

(14)

For k=1 we write

s_{0}^{r} = \sum_{r = 0}^{1} c_{n - r}^{1 - r} {(n)}^{r} ({- l_{0}^{q})}^{1 - r} l_{r - 1}^{q} = c_{n}^{1} {(n)}^{0} ({- l_{0}^{q})}^{1} l_{- 1}^{q} + c_{n}^{0} {(n)}^{1} ({- l_{0}^{q})}^{0} l_{0}^{q} = {- c}_{n}^{1} l_{0}^{q} l_{- 1}^{q} + n l_{0}^{q} = (n - n) l_{0}^{q} = 0

(15)

{s_{0}^{r} = - c}_{8}^{1} (16) + 8 (16) = - 128 + 128 = 0

(16)

For k=2 we find

s_{1}^{r} = \sum_{r = 0}^{2} c_{n - r}^{2 - r} {(n)}^{r} ({- l_{0}^{q})}^{2 - r} l_{r - 1}^{q} = c_{n}^{2} {(n)}^{0} ({l_{0}^{q})}^{2} l_{- 1}^{q} - c_{n - 1}^{1} {(n)}^{1} ({l_{0}^{q})}^{1} l_{0}^{q} + n^{2} l_{1}^{q} = (c_{n}^{2} - n c_{n - 1}^{1}) ({l_{0}^{q})}^{2} + n^{2} l_{1}^{q}

= {- c_{1}^{1} c}_{n}^{2}) ({l_{0}^{q})}^{2} + n^{2} l_{1}^{q}

(17)

{s_{1}^{r} = - c_{1}^{1} c}_{8}^{2} ({16)}^{2} + 8^{2} (\frac{7204}{8^{2}}) = 36

(18)

For k=3 we evaluate

s_{2}^{r} = \sum_{r = 0}^{3} c_{n - r}^{3 - r} {(n)}^{r} ({- l_{0}^{q})}^{3 - r} l_{r - 1}^{q} = {- c}_{n}^{3} {(n)}^{0} ({l_{0}^{q})}^{3} l_{- 1}^{q} + c_{n - 1}^{2} {(n)}^{1} ({l_{0}^{q})}^{2} l_{0}^{q} - c_{n - 2}^{1} {(n)}^{2} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q}

{= - c}_{n}^{3} ({l_{0}^{q})}^{3} + {nc}_{n - 1}^{2} ({l_{0}^{q})}^{3} - n^{2} c_{n - 2}^{1} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q} = {(- c}_{n}^{3} + {nc}_{n - 1}^{2}) ({l_{0}^{q})}^{3} - n^{2} c_{n - 2}^{1} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q}

= c_{n}^{2} c_{n}^{3} ({l_{0}^{q})}^{3} - n^{2} c_{n - 2}^{1} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q}

(19)

s_{2}^{r} = c_{2}^{1} c_{8}^{3} ({16)}^{3} - c_{6}^{1} {(8)}^{2} ({16)}^{1} (\frac{7204}{8^{2}}) + 8^{3} (\frac{232832}{8^{3}}) = 0

(20)

For k=4 we write

s_{3}^{r} = \sum_{r = 0}^{4} c_{n - r}^{4 - r} {(n)}^{r} ({- l_{0}^{q})}^{4 - r} l_{r - 1}^{q} = c_{n}^{4} {(n)}^{0} ({l_{0}^{q})}^{4} l_{- 1}^{q} - c_{n - 1}^{3} {(n)}^{1} ({l_{0}^{q})}^{3} l_{0}^{q} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

= (c_{n}^{4} - n c_{n - 1}^{3}) ({l_{0}^{q})}^{4} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

= - c_{3}^{1} c_{n}^{3} ({l_{0}^{q})}^{4} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

s_{3}^{r} = - c_{3}^{1} c_{n}^{4} ({l_{0}^{q})}^{4} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

(21)

s_{3}^{r} = - c_{3}^{1} c_{8}^{4} ({16)}^{4} + c_{6}^{2} {(8)}^{2} ({16)}^{2} (\frac{7204}{8^{2}}) - {c_{5}^{1} (8)}^{3} {16)}^{1} (\frac{232832}{8^{3}}) + {(8)}^{4} (\frac{4726206}{8^{4}} = 446

(22)

For k=5 we obtain

s_{4}^{r} = \sum_{r = 0}^{5} c_{n - r}^{5 - r} {(n)}^{r} ({- l_{0}^{q})}^{5 - r} l_{r - 1}^{q} = - c_{n}^{5} {(n)}^{0} ({l_{0}^{q})}^{5} l_{- 1}^{q} + c_{n - 1}^{4} {(n)}^{1} ({l_{0}^{q})}^{4} l_{0}^{q} - c_{n - 2}^{3} {(n)}^{2} ({l_{0}^{q})}^{3} l_{1}^{q} + c_{n - 3}^{2} {(n)}^{3} ({l_{0}^{q})}^{2} l_{2}^{q} - c_{n - 3}^{1} {(n)}^{4} ({l_{0}^{q})}^{1} l_{3}^{q} + n^{5} l_{4}^{q}

= (- c_{n}^{5} + n c_{n - 1}^{4}) ({l_{0}^{q})}^{5} - c_{n - 2}^{3} {(n)}^{2} ({l_{0}^{q})}^{3} l_{1}^{q} + c_{n - 3}^{2} {(n)}^{3} ({l_{0}^{q})}^{2} l_{2}^{q} - c_{n - 3}^{1} {(n)}^{4} ({l_{0}^{q})}^{1} l_{3}^{q} + n^{5} l_{4}^{q}

s_{4}^{r} = c_{4}^{1} c_{n}^{5} ({l_{0}^{q})}^{5} - c_{n - 2}^{3} {(n)}^{2} ({l_{0}^{q})}^{3} l_{1}^{q} + c_{n - 3}^{2} {(n)}^{3} ({l_{0}^{q})}^{2} l_{2}^{q} - c_{n - 4}^{1} {(n)}^{4} ({l_{0}^{q})}^{1} l_{3}^{q} + n^{5} l_{4}^{q}

(23)

s_{4}^{r} = c_{4}^{1} c_{8}^{5} ({16)}^{5} - c_{6}^{3} {(8)}^{2} ({16)}^{3} (\frac{7204}{8^{2}}) + {c_{5}^{2} (8)}^{3} {(16)}^{2} (\frac{232832}{8^{3}}) {- c_{4}^{1} (8)}^{4} {16)}^{1} (\frac{4726206}{8^{4}}) + (8)^{5} (\frac{61697920}{8^{5}}) = 0

(24)

For k=6 we write

s_{5}^{r} = \sum_{r = 0}^{6} c_{n - r}^{6 - r} {(n)}^{r} ({- l_{0}^{q})}^{6 - r} l_{r - 1}^{q} = c_{n}^{6} {(n)}^{0} ({l_{0}^{q})}^{6} l_{- 1}^{q} - c_{n - 1}^{5} {(n)}^{1} ({l_{0}^{q})}^{5} l_{0}^{q} + c_{n - 2}^{4} {(n)}^{2} ({l_{0}^{q})}^{4} l_{1}^{q} - c_{n - 3}^{3} {(n)}^{3} ({l_{0}^{q})}^{3} l_{2}^{q} +

c_{n - 4}^{2} {(n)}^{4} ({l_{0}^{q})}^{2} l_{3}^{q} {- c}_{n - 5}^{1} {(n)}^{5} ({l_{0}^{q})}^{1} l_{4}^{q} + n^{6} l_{5}^{q}

= (c_{n}^{6} - n c_{n - 1}^{5}) ({l_{0}^{q})}^{6} + c_{n - 2}^{4} {(n)}^{2} ({l_{0}^{q})}^{4} l_{1}^{q} - c_{n - 3}^{3} {(n)}^{3} ({l_{0}^{q})}^{3} l_{2}^{q} + c_{n - 4}^{2} {(n)}^{4} ({l_{0}^{q})}^{2} l_{3}^{q} {- c}_{n - 5}^{1} {(n)}^{5} ({l_{0}^{q})}^{1} l_{4}^{q} + n^{6} l_{5}^{q}

= c_{5}^{1} c_{n}^{6}) ({l_{0}^{q})}^{6} + c_{n - 2}^{4} {(n)}^{2} ({l_{0}^{q})}^{4} l_{1}^{q} - c_{n - 3}^{3} {(n)}^{3} ({l_{0}^{q})}^{3} l_{2}^{q} + c_{n - 4}^{2} {(n)}^{4} ({l_{0}^{q})}^{2} l_{3}^{q} {- c}_{n - 5}^{1} {(n)}^{5} ({l_{0}^{q})}^{1} l_{4}^{q} + n^{6} l_{5}^{q}

s_{5}^{r} = {- c}_{5}^{1} c_{n}^{6} ({l_{0}^{q})}^{6} + c_{n - 2}^{4} {(n)}^{2} ({l_{0}^{q})}^{4} l_{1}^{q} - c_{n - 3}^{3} {(n)}^{3} ({l_{0}^{q})}^{3} l_{2}^{q} + c_{n - 4}^{2} {(n)}^{4} ({l_{0}^{q})}^{2} l_{3}^{q} {- c}_{n - 5}^{1} {(n)}^{5} ({l_{0}^{q})}^{1} l_{4}^{q} + n^{6} l_{5}^{q}

(25)

s_{5}^{r} = - c_{5}^{1} c_{8}^{6} ({16)}^{6} + c_{6}^{4} {(8)}^{2} ({16)}^{4} (\frac{7204}{8^{2}}) - {c_{5}^{3} (8)}^{3} {(16)}^{3} (\frac{232832}{8^{3}}) {+ c_{4}^{2} (8)}^{4} {16)}^{2} (\frac{4726206}{8^{4}}) {- c_{3}^{1} (8)}^{5} {16)}^{1} (\frac{61697920}{8^{5}})

+ (8)^{6} (\frac{505838740}{8^{6}}) = 2196

(26)

