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 Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

An algebraic proof of the fundamental theorem of algebra

2017 ◽  
Vol 11 ◽  
pp. 343-347
Author(s):  
Ruben Puente
Keyword(s):  
Fundamental Theorem ◽  
Algebraic Proof ◽  
Fundamental Theorem Of Algebra

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 .

Microcomputer-assisted mathematics: Roots: Half-Interval Search

1985 ◽  
Vol 78 (2) ◽  
pp. 120-123
Author(s):  
Clark Kimberling
Keyword(s):  
Fundamental Theorem ◽  
Complex Numbers ◽  
Real Numbers ◽  
Fundamental Theorem Of Algebra ◽  
Roots Of A Polynomial

According to the fundamental theorem of algebra, the roots of a polynomial all lie in the set of complex numbers. Some of the roots may be real numbers, and in many applications, only these need be found.

Addendum to: "An algebraic proof of the fundamental theorem of algebra"

2018 ◽  
Vol 12 (2) ◽  
pp. 91-94
Author(s):  
Ruben Puente
Keyword(s):  
Fundamental Theorem ◽  
Algebraic Proof ◽  
Fundamental Theorem Of Algebra

scholarly journals Quantitative fundamental theorem of algebra

2019 ◽  
Vol 70 (3) ◽  
pp. 1009-1037 ◽  
Author(s):  
Daniel Perrucci ◽  
Marie-Françoise Roy
Keyword(s):  
Fundamental Theorem ◽  
Quantitative Information ◽  
Algebraic Proof ◽  
Intermediate Value Theorem ◽  
Real Polynomials ◽  
Fundamental Theorem Of Algebra ◽  
Quantitative Results ◽  
The Difference ◽  
Intermediate Value ◽  
Classical Proof

Abstract Using subresultants, we modify a real-algebraic proof due to Eisermann of the fundamental theorem of Algebra (FTA) to obtain the following quantitative information: in order to prove the FTA for polynomials of degree d, the intermediate value theorem (IVT) is required to hold only for real polynomials of degree at most d2. We also explain that the classical proof due to Laplace requires IVT for real polynomials of exponential degree. These quantitative results highlight the difference in nature of these two proofs.

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