scholarly journals Real Root Polynomials and Real Root Preserving Transformations

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Suchada Pongprasert ◽  
Kanyarat Chaengsisai ◽  
Wuttichai Kaewleamthong ◽  
Puttarawadee Sriphrom
Keyword(s):  
Real World ◽  
Real Root ◽  
Analogous Result ◽  
Complex Root ◽  
Linear Transformations ◽  
Real Numbers ◽  
Real Roots ◽  
Fundamental Theorem Of Algebra ◽  
Roots Of Polynomials ◽  
Complex Coefficients

Polynomials can be used to represent real-world situations, and their roots have real-world meanings when they are real numbers. The fundamental theorem of algebra tells us that every nonconstant polynomial p with complex coefficients has a complex root. However, no analogous result holds for guaranteeing that a real root exists to p if we restrict the coefficients to be real. Let n ≥ 1 and P n be the vector space of all polynomials of degree n or less with real coefficients. In this article, we give explicit forms of polynomials in P n such that all of their roots are real. Furthermore, we present explicit forms of linear transformations on P n which preserve real roots of polynomials in a certain subset of P n .

Activities for Students: A Graphical Approach to Understanding the Fundamental Theorem of Algebra

2001 ◽  
Vol 94 (9) ◽  
pp. 749-756
Author(s):  
Sudhir Kumar Goel ◽  
Denise T. Reid
Keyword(s):  
Fundamental Theorem ◽  
Polynomial Equation ◽  
Complex Numbers ◽  
Multiple Roots ◽  
Complex Root ◽  
Graphical Approach ◽  
Fundamental Theorem Of Algebra ◽  
Complex Coefficients

The fundamental theorem of algebra states, Every polynomial equation of degree n ≥ 1 with complex coefficients has at least one complex root. This fact implies that these equations have exactly n roots, counting multiple roots, in the set of complex numbers.

A NOTE ON THE FUNDAMENTAL THEOREM OF ALGEBRA

2018 ◽  
Vol 97 (3) ◽  
pp. 382-385
Author(s):  
MOHSEN ALIABADI
Keyword(s):  
Real Number ◽  
Fundamental Theorem ◽  
Real Root ◽  
Nonnegative Real Number ◽  
Square Root ◽  
Algebraic Proof ◽  
Real Numbers ◽  
Prime Degree ◽  
Fundamental Theorem Of Algebra ◽  
Odd Degree

The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First, every polynomial with odd degree and real coefficients has a real root. Second, every nonnegative real number has a square root. Shipman [‘Improving the fundamental theorem of algebra’, Math. Intelligencer29(4) (2007), 9–14] showed that the assumption about odd degree polynomials is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed. In this paper, we give a simpler proof of this result of Shipman.

scholarly journals Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 169
Author(s):  
Avram Sidi
Keyword(s):  
Nonlinear Equations ◽  
Local Convergence ◽  
Real Root ◽  
Numerical Procedure ◽  
Secant Method ◽  
Convergence Theory ◽  
Simple Complex ◽  
Real Roots ◽  
Complex Roots ◽  
Appl Math

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

scholarly journals On the Constructibility of Real 5th Roots of Rational Numbers with Marked Ruler and Compass

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Elliot Benjamin
Keyword(s):  
Rational Numbers ◽  
Real Numbers ◽  
Real Roots ◽  
Open Question

We demonstrate that there are infinitely many real numbers constructible by marked ruler and compass which are unique real roots of irreducible quintic polynomials over the field of rational numbers. This result can be viewed as a generalization of the historical open question of the constructibility by marked ruler and compass of real 5th roots of rational numbers. We obtain our results through marked ruler and compass constructions involving the intersection of conchoids and circles, and the application of number theoretic divisibility criteria.

scholarly journals Visualising Complex Polynomials: A Parabola Is but a Drop in the Ocean of Quadratics

2018 ◽  
Vol 10 (6) ◽  
pp. 91
Author(s):  
Harry Wiggins ◽  
Ansie Harding ◽  
Johann Engelbrecht
Keyword(s):  
Complex Function ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Complex Polynomials ◽  
Singular Case ◽  
Hyperbolic Paraboloid ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Complex Coefficients ◽  
Roots Of A Polynomial