For k=7 we write

s_{6}^{r} = \sum_{r = 0}^{7} c_{n - r}^{7 - r} {(n)}^{r} ({- l_{0}^{q})}^{7 - r} l_{r - 1}^{q} = {- c}_{n}^{7} {(n)}^{0} ({l_{0}^{q})}^{7} l_{- 1}^{q} + c_{n - 1}^{6} {(n)}^{1} ({l_{0}^{q})}^{6} l_{0}^{q} - c_{n - 2}^{5} {(n)}^{2} ({l_{0}^{q})}^{5} l_{1}^{q}

+ c_{n - 3}^{4} {(n)}^{3} ({l_{0}^{q})}^{4} l_{2}^{q} - c_{n - 4}^{3} {(n)}^{4} ({l_{0}^{q})}^{3} l_{3}^{q}

{+ c}_{n - 5}^{2} {(n)}^{5} ({l_{0}^{q})}^{2} l_{4}^{q} {- c}_{n - 6}^{1} {(n)}^{6} ({l_{0}^{q})}^{1} l_{5}^{q} + n^{7} l_{6}^{q}

= {(- c}_{n}^{7} + n c_{n - 1}^{6}) ({l_{0}^{q})}^{7} l_{- 1}^{q} - c_{n - 2}^{5} {(n)}^{2} ({l_{0}^{q})}^{5} l_{1}^{q} + c_{n - 3}^{4} {(n)}^{3} ({l_{0}^{q})}^{4} l_{2}^{q} - c_{n - 4}^{3} {(n)}^{4} ({l_{0}^{q})}^{3} l_{3}^{q} {+ c}_{n - 5}^{2} {(n)}^{5} ({l_{0}^{q})}^{2} l_{4}^{q}

{- c}_{n - 6}^{1} {(n)}^{6} ({l_{0}^{q})}^{1} l_{5}^{q} + n^{7} l_{6}^{q}

= c_{6}^{1} c_{n}^{7} ({l_{0}^{q})}^{7} - c_{n - 2}^{5} {(n)}^{2} ({l_{0}^{q})}^{5} l_{1}^{q} + c_{n - 3}^{4} {(n)}^{3} ({l_{0}^{q})}^{4} l_{2}^{q} - c_{n - 4}^{3} {(n)}^{4} ({l_{0}^{q})}^{3} l_{3}^{q} {+ c}_{n - 5}^{2} {(n)}^{5} ({l_{0}^{q})}^{2} l_{4}^{q} {- c}_{n - 6}^{1} {(n)}^{6} ({l_{0}^{q})}^{1} l_{5}^{q} + n^{7} l_{6}^{q}

s_{6}^{r} = c_{6}^{1} c_{n}^{7} ({l_{0}^{q})}^{7} - c_{n - 2}^{5} {(n)}^{2} ({l_{0}^{q})}^{5} l_{1}^{q} + c_{n - 3}^{4} {(n)}^{3} ({l_{0}^{q})}^{4} l_{2}^{q} - c_{n - 4}^{3} {(n)}^{4} ({l_{0}^{q})}^{3} l_{3}^{q} {+ c}_{n - 5}^{2} {(n)}^{5} ({l_{0}^{q})}^{2} l_{4}^{q} {- c}_{n - 6}^{1} {(n)}^{6} ({l_{0}^{q})}^{1} l_{5}^{q} + n^{7} l_{6}^{q}

(27)

s_{6}^{r} = c_{6}^{1} c_{8}^{7} ({16)}^{7} - c_{6}^{5} {(8)}^{2} ({16)}^{5} (\frac{7204}{8^{2}}) + {c_{5}^{4} (8)}^{3} {(16)}^{4} (\frac{232832}{8^{3}}) {- c_{4}^{3} (8)}^{4} {16)}^{3} (\frac{4726206}{8^{4}}) {+ c_{3}^{2} (8)}^{5} {16)}^{2} (\frac{61697920}{8^{5}})

- {c_{2}^{1} (8)}^{6} {16)}^{1} (\frac{505838740}{8^{6}}) + (8)^{7} (\frac{2381353600}{8^{7}}) = 0

(28)

For k=8 we write

s_{7}^{r} = \sum_{r = 0}^{8} c_{n - r}^{8 - r} {(n)}^{r} ({- l_{0}^{q})}^{8 - r} l_{r - 1}^{q} = c_{n}^{8} {(n)}^{0} ({l_{0}^{q})}^{8} l_{- 1}^{q} - c_{n - 1}^{7} {(n)}^{1} ({l_{0}^{q})}^{7} l_{0}^{q} + c_{n - 2}^{6} {(n)}^{2} ({l_{0}^{q})}^{6} l_{1}^{q} - c_{n - 3}^{5} {(n)}^{3} ({l_{0}^{q})}^{5} l_{2}^{q} + c_{n - 4}^{4} {(n)}^{4} ({l_{0}^{q})}^{4} l_{3}^{q}

{- c}_{n - 5}^{3} {(n)}^{5} ({l_{0}^{q})}^{3} l_{4}^{q} {+ c_{n - 6}^{2} {(n)}^{6} ({l_{0}^{q})}^{2} l_{5}^{q} - c}_{n - 7}^{1} {(n)}^{7} ({l_{0}^{q})}^{1} l_{6}^{q} + n^{8} l_{7}^{q}

= (c_{n}^{8} - n c_{n - 1}^{7}) ({l_{0}^{q})}^{8} + c_{n - 2}^{6} {(n)}^{2} ({l_{0}^{q})}^{6} l_{1}^{q} - c_{n - 3}^{5} {(n)}^{3} ({l_{0}^{q})}^{5} l_{2}^{q} + c_{n - 4}^{4} {(n)}^{4} ({l_{0}^{q})}^{4} l_{3}^{q} {- c}_{n - 5}^{3} {(n)}^{5} ({l_{0}^{q})}^{3} l_{4}^{q}

{+ c_{n - 6}^{2} {(n)}^{6} ({l_{0}^{q})}^{2} l_{5}^{q} - c}_{n - 7}^{1} {(n)}^{7} ({l_{0}^{q})}^{1} l_{6}^{q} + n^{8} l_{7}^{q}

= - c_{7}^{1} c_{n}^{8} ({l_{0}^{q})}^{8} + c_{n - 2}^{6} {(n)}^{2} ({l_{0}^{q})}^{6} l_{1}^{q} - c_{n - 3}^{5} {(n)}^{3} ({l_{0}^{q})}^{5} l_{2}^{q} + c_{n - 4}^{4} {(n)}^{4} ({l_{0}^{q})}^{4} l_{3}^{q} {- c}_{n - 5}^{3} {(n)}^{5} ({l_{0}^{q})}^{3} l_{4}^{q}

{+ c_{n - 6}^{2} {(n)}^{6} ({l_{0}^{q})}^{2} l_{5}^{q} - c}_{n - 7}^{1} {(n)}^{7} ({l_{0}^{q})}^{1} l_{6}^{q} + n^{8} l_{7}^{q}

s_{7}^{r} = - c_{7}^{1} c_{n}^{8} ({l_{0}^{q})}^{8} + c_{n - 2}^{6} {(n)}^{2} ({l_{0}^{q})}^{6} l_{1}^{q} - c_{n - 3}^{5} {(n)}^{3} ({l_{0}^{q})}^{5} l_{2}^{q} + c_{n - 4}^{4} {(n)}^{4} ({l_{0}^{q})}^{4} l_{3}^{q} {- c}_{n - 5}^{3} {(n)}^{5} ({l_{0}^{q})}^{3} l_{4}^{q} + c_{n - 6}^{2} {(n)}^{6} ({l_{0}^{q})}^{2} l_{5}^{q}

{- c}_{n - 7}^{1} {(n)}^{7} ({l_{0}^{q})}^{1} l_{6}^{q} + n^{8} l_{7}^{q}

(29)

s_{7}^{r} = - c_{7}^{1} c_{8}^{8} ({16)}^{8} + c_{6}^{6} {(8)}^{2} ({16)}^{6} (\frac{7204}{8^{2}}) - {c_{5}^{5} (8)}^{3} {(16)}^{5} (\frac{232832}{8^{3}}) {+ c_{4}^{4} (8)}^{4} {16)}^{4} (\frac{4726206}{8^{4}}) {- c_{3}^{3} (8)}^{5} {16)}^{3} (\frac{61697920}{8^{5}})

+ {c_{2}^{2} (8)}^{6} {16)}^{2} (\frac{505838740}{8^{6}}) - c_{1}^{1} (8)^{7} {(16)}^{1} (\frac{2381353600}{8^{7}}) + (8)^{8} (\frac{4928741769}{8^{8}}) = 3465

(30)

In other words

s_{0}^{r}

s_{2}^{r}

s_{4}^{r}

s_{6}^{r} = 0

Except

s_{1}^{r} = 36 s_{3}^{r} = 446 s_{5}^{r} = 2196 s_{7}^{r} = 3465

P (x) = x^{8} - s_{0}^{r} x^{7} + s_{1}^{r} x^{6} - s_{2}^{r} x^{5} + s_{3}^{r} x^{4} - s_{4}^{r} x^{3} + {s_{5}^{r} x^{2} - s}_{6}^{r} x + s_{7}^{r} = 0

(31)

{= x}^{8} + s_{1}^{r} x^{6} + s_{3}^{r} x^{4} + s_{5}^{r} x^{2} + s_{7}^{r} = 0

(32)

{= x}^{8} + 36 x^{6} + 446 x^{4} + 2196 x^{2} + 3465 = 0 let x^{2} = X

(33)

{= X}^{4} + 36 X^{3} + 446 X^{2} + 2196 X^{1} + 3465 = 0

(34)

s_{1}^{r} = {- c_{1}^{1} c}_{n}^{2}) ({l_{0}^{q})}^{2} + n^{2} l_{1}^{q}

(35)

s_{1}^{r} = - c_{1}^{1} c_{4}^{2} {(- 36)}^{2} + 4^{2} (446) = - 640

(36)

s_{2}^{r} = c_{2}^{1} c_{n}^{3} {(l_{0}^{q})}^{3} - c_{n - 2}^{1} (n)^{2} {(l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q}

(37)

s_{2}^{r} = c_{2}^{1} c_{4}^{3} {(- 36)}^{3} - c_{2}^{1} ({4)}^{2} {(- 36)}^{1} (446) + 4^{3} (- 2196) = 0

(38)

s_{3}^{r} = - c_{3}^{1} c_{n}^{4} ({l_{0}^{q})}^{4} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

(39)

s_{3}^{r} = - c_{3}^{1} c_{n}^{4} {(- 36)}^{3} + (4)^{2} {(- 36)}^{2} (446) - 4^{3} (- 36) (- 2196) + 4^{4} (3465) = 36864

(40)

\to P (x) = x^{4} + s_{1}^{r} x^{2} + s_{3}^{r} = 0 \to P (x) = x^{4} - 640 x^{2} + 36864 = 0

(41)

s_{1}^{q'} = {- s}_{1}^{q}^{2} + 4 s_{3}^{q} = {- (- 640)}_{}^{}^{2} + 4 \times 36864 = - 262144 \to {- s}_{1}^{q^{'}} = 262144 = 512^{2}

(42)

x^{2} = \frac{{- s}_{1}^{q} + \sqrt{{- s}_{1}^{q^{'}}}}{2} = \frac{640 + \sqrt{512^{2}}}{2} = \frac{640 + 512}{2} = 576 = 24^{2} \to x_{kϵ (0, 1,)} = \pm 24

(43)

α_{kϵ (0, 1,)} = \frac{l_{0}^{q} + x_{kϵ (0, 1)}}{4} = \frac{- 36 \pm 24}{4} = - 3 or - 15

(44)

x^{2} = \frac{{- s}_{1}^{q} - \sqrt{{- s}_{1}^{q^{'}}}}{2} = \frac{640 - \sqrt{512^{2}}}{2} = \frac{640 - 512}{2} = 64 = 8^{2} \to x_{kϵ (0, 1,)} = \pm 8

(45)

α_{kϵ (0, 1,)} = \frac{l_{0}^{q} + x_{kϵ (0, 1)}}{4} = \frac{- 36 \pm 8}{4} = - 7 or - 11

(46)

In other words

P (x) = x^{8} - s_{0}^{r} x^{7} + s_{1}^{r} x^{6} - s_{2}^{r} x^{5} + s_{3}^{r} x^{4} - s_{4}^{r} x^{3} + {s_{5}^{r} x^{2} - s}_{6}^{r} x + s_{7}^{r} = 0

(47)

{= x}^{8} + s_{1}^{r} x^{6} + s_{3}^{r} x^{4} + s_{5}^{r} x^{2} + s_{7}^{r} = 0

(48)

{= x}^{8} + 36 x^{6} + 446 x^{4} + 2196 x^{2} + 3465 = 0 let x^{2} = X

(49)

{= X}^{4} + 36 X^{3} + 446 X^{2} + 2196 X^{1} + 3465 = 0

(50)

x^{2} = X = - 3 or - 7 or - 11 or - 15 \to x = \pm \sqrt{3} i or \pm \sqrt{7} ior or \pm \sqrt{11} i or or \pm \sqrt{15} i

(51)

α_{k} = \frac{l_{0}^{q} + x_{k}}{8} = \frac{l_{0}^{q} \pm \sqrt{X_{k}}}{8} = \frac{16 \pm i \sqrt{3}}{8} or \frac{16 \pm i \sqrt{7}}{8} or \frac{16 \pm i \sqrt{11}}{8} or \frac{16 \pm i \sqrt{15}}{8}

(52)

4.2. Recurrence Approach via EMS’s Theorem: Second Example

Consider the following 8^th polynomial

P(x)=

x^{8} - 16 x^{7} + {\frac{7132}{8^{2}} x}^{6} - \frac{225920}{8^{3}} x^{5} + \frac{4449726}{8^{4}} x^{4} - \frac{55799680}{8^{5}} x^{3} + \frac{435055468}{8^{6}} x^{2} - \frac{1928228224}{8^{7}} x + \frac{3719657865}{8^{8}} = 0

P(x)=

x^{8} - l_{0}^{q} x^{7} + {l_{1}^{q} x}^{6} - l_{2}^{q} x^{5} + l_{3}^{q} x^{4} - l_{4}^{q} x^{3} {+ l_{5}^{q} x^{2} - l}_{6}^{q} x + l_{7}^{q} = 0

(53)

Let’s focus on this polynomial to the power eight. To find out the roots of this polynomial we will determine the values

l_{0}^{q} = 16, l_{1}^{q} = \frac{7132}{8^{2}} l_{2}^{q} = \frac{225920}{8^{3}}, l_{3}^{q} = \frac{4449726}{8^{4}}, l_{4}^{q} = \frac{55799680}{8^{5}} l_{5}^{q} = \frac{435055468}{8^{6}} l_{6}^{q}

\frac{1928228224}{8^{7}} l_{7}^{q}

\frac{3719657865}{8^{8}}

(54)

Using Ezouidi Mourad Sulta n^{'} s Theorem (EMST)

s_{k - 1}^{r} = \sum_{r = 0}^{k} c_{n - r}^{k - r} {(n)}^{r} ({- l_{0}^{q})}^{k - r} l_{r - 1}^{q}

(55)

For k=0 we write

s_{- 1}^{r} = \sum_{r = 0}^{0} c_{n - r}^{- r} {(n)}^{r} ({- l_{0}^{q})}^{- r} l_{r - 1}^{q} = c_{n}^{- 1 + 1} = c_{n}^{0} = 1

(56)

For k=1 we write

s_{0}^{r} = \sum_{r = 0}^{1} c_{n - r}^{1 - r} {(n)}^{r} ({- l_{0}^{q})}^{1 - r} l_{r - 1}^{q} = c_{n}^{1} {(n)}^{0} ({- l_{0}^{q})}^{1} l_{- 1}^{q} + c_{n}^{0} {(n)}^{1} ({- l_{0}^{q})}^{0} l_{0}^{q} = {- c}_{n}^{1} l_{0}^{q} l_{- 1}^{q} + n l_{0}^{q} = (n - n) l_{0}^{q} = 0

(57)

{s_{0}^{r} = - c}_{8}^{1} (16) + 8 (16) = - 128 + 128 = 0

(58)

For k=2 we find

s_{1}^{r} = \sum_{r = 0}^{2} c_{n - r}^{2 - r} {(n)}^{r} ({- l_{0}^{q})}^{2 - r} l_{r - 1}^{q} = c_{n}^{2} {(n)}^{0} ({l_{0}^{q})}^{2} l_{- 1}^{q} - c_{n - 1}^{1} {(n)}^{1} ({l_{0}^{q})}^{1} l_{0}^{q} + n^{2} l_{1}^{q} = (c_{n}^{2} - n c_{n - 1}^{1}) ({l_{0}^{q})}^{2} + n^{2} l_{1}^{q} = {- c_{1}^{1} c}_{n}^{2}) ({l_{0}^{q})}^{2} + n^{2} l_{1}^{q}

(59)

{s_{1}^{r} = - c_{1}^{1} c}_{8}^{2} ({16)}^{2} + 8^{2} (\frac{7132}{8^{2}}) = - 36

(60)

For k=3 we evaluate

s_{2}^{r} = \sum_{r = 0}^{3} c_{n - r}^{3 - r} {(n)}^{r} ({- l_{0}^{q})}^{3 - r} l_{r - 1}^{q} = {- c}_{n}^{3} {(n)}^{0} ({l_{0}^{q})}^{3} l_{- 1}^{q} + c_{n - 1}^{2} {(n)}^{1} ({l_{0}^{q})}^{2} l_{0}^{q} - c_{n - 2}^{1} {(n)}^{2} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q}

{= - c}_{n}^{3} ({l_{0}^{q})}^{3} + {nc}_{n - 1}^{2} ({l_{0}^{q})}^{3} - n^{2} c_{n - 2}^{1} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q} = {(- c}_{n}^{3} + {nc}_{n - 1}^{2}) ({l_{0}^{q})}^{3} - n^{2} c_{n - 2}^{1} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q}

= c_{n}^{2} c_{n}^{3} ({l_{0}^{q})}^{3} - n^{2} c_{n - 2}^{1} ({l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q}

(61)

s_{2}^{r} = c_{2}^{1} c_{8}^{3} ({16)}^{3} - c_{6}^{1} {(8)}^{2} ({16)}^{1} (\frac{7132}{8^{2}}) + 8^{3} (\frac{225920}{8^{3}}) = 0

(62)

For k=4 we write

s_{3}^{r} = \sum_{r = 0}^{4} c_{n - r}^{4 - r} {(n)}^{r} ({- l_{0}^{q})}^{4 - r} l_{r - 1}^{q} = c_{n}^{4} {(n)}^{0} ({l_{0}^{q})}^{4} l_{- 1}^{q} - c_{n - 1}^{3} {(n)}^{1} ({l_{0}^{q})}^{3} l_{0}^{q} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

= (c_{n}^{4} - n c_{n - 1}^{3}) ({l_{0}^{q})}^{4} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

= - c_{3}^{1} c_{n}^{3} ({l_{0}^{q})}^{4} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

s_{3}^{r} = - c_{3}^{1} c_{n}^{4} ({l_{0}^{q})}^{4} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

(63)

s_{3}^{r} = - c_{3}^{1} c_{8}^{4} ({16)}^{4} + c_{6}^{2} {(8)}^{2} ({16)}^{2} (\frac{7132}{8^{2}}) - {c_{5}^{1} (8)}^{3} {16)}^{1} (\frac{225920}{8^{3}}) + {(8)}^{4} (\frac{4449726}{8^{4}} = 446

(64)

For k=5 we obtain

s_{4}^{r} = \sum_{r = 0}^{5} c_{n - r}^{5 - r} {(n)}^{r} ({- l_{0}^{q})}^{5 - r} l_{r - 1}^{q} = - c_{n}^{5} {(n)}^{0} ({l_{0}^{q})}^{5} l_{- 1}^{q} + c_{n - 1}^{4} {(n)}^{1} ({l_{0}^{q})}^{4} l_{0}^{q} - c_{n - 2}^{3} {(n)}^{2} ({l_{0}^{q})}^{3} l_{1}^{q} + c_{n - 3}^{2} {(n)}^{3} ({l_{0}^{q})}^{2} l_{2}^{q}