One of the problems encountered when teaching complex numbers arises from an inability to visualise the complex roots, the so-called &quot;imaginary&quot; roots of a polynomial. Being four dimensional, it is problematic to visualize graphs and roots of polynomials with complex coefficients in spite of many attempts through centuries. An innovative way is described to visualize the graphs and roots of functions, by restricting the domain of the complex function to those complex numbers that map onto real values, leading to the concept of three dimensional sibling curves. Using this approach we see that a parabola is but a singular case of a complex quadratic.&nbsp; We see that sibling curves of a complex quadratic lie on a three-dimensional hyperbolic paraboloid. Finally, we show that the restriction to a real range causes no loss of generality.

scholarly journals Stability Regions of Fractional First Order Controllers Applied to Fractional Order Delay Systems

2021 ◽  
Vol 15 ◽  
pp. 86-90
Author(s):  
Karim Saadaoui
Keyword(s):  
Fractional Order ◽  
Real Root ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Complex Root ◽  
Stability Regions ◽  
First Order ◽  
The Stability ◽  
Fractional Order Controller ◽  
Fractional Controller

This paper focuses on the problem of stabilizing fractional order time delay systems by fractional first order controllers. A solution is proposed to find the set of all stability regions in the controller’s parameter space. The D-decomposition method is employed to find the real root boundary and complex root boundaries which are used to identify the stability regions. Illustrative examples are given to show the effectiveness of the proposed approach, and it is remarked that the stability region obtained for the fractional order controller is larger than the non-fractional controller.

scholarly journals On Roots of Polynomials and Algebraically Closed Fields

2017 ◽  
Vol 25 (3) ◽  
pp. 185-195 ◽  
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Algebraic Theory ◽  
Polynomial Rings ◽  
Multiple Roots ◽  
Real Numbers ◽  
The Real ◽  
Closed Fields ◽  
Roots Of Polynomials ◽  
Algebraically Closed Fields ◽  
Finite Domains ◽  
Identity Theorem

Summary In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].

scholarly journals Lower bound for the number of real roots of a random algebraic polynomial

1983 ◽  
Vol 35 (1) ◽  
pp. 18-27 ◽  
Author(s):  
M. N. Mishra ◽  
N. N. Nayak ◽  
S. Pattanayak
Keyword(s):  
Lower Bound ◽  
Algebraic Polynomial ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Independent Random Variables ◽  
Real Numbers ◽  
Exceptional Set ◽  
Real Roots ◽  
Normal Law ◽  
Number Of Real Roots

AbstractLet X1, X2, …, Xn be identically distributed independent random variables belonging to the domain of attraction of the normal law, have zero means and Pr{Xr ≠ 0} > 0. Suppose a0, a1, …, an are non-zero real numbers and max and εn is such that as n → ∞, εn. If Nn be the number of real roots of the equation then for n > n0, Nn > εn log n outside an exceptional set of measure at most provided limn→∞ (kn/tn) is finite.

Notes on the Theory of Equations

1939 ◽  
Vol 23 (256) ◽  
pp. 376-379
Author(s):  
E. P. Lewis
Keyword(s):  
Real Root ◽  
Real Roots ◽  
The Third ◽  
Complex Roots ◽  
Image Position

Multiply throughout by a 2 and write y for ax+ b ; the equation becomes where H ≡ ac − b 2, G ≡ a2d − 3abc + 2b3. Since in an equation with real coefficients complex roots occur in conjugate pairs, (i) must have at least one real root; so if α is this root, (i) may be written Accordingly the two remaining roots are also real if But since α satisfies (i), and so Hence if (i) has three real roots, G2 +4H 3 ≤ 0; and clearly, when G2 +4H 3 = 0, two roots are numerically equal to and the third to .

Oscillations of Second Order Neutral Equations

1988 ◽  
Vol 40 (6) ◽  
pp. 1301-1314 ◽  
Author(s):  
G. Ladas ◽  
E. C. Partheniadis ◽  
Y. G. Sficas
Keyword(s):  
Second Order ◽  
List Type ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
Neutral Equations ◽  
Deviating Arguments ◽  
Real Roots ◽  
All Solutions ◽  
Image Position ◽  
Necessary And Sufficient

Consider the second order neutral differential equation1where the coefficients p and q and the deviating arguments τ and σ are real numbers. The characteristic equation of Eq. (1) is2The main result in this paper is the following necessary and sufficient condition for all solutions of Eq. (1) to oscillate.THEOREM. The following statements are equivalent:(a) Every solution of Eq. (1) oscillates.(b) Equation (2) has no real roots.

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