- c_{n - 3}^{1} {(n)}^{4} ({l_{0}^{q})}^{1} l_{3}^{q} + n^{5} l_{4}^{q}

= (- c_{n}^{5} + n c_{n - 1}^{4}) ({l_{0}^{q})}^{5} - c_{n - 2}^{3} {(n)}^{2} ({l_{0}^{q})}^{3} l_{1}^{q} + c_{n - 3}^{2} {(n)}^{3} ({l_{0}^{q})}^{2} l_{2}^{q} - c_{n - 3}^{1} {(n)}^{4} ({l_{0}^{q})}^{1} l_{3}^{q} + n^{5} l_{4}^{q} s_{4}^{r}

= c_{4}^{1} c_{n}^{5} ({l_{0}^{q})}^{5} - c_{n - 2}^{3} {(n)}^{2} ({l_{0}^{q})}^{3} l_{1}^{q} + c_{n - 3}^{2} {(n)}^{3} ({l_{0}^{q})}^{2} l_{2}^{q} - c_{n - 4}^{1} {(n)}^{4} ({l_{0}^{q})}^{1} l_{3}^{q} + n^{5} l_{4}^{q}

(65)

s_{4}^{r} = c_{4}^{1} c_{8}^{5} ({16)}^{5} - c_{6}^{3} {(8)}^{2} ({16)}^{3} (\frac{7132}{8^{2}}) + {c_{5}^{2} (8)}^{3} {(16)}^{2} (\frac{225920}{8^{3}}) {- c_{4}^{1} (8)}^{4} {16)}^{1} (\frac{4449726}{8^{4}}) + (8)^{5} (\frac{55799680}{8^{5}}) = 0

(66)

For k=6 we write

s_{5}^{r} = \sum_{r = 0}^{6} c_{n - r}^{6 - r} {(n)}^{r} ({- l_{0}^{q})}^{6 - r} l_{r - 1}^{q} = c_{n}^{6} {(n)}^{0} ({l_{0}^{q})}^{6} l_{- 1}^{q} - c_{n - 1}^{5} {(n)}^{1} ({l_{0}^{q})}^{5} l_{0}^{q} + c_{n - 2}^{4} {(n)}^{2} ({l_{0}^{q})}^{4} l_{1}^{q} - c_{n - 3}^{3} {(n)}^{3} ({l_{0}^{q})}^{3} l_{2}^{q} + c_{n - 4}^{2} {(n)}^{4} ({l_{0}^{q})}^{2} l_{3}^{q} {- c}_{n - 5}^{1} {(n)}^{5} ({l_{0}^{q})}^{1} l_{4}^{q} + n^{6} l_{5}^{q}

= (c_{n}^{6} - n c_{n - 1}^{5}) ({l_{0}^{q})}^{6} + c_{n - 2}^{4} {(n)}^{2} ({l_{0}^{q})}^{4} l_{1}^{q} - c_{n - 3}^{3} {(n)}^{3} ({l_{0}^{q})}^{3} l_{2}^{q} + c_{n - 4}^{2} {(n)}^{4} ({l_{0}^{q})}^{2} l_{3}^{q} {- c}_{n - 5}^{1} {(n)}^{5} ({l_{0}^{q})}^{1} l_{4}^{q} + n^{6} l_{5}^{q}

= c_{5}^{1} c_{n}^{6}) ({l_{0}^{q})}^{6} + c_{n - 2}^{4} {(n)}^{2} ({l_{0}^{q})}^{4} l_{1}^{q} - c_{n - 3}^{3} {(n)}^{3} ({l_{0}^{q})}^{3} l_{2}^{q} + c_{n - 4}^{2} {(n)}^{4} ({l_{0}^{q})}^{2} l_{3}^{q} {- c}_{n - 5}^{1} {(n)}^{5} ({l_{0}^{q})}^{1} l_{4}^{q} + n^{6} l_{5}^{q}

s_{5}^{r} = {- c}_{5}^{1} c_{n}^{6} ({l_{0}^{q})}^{6} + c_{n - 2}^{4} {(n)}^{2} ({l_{0}^{q})}^{4} l_{1}^{q} - c_{n - 3}^{3} {(n)}^{3} ({l_{0}^{q})}^{3} l_{2}^{q} + c_{n - 4}^{2} {(n)}^{4} ({l_{0}^{q})}^{2} l_{3}^{q} {- c}_{n - 5}^{1} {(n)}^{5} ({l_{0}^{q})}^{1} l_{4}^{q} + n^{6} l_{5}^{q}

(67)

s_{5}^{r} = - c_{5}^{1} c_{8}^{6} ({16)}^{6} + c_{6}^{4} {(8)}^{2} ({16)}^{4} (\frac{7132}{8^{2}}) - {c_{5}^{3} (8)}^{3} {(16)}^{3} (\frac{225920}{8^{3}}) {+ c_{4}^{2} (8)}^{4} {16)}^{2} (\frac{4449726}{8^{4}}) {- c_{3}^{1} (8)}^{5} {16)}^{1} (\frac{55799680}{8^{5}})

+ (8)^{6} (\frac{435055468}{8^{6}}) = - 2196

(68)

For k=7 we write

s_{6}^{r} = \sum_{r = 0}^{7} c_{n - r}^{7 - r} {(n)}^{r} ({- l_{0}^{q})}^{7 - r} l_{r - 1}^{q} = {- c}_{n}^{7} {(n)}^{0} ({l_{0}^{q})}^{7} l_{- 1}^{q} + c_{n - 1}^{6} {(n)}^{1} ({l_{0}^{q})}^{6} l_{0}^{q} - c_{n - 2}^{5} {(n)}^{2} ({l_{0}^{q})}^{5} l_{1}^{q} + c_{n - 3}^{4} {(n)}^{3} ({l_{0}^{q})}^{4} l_{2}^{q} - c_{n - 4}^{3} {(n)}^{4} ({l_{0}^{q})}^{3} l_{3}^{q} {+ c}_{n - 5}^{2} {(n)}^{5} ({l_{0}^{q})}^{2} l_{4}^{q} {- c}_{n - 6}^{1} {(n)}^{6} ({l_{0}^{q})}^{1} l_{5}^{q} + n^{7} l_{6}^{q}

= {(- c}_{n}^{7} + n c_{n - 1}^{6}) ({l_{0}^{q})}^{7} l_{- 1}^{q} - c_{n - 2}^{5} {(n)}^{2} ({l_{0}^{q})}^{5} l_{1}^{q} + c_{n - 3}^{4} {(n)}^{3} ({l_{0}^{q})}^{4} l_{2}^{q} - c_{n - 4}^{3} {(n)}^{4} ({l_{0}^{q})}^{3} l_{3}^{q} {+ c}_{n - 5}^{2} {(n)}^{5} ({l_{0}^{q})}^{2} l_{4}^{q}

{- c}_{n - 6}^{1} {(n)}^{6} ({l_{0}^{q})}^{1} l_{5}^{q} + n^{7} l_{6}^{q}

= c_{6}^{1} c_{n}^{7} ({l_{0}^{q})}^{7} - c_{n - 2}^{5} {(n)}^{2} ({l_{0}^{q})}^{5} l_{1}^{q} + c_{n - 3}^{4} {(n)}^{3} ({l_{0}^{q})}^{4} l_{2}^{q} - c_{n - 4}^{3} {(n)}^{4} ({l_{0}^{q})}^{3} l_{3}^{q} {+ c}_{n - 5}^{2} {(n)}^{5} ({l_{0}^{q})}^{2} l_{4}^{q}

{- c}_{n - 6}^{1} {(n)}^{6} ({l_{0}^{q})}^{1} l_{5}^{q} + n^{7} l_{6}^{q}

s_{6}^{r} = c_{6}^{1} c_{n}^{7} ({l_{0}^{q})}^{7} - c_{n - 2}^{5} {(n)}^{2} ({l_{0}^{q})}^{5} l_{1}^{q} + c_{n - 3}^{4} {(n)}^{3} ({l_{0}^{q})}^{4} l_{2}^{q} - c_{n - 4}^{3} {(n)}^{4} ({l_{0}^{q})}^{3} l_{3}^{q} {+ c}_{n - 5}^{2} {(n)}^{5} ({l_{0}^{q})}^{2} l_{4}^{q} {- c}_{n - 6}^{1} {(n)}^{6} ({l_{0}^{q})}^{1} l_{5}^{q} + n^{7} l_{6}^{q}

(69)

s_{6}^{r} = c_{6}^{1} c_{8}^{7} ({16)}^{7} - c_{6}^{5} {(8)}^{2} ({16)}^{5} (\frac{7132}{8^{2}}) + {c_{5}^{4} (8)}^{3} {(16)}^{4} (\frac{225920}{8^{3}}) {- c_{4}^{3} (8)}^{4} {16)}^{3} (\frac{4449726}{8^{4}}) {+ c_{3}^{2} (8)}^{5} {16)}^{2} (\frac{55799680}{8^{5}})

- {c_{2}^{1} (8)}^{6} {16)}^{1} (\frac{435055468}{8^{6}}) + (8)^{7} (\frac{1928228224}{8^{7}}) = 0

(70)

For k=8 we write

s_{7}^{r} = \sum_{r = 0}^{8} c_{n - r}^{8 - r} ({(n)}^{r} {- l_{0}^{q})}^{8 - r} l_{r - 1}^{q} = c_{n}^{8} {(n)}^{0} ({l_{0}^{q})}^{8} l_{- 1}^{q} - c_{n - 1}^{7} {(n)}^{1} ({l_{0}^{q})}^{7} l_{0}^{q} + c_{n - 2}^{6} {(n)}^{2} ({l_{0}^{q})}^{6} l_{1}^{q} - c_{n - 3}^{5} {(n)}^{3} ({l_{0}^{q})}^{5} l_{2}^{q} + c_{n - 4}^{4} {(n)}^{4} ({l_{0}^{q})}^{4} l_{3}^{q}

{- c}_{n - 5}^{3} {(n)}^{5} ({l_{0}^{q})}^{3} l_{4}^{q} {+ c_{n - 6}^{2} {(n)}^{6} ({l_{0}^{q})}^{2} l_{5}^{q} - c}_{n - 7}^{1} {(n)}^{7} ({l_{0}^{q})}^{1} l_{6}^{q} + n^{8} l_{7}^{q}

= (c_{n}^{8} - n c_{n - 1}^{7}) ({l_{0}^{q})}^{8} + c_{n - 2}^{6} {(n)}^{2} ({l_{0}^{q})}^{6} l_{1}^{q} - c_{n - 3}^{5} {(n)}^{3} ({l_{0}^{q})}^{5} l_{2}^{q} + c_{n - 4}^{4} {(n)}^{4} ({l_{0}^{q})}^{4} l_{3}^{q} {- c}_{n - 5}^{3} {(n)}^{5} ({l_{0}^{q})}^{3} l_{4}^{q}

{+ c_{n - 6}^{2} {(n)}^{6} ({l_{0}^{q})}^{2} l_{5}^{q} - c}_{n - 7}^{1} {(n)}^{7} ({l_{0}^{q})}^{1} l_{6}^{q} + n^{8} l_{7}^{q}

= - c_{7}^{1} c_{n}^{8} ({l_{0}^{q})}^{8} + c_{n - 2}^{6} {(n)}^{2} ({l_{0}^{q})}^{6} l_{1}^{q} - c_{n - 3}^{5} {(n)}^{3} ({l_{0}^{q})}^{5} l_{2}^{q} + c_{n - 4}^{4} {(n)}^{4} ({l_{0}^{q})}^{4} l_{3}^{q} {- c}_{n - 5}^{3} {(n)}^{5} ({l_{0}^{q})}^{3} l_{4}^{q}

{+ c_{n - 6}^{2} {(n)}^{6} ({l_{0}^{q})}^{2} l_{5}^{q} - c}_{n - 7}^{1} {(n)}^{7} ({l_{0}^{q})}^{1} l_{6}^{q} + n^{8} l_{7}^{q}

s_{7}^{r} = - c_{7}^{1} c_{n}^{8} ({l_{0}^{q})}^{8} + c_{n - 2}^{6} {(n)}^{2} ({l_{0}^{q})}^{6} l_{1}^{q} - c_{n - 3}^{5} {(n)}^{3} ({l_{0}^{q})}^{5} l_{2}^{q} + c_{n - 4}^{4} {(n)}^{4} ({l_{0}^{q})}^{4} l_{3}^{q} {- c}_{n - 5}^{3} {(n)}^{5} ({l_{0}^{q})}^{3} l_{4}^{q} + c_{n - 6}^{2} {(n)}^{6} ({l_{0}^{q})}^{2} l_{5}^{q}

{- c}_{n - 7}^{1} {(n)}^{7} ({l_{0}^{q})}^{1} l_{6}^{q} + n^{8} l_{7}^{q}

(71)

s_{7}^{r} = - c_{7}^{1} c_{8}^{8} ({16)}^{8} + c_{6}^{6} {(8)}^{2} ({16)}^{6} (\frac{7132}{8^{2}}) - {c_{5}^{5} (8)}^{3} {(16)}^{5} (\frac{225920}{8^{3}}) {+ c_{4}^{4} (8)}^{4} {16)}^{4} (\frac{4449726}{8^{4}}) {- c_{3}^{3} (8)}^{5} {16)}^{3} (\frac{55799680}{8^{5}})

+ {c_{2}^{2} (8)}^{6} {16)}^{2} (\frac{435055468}{8^{6}}) - c_{1}^{1} (8)^{7} {(16)}^{1} (\frac{1928228224}{8^{7}}) + (8)^{8} (\frac{3719657865}{8^{8}}) = 3465

(72)

In other words

s_{0}^{r}

s_{2}^{r}

s_{4}^{r}

s_{6}^{r} = 0

Except

s_{1}^{r} = - 36 s_{3}^{r} = 446 s_{5}^{r} = - 2196 s_{7}^{r} = 3465

P (x) = x^{8} - s_{0}^{r} x^{7} + s_{1}^{r} x^{6} - s_{2}^{r} x^{5} + s_{3}^{r} x^{4} - s_{4}^{r} x^{3} + s_{5}^{r} x^{2} - s_{6}^{r} x + s_{7}^{r} = 0

(73)

{= x}^{8} + s_{1}^{r} x^{6} + s_{3}^{r} x^{4} + s_{5}^{r} x^{2} + s_{7}^{r} = 0

(74)

{= x}^{8} - 36 x^{6} + 446 x^{4} - 2196 x^{2} + 3465 = 0 let x^{2} = X

(75)

{= X}^{4} - 36 X^{3} + 446 X^{2} - 2196 X^{1} + 3465 = 0

(76)

s_{1}^{r} = {- c_{1}^{1} c}_{n}^{2}) ({l_{0}^{q})}^{2} + n^{2} l_{1}^{q}

(77)

s_{1}^{r} = - c_{1}^{1} c_{4}^{2} {(36)}^{2} + 4^{2} (446) = - 640

(78)

s_{2}^{r} = c_{2}^{1} c_{n}^{3} {(l_{0}^{q})}^{3} - c_{n - 2}^{1} (n)^{2} {(l_{0}^{q})}^{1} l_{1}^{q} + n^{3} l_{2}^{q}

(79)

s_{2}^{r} = c_{2}^{1} c_{4}^{3} {(36)}^{3} - c_{2}^{1} ({4)}^{2} {(36)}^{1} (446) + 4^{3} (2196) = 0

(80)

s_{3}^{r} = - c_{3}^{1} c_{n}^{4} ({l_{0}^{q})}^{4} + c_{n - 2}^{2} {(n)}^{2} ({l_{0}^{q})}^{2} l_{1}^{q} - c_{n - 3}^{1} {(n)}^{3} ({l_{0}^{q})}^{1} l_{2}^{q} + n^{4} l_{3}^{q}

(81)

s_{3}^{r} = - c_{3}^{1} c_{n}^{4} {(36)}^{4} + (4)^{2} {(36)}^{2} (446) - 4^{3} (36) (2196) + 4^{4} (3465) = 36864

(82)

\to P (x) = x^{4} + s_{1}^{r} x^{2} + s_{3}^{r} = 0 \to P (x) = x^{4} - 640 x^{2} + 36864 = 0

(83)

s_{1}^{q'} = {- s}_{1}^{q}^{2} + 4 s_{3}^{q} = {- (- 640)}_{}^{}^{2} + 4 \times 36864 = - 262144 \to {- s}_{1}^{q^{'}} = 262144 = 512^{2}

(84)

x^{2} = \frac{{- s}_{1}^{q} + \sqrt{{- s}_{1}^{q^{'}}}}{2} = \frac{640 + \sqrt{512^{2}}}{2} = \frac{640 + 512}{2} = 576 = 24^{2} \to x_{kϵ (0, 1,)} = \pm 24

(85)

α_{kϵ (0, 1,)} = \frac{l_{0}^{q} + x_{kϵ (0, 1)}}{4} = \frac{36 \pm 24}{4} = 3 or 15

(86)

x^{2} = \frac{{- s}_{1}^{q} - \sqrt{{- s}_{1}^{q^{'}}}}{2} = \frac{640 - \sqrt{512^{2}}}{2} = \frac{640 - 512}{2} = 64 = 8^{2} \to x_{kϵ (0, 1,)} = \pm 8

(87)

α_{kϵ (0, 1,)} = \frac{l_{0}^{q} + x_{kϵ (0, 1)}}{4} = \frac{36 \pm 8}{4} = 7 or 11

(88)

x^{2} = X = 3 or 7 or 11 or 15 \to x = \pm \sqrt{3} or \pm \sqrt{7} or or \pm \sqrt{11} or or \pm \sqrt{15}

(89)

α_{k} = \frac{l_{0}^{q} + y_{k}}{8} = \frac{l_{0}^{q} \pm \sqrt{x_{k}}}{8} = \frac{16 \pm \sqrt{3}}{8} or \frac{16 \pm \sqrt{7}}{8} or \frac{16 \pm \sqrt{11}}{8} or \frac{16 \pm \sqrt{15}}{8}

(90)

Table 1. Cardano’s, Newton’s, and EMST Methods for n-th Degree Polynomial solutions.

$n^{th}$ degree	Polynomial equations	Cardano's method	Newton's method	Present work Ezouidi Mourad Sultan’s Theorem (EMST)
3	${P (x) = x}^{3} + 86.4 x + 525.312 = 0$	$x_{1}$ ≈−4.7669 $x_{2}$ ≈2.3835+10.1570i,; $x_{3}$ ≈2.3835−10.1570i	$x_{1} \approx -$ 4.8018, $x_{2} \approx$ 2.4009+10.202i $x_{3} \approx$ 2.4009 $-$ 10.202i	$4.8 \pm 6 \sqrt{2.88} i$ or -4.8
5	${P (x) = x}^{5} - 12 x^{4} + \frac{1440}{5^{2}} x^{3} - \frac{17280}{5^{3}} x^{2} + \frac{103936}{5^{4}} x^{1} - \frac{251904}{5^{5}} = 0$	Not applicable	Not applicable	$\frac{12 + \sqrt[4]{256} e^{\frac{(2 k + 1) iπ}{5}}}{5} = \frac{12 + 4 e^{\frac{(2 k + 1) iπ}{5}}}{5}$
6	${P (x) = x}^{6} - x^{5} + \frac{15}{6^{2}} x^{4} - x^{3} + \frac{603}{6^{4}} x^{2} - \frac{624}{6^{5}} x + \frac{7916}{6^{6}} = 0$	Not applicable	Not applicable
7	${P (x) = x}^{7} - x^{6} + \frac{18}{8^{2}} x^{5} + \frac{4}{8^{3}} x^{4} - \frac{45}{8^{4}} x^{3} + \frac{4}{8^{5}} x^{2} + \frac{38}{8^{6}} x + \frac{12}{8^{7}} = 0$	Not applicable	Not applicable	$\frac{1 + 1}{8}; \frac{1 \pm 2}{8}; \frac{1 \pm \sqrt{2}}{8}; \frac{1 \pm \sqrt{3}}{8}$
8	${P (x) = x}^{8} - 16 x^{7} + {\frac{7204}{8^{2}} x}^{6} - \frac{232832}{8^{3}} x^{5} + \frac{4726206}{8^{4}} x^{4} - \frac{61697920}{8^{5}} x^{3} + \frac{505838740}{8^{6}} x^{2} - \frac{2381353600}{8^{7}} x + \frac{4928741769}{8^{8}} = 0$	Not applicable	Not applicable	$\frac{16 \pm i \sqrt{3}}{8} or \frac{16 \pm i \sqrt{7}}{8} or \frac{16 \pm i \sqrt{11}}{8}$ $or \frac{16 \pm i \sqrt{15}}{8}$
8	$P (x) = x^{8} - 16 x^{7} + {\frac{7132}{8^{2}} x}^{6} - \frac{225920}{8^{3}} x^{5} + \frac{4449726}{8^{4}} x^{4} - \frac{55799680}{8^{5}} x^{3} + \frac{435055468}{8^{6}} x^{2} - \frac{1928228224}{8^{7}} x + \frac{3719657865}{8^{8}} = 0$	Not applicable	Not applicable	$\frac{16 \pm \sqrt{3}}{8} or \frac{16 \pm \sqrt{7}}{8} or \frac{16 \pm \sqrt{11}}{8}$ $or \frac{16 \pm \sqrt{15}}{8}$

Table 1 shows that the solutions obtained using the proposed recurrence method grounded in Mourad Sultan Ezouidi’s Theorem (EMST) correspond to explicit, exact roots for high-degree polynomials, including complex and irrational roots. In contrast, traditional methods such as Cardano’s method and Newton’s method are either inapplicable or yield only approximate solutions. For example, for the eighth-degree polynomial equations considered, the EMST approach provides precise roots expressed in radical and complex form, such as

\frac{16 \pm i \sqrt{3}}{8} or \frac{16 \pm i \sqrt{7}}{8} or \frac{16 \pm i \sqrt{11}}{8} or \frac{16 \pm i \sqrt{15}}{8}

. Conversely, classical methods do not furnish explicit roots for these high-degree polynomials, highlighting their limitations in handling such complex cases.

This comparison underscores that EMST offers a systematic way to compute exact solutions for polynomial roots of any degree, including irreducible and highly complex cases. Traditional techniques, on the other hand, are often restricted to low-degree or specific polynomial forms, and tend to be impractical or inapplicable for higher degrees. The results demonstrate the superior applicability, precision, and completeness of the EMST-based recurrence method in solving polynomial equations explicitly, representing a significant advancement over existing approaches.

Table 2. Root-finding methodology comparison: Mean Absolute Error (MAE) analysis for EMST versus classical methods.

Roots

${P (x) = x}^{3} + 86.4 x + 525.312 = 0$

$MAE$

EMST

$2.4 \pm 6 \sqrt{2.88} i or - 4.8$

$P (- 4.8) = 0$

$\{\begin{matrix} EMST vs Cardano : \\ MAE = 33.73 % \end{matrix}$

$\{\begin{matrix} EMST vs Newton : \\ MAE = 28 % \end{matrix}$

Cardano

x1≈−4.7669, x2 ≈2.3835+10.1570i, x3 ≈2.3835−10.1570i

$P (- 4.802168336) = - 0.3372873291$

Newton

x1 ≈−4.8018, x2,3 ≈2.4009±10.202i

$P (- 4.8018) = - 0.2799826618$

Table 2 presents a comparative error evaluation of three root-finding methodologies—Ezouidi Mourad Sultan’s Theorem (EMST), Cardano’s method, and Newton’s method—applied to the cubic polynomial

{P (x) = x}^{3} + 86.4 x + 525.312 = 0

. The table reports the roots obtained by each method, as well as the Mean Absolute Error (MAE) associated with each approach.

The EMST method yields the exact roots, specifically

x = - 4.8

(real root) and

x = 2.4 \pm 6 \sqrt{2.88} i

(complex roots), with the polynomial evaluated at

x = - 4.8

resulting in zero, confirming the method’s exactness. In contrast, Cardano’s method provides approximate roots (x1≈−4.7669, x2 ≈2.3835+10.1570i, x3 ≈2.3835−10.1570i, and evaluation of the polynomial at x1 produces a small residual (P(−4.802)≈−0.337), indicating a slight deviation from the exact solution.

Similarly, Newton’s method yields an approximate real root (x≈−4.8018), with a corresponding polynomial value of approximately −0.280-−0.280, again reflecting a minor error relative to the exact root.

The Mean Absolute Error between EMST and Cardano’s method is calculated as 33.73%, while the MAE between EMST and Newton’s method is 28%. These results quantitatively demonstrate the superior accuracy and exactness of the EMST approach in determining the roots of the cubic polynomial, whereas classical methods introduce measurable errors due to their reliance on approximation or iterative procedures.

Overall, the table highlights the ability of the EMST recurrence method to deliver exact solutions where classical methods yield only approximate results, as evidenced by the lower (zero) error at the EMST root and the higher MAE values observed when comparing EMST to Cardano and Newton approaches.

5. Conclusion

The recurrence-based methodology introduced in this paper, embodied by the Ezouidi Mourad Sultan’s Theorem, marks a significant advancement in the theory and practice of solving higher-degree polynomial equations. By leveraging recursive relationships among the coefficients, the EMST framework delivers exact solutions for all types of roots and for polynomials of any degree, thereby overcoming the fundamental limitations of classical and numerical methods.

Our detailed examples and comparisons illustrate that this unified approach not only generalizes established techniques but also provides new analytical tools for modern applications in mathematics and engineering. The recurrence structure at the heart of EMST offers a scalable and robust pathway to explicit solutions, opening new avenues for research and teaching in algebraic problem-solving.

With its capacity for both generality and precision, the recurrence approach via EMS’s Theorem stands poised to reshape how mathematicians and practitioners address the enduring challenge of polynomial root-finding.

Abbreviations

EMST	Ezouidi Mourad Sultan's Theorem
MAE	Mean Absolute Error

Author Contributions

Mourad Sultan Ezouidi: Conceptualization, Methodology, Formal Analysis, Writing – original draft, Writing – review & editing

Taoufik Gassoumi: Formal Analysis, Supervision, Validation, Writing – review & editing

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	I. M. Isaacs, Roots of Polynomials in Algebraic Extensions of Fields. The American Mathematical Monthly, 87(7) (1980) 543–544. https://doi.org/10.1080/00029890.1980.11995085
[2]	L. L. Pennisi, A Method for Finding the Real Roots of Cubic Equations by Using the Slide Rule. Mathematics Magazine, 31(4) (1958) 211–214. https://doi.org/10.1080/19300980.1958.12467565
[3]	D. Redmond, Finding Rational Roots of Polynomials. The College Mathematics Journal, 20(2) (1989) 139–141. https://doi.org/10.1080/07468342.1989.11973222
[4]	A. Borzì, X. Chen, H. J. Motwani, L. Venturello, M. Vodička, The leading coefficient of Lascoux polynomials, Discrete Mathematics, 346 (2) (2023), 113217, https://doi.org/10.1016/j.disc.2022.113217
[5]	S. Steinerberger, Free Convolution Powers Via Roots of Polynomials. Experimental Mathematics, 32(4) (2021) 567–572. https://doi.org/10.1080/10586458.2021.1980751
[6]	C. C. Lee, H. P. Niu, Determination of zeros of polynomials by synthetic division International Journal of Computer Mathematics, 7(2), (1979) 131–139. https://doi.org/10.1080/00207167908803164
[7]	K. Bryś, Z. Lonc, Polynomial cases of graph decomposition: A complete solution of Holyer’s problem, Discrete Mathematics, 309(6) (2009) 1294-1326, https://doi.org/10.1016/j.disc.2008.01.054
[8]	M. Sezer, A method for the approximate solution of the second‐order linear differential equations in terms of Taylor polynomials. International Journal of Mathematical Education in Science and Technology, 27(6) (1996) 821–834. https://doi.org/10.1080/0020739960270606
[9]	S. Beji†, Investigations on cubic polynomials. International Journal of Mathematical Education in Science and Technology, 23(2) (1992) 167–173. https://doi.org/10.1080/0020739920230201
[10]	G. B. Costa, L. E. Levine, Polynomial solutions of certain classes of ordinary differential equations. International Journal of Mathematical Education in Science and Technology, 20(1) (1989) 1–11. https://doi.org/10.1080/0020739890200101
[11]	M. Sezer, M. Kaynak, Chebyshev polynomial solutions of linear differential equations. International Journal of Mathematical Education in Science and Technology, 27(4) (1996) 607–618. https://doi.org/10.1080/0020739960270414
[12]	J. Gal‐Ezer, G. Zwas, Computational aspects of rational versus polynomial interpolations. International Journal of Mathematical Education in Science and Technology, 19(4) (1988) 567–579. https://doi.org/10.1080/0020739880190408
[13]	D. A. Wolfram, Solving change of basis from Bernstein to Chebyshev polynomials, Examples and Counterexamples, 7 (2025) 100178, https://doi.org/10.1016/j.exco.2025.100178.
[14]	L. E. Levine, R. Maleh, Polynomial solutions of the classical equations of Hermite, Legendre, and Chebyshev. International Journal of Mathematical Education in Science and Technology, 34(1) (2003) 95–103. https://doi.org/10.1080/0020739021000053891
[15]	C. Henriksen, Roots of polynomial sequences in root-sparse regions January (2025) https://doi.org/10.48550/arXiv.2501.05203
[16]	M. Ito, Generalized Schröder paths arising from a combinatorial interpretation of generalized Laurent bi-orthogonal polynomials, Discrete Mathematics, 348 (1) (2025) 114230 https://doi.org/10.1016/j.disc.2024.114230
[17]	T. Wang, B. Wu, Induced subgraphs of a tree with constraint degree, Applied Mathematics and Computation, 440 (2023) 127657, https://doi.org/10.1016/j.amc.2022.127657
[18]	T. Gassoumi, U. F. Alqsair, R. Said, Enhanced discrete ordinates approach for mitigating shadowing effects in obstructed gas-filled spaces: Implications for astrophysical and industrial applications, Results in Engineering, 23 (2024) 102728, https://doi.org/10.1016/j.rineng.2024.102728
[19]	R. C. Mittal, R. Bhatia, Numerical solution of nonlinear system of Klein–Gordon equations by cubic B-spline collocation method. International Journal of Computer Mathematics, 92(10), (2014) 2139–2159. https://doi.org/10.1080/00207160.2014.970182
[20]	P. Cheewaphutthisakun, J. Shiraishi, K. Wiboonton, Quantum Corner Polynomials: A Generalization of Super Macdonald Polynomials and Their VOA Correspondence August (2025) https://doi.org/10.48550/arXiv.2508.12267
[21]	B. Heim, M. Neuhauser, R. Tröger, Zeros of recursively defined polynomials. Journal of Difference Equations and Applications, 26(4) (2020) 510–531. https://doi.org/10.1080/10236198.2020.1748022
[22]	C. S. Withers, S. Nadarajah, β-reciprocal polynomials. International Journal of Mathematical Education in Science and Technology, 47(5) (2015) 762–766. https://doi.org/10.1080/0020739X.2015.1112043
[23]	H. Wiggins, Sibling curves of cubic polynomials. Quaestiones Mathematicae, 43(8) (2019) 1083–1090. https://doi.org/10.2989/16073606.2019.1600595
[24]	S. P. Gordon, ON SYMMETRIES OF POLYNOMIALS. PRIMUS, 9(1) (1999) 13–20. https://doi.org/10.1080/10511979908965912
[25]	U. Duran, M. Acikgoz, S. Araci, New Families of Certain Special Polynomials: A Kaniadakis Calculus Viewpoint. Symmetry. 17(9) (2025) 1534. https://doi.org/10.3390/sym17091534
[26]	F. Taşdemir, V. Ş. Durusoy, Gaussian Chebyshev Polynomials and Their Properties Symmetry July (2025) 17(7): 1040 https://doi.org/10.3390/sym17071040
[27]	G. Károlyi, A compactness argument in the additive theory and the polynomial method, Discrete Mathematics, 302(1–3) (2005) 124-144, https://doi.org/10.1016/j.disc.2004.07.030
[28]	M. Hadjiat, J. F. Maurras, A strongly polynomial algorithm for the minimum cost tension problem, Discrete Mathematics, 165–166 (1997) 377-394, https://doi.org/10.1016/S0012-365X(96)00185-9
[29]	I. Lamiri, J. Weslati, Limit relations involving 2-orthogonal polynomials. Integral Transforms and Special Functions, 32(5–8) (2021) 545–559. https://doi.org/10.1080/10652469.2020.1798948
[30]	P. Tiwari, A. K. Pathak, ‘Incomplete’ Pál type interpolation problems on zeros of polynomials with complex coefficients, Examples and Counterexamples, 5 (2024) 100132, https://doi.org/10.1016/j.exco.2023.100132
[31]	S. A. Bhanotar, Exploration of novel analytical solutions of boundary layer equation via the modified sumudu transform, Examples and Counterexamples, 5 (2024) 100140, https://doi.org/10.1016/j.exco.2024.100140
[32]	C. V. Valencia-Negrete, Example of a solution for Dorodnitzyn’s limit formula, Examples and Counterexamples, 3 (2023) 100114 https://doi.org/10.1016/j.exco.2023.100114
[33]	A. Kyriakoussis, M. G. Vamvakari, Generalization of matching extensions in graphs—combinatorial interpretation of orthogonal and q-orthogonal polynomials, Discrete Mathematics, 296 2–3 (2005) 199-209 https://doi.org/10.1016/j.disc.2005.02.013
[34]	A. Kyriakoussis, M. G. Vamvakari, Asymptotic normality of the coefficients of polynomials related to the classical system orthogonal ones, Discrete Mathematics, 205 (1–3) (1999) 145-169 https://doi.org/10.1016/S0012-365X(98)00290-8
[35]	Y. Zhou, Y. Zhong, A combinatorial proof of the parity unimodality of the (m,n)-rational q-Catalan polynomial for m=3, Applied Mathematics and Computation, 440 (2023) 127616 https://doi.org/10.1016/j.amc.2022.127616
[36]	A. Leriche, Cubic, quartic and sextic Pólya fields, Journal of Number Theory, 133 (1) (2013) 59-71, https://doi.org/10.1016/j.jnt.2012.06.016
[37]	S. P. Trofimov, Optimization of complex functions and the algorithm for exact geometric search for complex roots of a polynomial, IFAC-PapersOnLine, 51 (32) (2018) 883-888, https://doi.org/10.1016/j.ifacol.2018.11.433
[38]	S. Monnet, Counting wild quartics with prescribed discriminant and Galois closure group, Journal of Number Theory, 269 (2025) 157-202, https://doi.org/10.1016/j.jnt.2024.10.008
[39]	W. Yang, C. Wang, Y. Shi, X. Xin, Construction and exact solution of the nonlocal Kuralay-II equation via Darboux transformation, Applied Mathematics Letters, (2025) 109758, https://doi.org/10.1016/j.aml.2025.109758
[40]	L. E. Levine, R. Maleh, Polynomial solutions of N th order non-homogeneous differential equations. International Journal of Mathematical Education in Science and Technology, 33(6), (2002) 898–906. https://doi.org/10.1080/002073902321093577
[41]	L. E. Levine, R. Maleh, Classroom note: Polynomial solutions of the Laguerre equation and other differential equations near a singular point. International Journal of Mathematical Education in Science and Technology, 34(5) (2003) 781–786. https://doi.org/10.1080/00207390310001595447
[42]	T. H. Fay, P. H. Kloppers, The Gibbs phenomenon for series of orthogonal polynomials. International Journal of Mathematical Education in Science and Technology, 37(8) (2006) 973–989. https://doi.org/10.1080/00207390601083617
[43]	R. G. Kulkarni, Solving cubic equations by polynomial decomposition. International Journal of Mathematical Education in Science and Technology, 42(1) (2010) 105–108. https://doi.org/10.1080/0020739X.2010.510220
[44]	H. P. Hirst, W. T. Macey, Bounding the Roots of Polynomials. The College Mathematics Journal, 28(4) (1997) 292–295. https://doi.org/10.1080/07468342.1997.11973878
[45]	N. Tsirivas, A solution of polynomial equations May (2020) https://doi.org/10.48550/arXiv.2005.00650
[46]	Joab R. Winkler, Xin Lao, Madina Hasan, The computation of multiple roots of a polynomial, Journal of Computational and Applied Mathematics, Volume 236, Issue 14, 2012, 3478-3497, https://doi.org/10.1016/j.cam.2012.02.018
[47]	Fiza Zafar, Sofia Iqbal, Tahira Nawaz, A Steffensen type optimal eighth order multiple root finding scheme for nonlinear equations, Journal of Computational Mathematics and Data Science, 7, 2023, 100079, https://doi.org/10.1016/j.jcmds.2023.100079
[48]	Chacha Stephen Chacha, On exact line search method for a polynomial matrix equation, Journal of Computational Mathematics and Data Science, 15, 2025, 100120, https://doi.org/10.1016/j.jcmds.2025.100120
[49]	Emil M. Prodanov, On Newton’s Rule of signs, Journal of Computational Mathematics and Data Science, Volume 6, 2023, 100076, https://doi.org/10.1016/j.jcmds.2023.100076
[50]	William H. Foster, Ilia Krasikov, Inequalities for real-root polynomials and entire functions, Advances in Applied Mathematics, 29, Issue 1, 2002, 102-114, https://doi.org/10.1016/S0196-8858(02)00005-2
[51]	Guoce Xin, Chen Zhang, A polynomial time algorithm for Sylvester waves when entries are bounded, Advances in Applied Mathematics, 170, 2025, 102931, https://doi.org/10.1016/j.aam.2025.102931
[52]	Jun Ma, Kaiying Pan, (M,i)-multiset Eulerian polynomials, Advances in Applied Mathematics, 149, 2023, 102547, https://doi.org/10.1016/j.aam.2023.102547
[53]	Tiago Fonseca, Philippe Nadeau, On some polynomials enumerating Fully Packed Loop configurations, Advances in Applied Mathematics, 47, Issue 3, 2011, 434-462, https://doi.org/10.1016/j.aam.2010.11.003
[54]	Christoph Koutschan, Martin Neumüller, Cristian-Silviu Radu, Inverse inequality estimates with symbolic computation, Advances in Applied Mathematics, 80, 2016, 1-23, https://doi.org/10.1016/j.aam.2016.04.005
[55]	A. Gheffar, S. Abramov, Valuations of rational solutions of linear difference equations at irreducible polynomials, Advances in Applied Mathematics, 47, Issue 2, 2011, 352-364, https://doi.org/10.1016/j.aam.2010.07.002
[56]	Alexander P Morgan, Andrew J Sommese, Charles W Wampler, Computing singular solutions to polynomial systems, Advances in Applied Mathematics, 13, Issue 3, 1992, 305-327, https://doi.org/10.1016/0196-8858(92)90014-N
[57]	A. J. Sommese and C. W. Wampler, II., The Numerical Solution of Systems of Polynomials, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. Arising in engineering and science, https://doi.org/10.1142/5763
[58]	Danish Zaidi, Imran Talib, Muhammad Bilal Riaz, Md. Nur Alam, Extending spectral methods to solve time fractional-order Bloch equations using generalized Laguerre polynomials, Partial Differential Equations in Applied Mathematics, Volume 13, 2025, 101049, https://doi.org/10.1016/j.padiff.2024.101049
[59]	Sandeep Kumar Yadav, Giriraj Methi, Application of fractional differential transform method and Bell polynomial for solving system of fractional delay differential equations, Partial Differential Equations in Applied Mathematics, Volume 12, 2024, 100971, https://doi.org/10.1016/j.padiff.2024.100971
[60]	Imane Zemmouri, Amor Menaceur, Abdelhamid Laouar, Salah Boulaaras, Limit cycles of septic polynomial differential systems bifurcating from the periodic annulus of cubic center, Partial Differential Equations in Applied Mathematics, Volume 9, 2024, 100622, https://doi.org/10.1016/j.padiff.2024.100622
[61]	T. Dangal, B. Khatri Ghimire, A. R. Lamichhane, Localized method of particular solutions using polynomial basis functions for solving two-dimensional nonlinear partial differential equations, Partial Differential Equations in Applied Mathematics, Volume 4, 2021, 100114, https://doi.org/10.1016/j.padiff.2021.100114

Cite This Article

Plain Text BibTeX RIS

APA Style

Ezouidi, M. S., Gassoumi, T. (2026). Advancing Solutions to Higher-Degree Polynomials: A Novel Recurrence Approach via EMS’s Theorem. American Journal of Astronomy and Astrophysics, 13(2), 59-73. https://doi.org/10.11648/j.ajaa.20261302.11

Copy | Download

ACS Style

Ezouidi, M. S.; Gassoumi, T. Advancing Solutions to Higher-Degree Polynomials: A Novel Recurrence Approach via EMS’s Theorem. Am. J. Astron. Astrophys. 2026, 13(2), 59-73. doi: 10.11648/j.ajaa.20261302.11

Copy | Download

AMA Style

Ezouidi MS, Gassoumi T. Advancing Solutions to Higher-Degree Polynomials: A Novel Recurrence Approach via EMS’s Theorem. Am J Astron Astrophys. 2026;13(2):59-73. doi: 10.11648/j.ajaa.20261302.11

Copy | Download

@article{10.11648/j.ajaa.20261302.11,
  author = {Mourad Sultan Ezouidi and Taoufik Gassoumi},
  title = {Advancing Solutions to Higher-Degree Polynomials: 
A Novel Recurrence Approach via EMS’s Theorem},
  journal = {American Journal of Astronomy and Astrophysics},
  volume = {13},
  number = {2},
  pages = {59-73},
  doi = {10.11648/j.ajaa.20261302.11},
  url = {https://doi.org/10.11648/j.ajaa.20261302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajaa.20261302.11},
  abstract = {Solving higher-degree polynomial equations remains a fundamental challenge in both pure and applied mathematics. While quadratic, cubic, and quartic equations have known algebraic solutions, no general radical solution exists for degree five and higher (Abel–Ruffini theorem). This work introduces a novel recurrence-based methodology for deriving exact roots of polynomials of arbitrary degree, based on Ezouidi Mourad Sultan's Theorem (EMST). Unlike traditional algebraic techniques that are often restricted to degrees four or less or rely on numerical approximations, this framework allows for the explicit determination of roots, including irrational, complex, and multiple roots, across any polynomial degree. By systematically leveraging the structure of polynomial coefficients through recursive relationships, this approach extends the capabilities of classical methods and enhances their precision. The method is demonstrated through comprehensive examples involving irreducible and high-degree polynomials of degree 8, producing exact roots in closed form. Comparative analyses with established techniques such as Cardano's method, Newton's Method, and the Rational Root Theorem highlight the advantages of this recurrence formulation, including exactness, no reliance on initial guesses, and applicability to any degree. The EMST-based methodology offers a unified pathway toward exact solutions for longstanding algebraic problems.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Advancing Solutions to Higher-Degree Polynomials: 
A Novel Recurrence Approach via EMS’s Theorem
AU  - Mourad Sultan Ezouidi
AU  - Taoufik Gassoumi
Y1  - 2026/04/29
PY  - 2026
N1  - https://doi.org/10.11648/j.ajaa.20261302.11
DO  - 10.11648/j.ajaa.20261302.11
T2  - American Journal of Astronomy and Astrophysics
JF  - American Journal of Astronomy and Astrophysics
JO  - American Journal of Astronomy and Astrophysics
SP  - 59
EP  - 73
PB  - Science Publishing Group
SN  - 2376-4686
UR  - https://doi.org/10.11648/j.ajaa.20261302.11
AB  - Solving higher-degree polynomial equations remains a fundamental challenge in both pure and applied mathematics. While quadratic, cubic, and quartic equations have known algebraic solutions, no general radical solution exists for degree five and higher (Abel–Ruffini theorem). This work introduces a novel recurrence-based methodology for deriving exact roots of polynomials of arbitrary degree, based on Ezouidi Mourad Sultan's Theorem (EMST). Unlike traditional algebraic techniques that are often restricted to degrees four or less or rely on numerical approximations, this framework allows for the explicit determination of roots, including irrational, complex, and multiple roots, across any polynomial degree. By systematically leveraging the structure of polynomial coefficients through recursive relationships, this approach extends the capabilities of classical methods and enhances their precision. The method is demonstrated through comprehensive examples involving irreducible and high-degree polynomials of degree 8, producing exact roots in closed form. Comparative analyses with established techniques such as Cardano's method, Newton's Method, and the Rational Root Theorem highlight the advantages of this recurrence formulation, including exactness, no reliance on initial guesses, and applicability to any degree. The EMST-based methodology offers a unified pathway toward exact solutions for longstanding algebraic problems.
VL  - 13
IS  - 2
ER  -

Copy | Download

Author Information

Mourad Sultan Ezouidi

Faculty of Sciences of Gafsa, University of Gafsa, Gafsa, Tunisia

Contact Email

http://orcid.org/0009-0003-6245-8324
Taoufik Gassoumi

Faculty of Sciences of Gafsa, University of Gafsa, Gafsa, Tunisia

Contact Email

http://orcid.org/0000-0002-8468-1442

Download PDF

Submit an Article

Figure 1

Figure 1. Graphical abstract of the EMST recurrence method for solving higher-degree polynomials.

Plain Text BibTeX RIS

APA Style

Ezouidi, M. S., Gassoumi, T. (2026). Advancing Solutions to Higher-Degree Polynomials: A Novel Recurrence Approach via EMS’s Theorem. American Journal of Astronomy and Astrophysics, 13(2), 59-73. https://doi.org/10.11648/j.ajaa.20261302.11

Copy | Download

ACS Style

Ezouidi, M. S.; Gassoumi, T. Advancing Solutions to Higher-Degree Polynomials: A Novel Recurrence Approach via EMS’s Theorem. Am. J. Astron. Astrophys. 2026, 13(2), 59-73. doi: 10.11648/j.ajaa.20261302.11

Copy | Download

AMA Style

Ezouidi MS, Gassoumi T. Advancing Solutions to Higher-Degree Polynomials: A Novel Recurrence Approach via EMS’s Theorem. Am J Astron Astrophys. 2026;13(2):59-73. doi: 10.11648/j.ajaa.20261302.11

Copy | Download

@article{10.11648/j.ajaa.20261302.11,
  author = {Mourad Sultan Ezouidi and Taoufik Gassoumi},
  title = {Advancing Solutions to Higher-Degree Polynomials: 
A Novel Recurrence Approach via EMS’s Theorem},
  journal = {American Journal of Astronomy and Astrophysics},
  volume = {13},
  number = {2},
  pages = {59-73},
  doi = {10.11648/j.ajaa.20261302.11},
  url = {https://doi.org/10.11648/j.ajaa.20261302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajaa.20261302.11},
  abstract = {Solving higher-degree polynomial equations remains a fundamental challenge in both pure and applied mathematics. While quadratic, cubic, and quartic equations have known algebraic solutions, no general radical solution exists for degree five and higher (Abel–Ruffini theorem). This work introduces a novel recurrence-based methodology for deriving exact roots of polynomials of arbitrary degree, based on Ezouidi Mourad Sultan's Theorem (EMST). Unlike traditional algebraic techniques that are often restricted to degrees four or less or rely on numerical approximations, this framework allows for the explicit determination of roots, including irrational, complex, and multiple roots, across any polynomial degree. By systematically leveraging the structure of polynomial coefficients through recursive relationships, this approach extends the capabilities of classical methods and enhances their precision. The method is demonstrated through comprehensive examples involving irreducible and high-degree polynomials of degree 8, producing exact roots in closed form. Comparative analyses with established techniques such as Cardano's method, Newton's Method, and the Rational Root Theorem highlight the advantages of this recurrence formulation, including exactness, no reliance on initial guesses, and applicability to any degree. The EMST-based methodology offers a unified pathway toward exact solutions for longstanding algebraic problems.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Advancing Solutions to Higher-Degree Polynomials: 
A Novel Recurrence Approach via EMS’s Theorem
AU  - Mourad Sultan Ezouidi
AU  - Taoufik Gassoumi
Y1  - 2026/04/29
PY  - 2026
N1  - https://doi.org/10.11648/j.ajaa.20261302.11
DO  - 10.11648/j.ajaa.20261302.11
T2  - American Journal of Astronomy and Astrophysics
JF  - American Journal of Astronomy and Astrophysics
JO  - American Journal of Astronomy and Astrophysics
SP  - 59
EP  - 73
PB  - Science Publishing Group
SN  - 2376-4686
UR  - https://doi.org/10.11648/j.ajaa.20261302.11
AB  - Solving higher-degree polynomial equations remains a fundamental challenge in both pure and applied mathematics. While quadratic, cubic, and quartic equations have known algebraic solutions, no general radical solution exists for degree five and higher (Abel–Ruffini theorem). This work introduces a novel recurrence-based methodology for deriving exact roots of polynomials of arbitrary degree, based on Ezouidi Mourad Sultan's Theorem (EMST). Unlike traditional algebraic techniques that are often restricted to degrees four or less or rely on numerical approximations, this framework allows for the explicit determination of roots, including irrational, complex, and multiple roots, across any polynomial degree. By systematically leveraging the structure of polynomial coefficients through recursive relationships, this approach extends the capabilities of classical methods and enhances their precision. The method is demonstrated through comprehensive examples involving irreducible and high-degree polynomials of degree 8, producing exact roots in closed form. Comparative analyses with established techniques such as Cardano's method, Newton's Method, and the Rational Root Theorem highlight the advantages of this recurrence formulation, including exactness, no reliance on initial guesses, and applicability to any degree. The EMST-based methodology offers a unified pathway toward exact solutions for longstanding algebraic problems.
VL  - 13
IS  - 2
ER  -

Copy | Download