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 forgotten paper on the fundamental theorem of algebra

2000 ◽  
Vol 54 (3) ◽  
pp. 333-341 ◽  
Author(s):  
F. Smithies
Keyword(s):  
Fundamental Theorem ◽  
Royal Society ◽  
Complex Form ◽  
Careful Examination ◽  
Complex Field ◽  
Complex Zero ◽  
Original Idea ◽  
Fundamental Theorem Of Algebra ◽  
Complex Coefficients

In 1798, there appeared in the Philosophical Transactions of the Royal Society a paper by James Wood, purporting to prove the fundamental theorem of algebra, to the effect that every non-constant polynomial with real coefficients has at least one real or complex zero. Since the first generally accepted proof of this result was given by Gauss in 1799, Wood's paper deserves careful examination. After giving a brief outline of Wood's career, I describe the argument of his paper. His proof turns out to be incomplete as it stands, but it contains an original idea, which was to be used later, in the same context, by von Staudt, Gordan and others, without knowledge of Wood's work. After putting Wood's work in context, I conclude by showing how his idea can be used to prove the complex form of the fundamental theorem of algebra, stating that every non-constant polynomial with complex coefficients has at least one zero in the complex field.

scholarly journals STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING

2021 ◽  
pp. 1-7
Author(s):  
SOHAM BASU
Keyword(s):  
Fundamental Theorem ◽  
Quadratic Polynomial ◽  
Complex Numbers ◽  
Real Analysis ◽  
Real Polynomial ◽  
Bivariate Polynomials ◽  
Real Polynomials ◽  
Fundamental Theorem Of Algebra ◽  
Polynomial Factor ◽  
Real Quadratic

Abstract Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss's first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

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.

Polynomiography: From the Fundamental Theorem of Algebra to Art

Leonardo ◽  
2005 ◽  
Vol 38 (3) ◽  
pp. 233-238 ◽  
Author(s):  
Bahman Kalantari
Keyword(s):  
Fundamental Theorem ◽  
Polynomial Equation ◽  
Computer Visualization ◽  
Iteration Functions ◽  
Fundamental Theorem Of Algebra

The author introduces polynomiography, a bridge between the Fundamental Theorem of Algebra and art. Polynomiography provides a tool for artists to create a 2D image—a polynomiograph—based on the computer visualization of a polynomial equation. The image is dependent upon the solutions of a polynomial equation, various interactive coloring schemes driven by iteration functions and several other parameters under the control of the polynomiographer's choice and creativity. Polynomiography software can mask all of the underlying mathematics, offering a tool that, although easy to use, affords the polynomiographer infinite artistic capabilities.

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 .

scholarly journals Relative equilibria of point vortices and the fundamental theorem of algebra

2011 ◽  
Vol 467 (2132) ◽  
pp. 2168-2184 ◽  
Author(s):  
Hassan Aref
Keyword(s):  
Fundamental Theorem ◽  
Relative Equilibrium ◽  
Polynomial Equation ◽  
Relative Equilibria ◽  
Point Vortices ◽  
Generating Polynomial ◽  
Second Derivatives ◽  
Fundamental Theorem Of Algebra ◽  
Vortex Systems ◽  
Derivatives Of

Relative equilibria of identical point vortices may be associated with a generating polynomial that has the vortex positions as its roots. A formula is derived that relates the first and second derivatives of this polynomial evaluated at a vortex position. Using this formula, along with the fundamental theorem of algebra, one can sometimes write a general polynomial equation. In this way, results about relative equilibria of point vortices may be proved in a compact and elegant way. For example, the classical result of Stieltjes, that if the vortices are on a line they must be situated at the zeros of the N th Hermite polynomial, follows easily. It is also shown that if in a relative equilibrium the vortices are all situated on a circle, they must form a regular N -gon. Several other results are proved using this approach. An ordinary differential equation for the generating polynomial when the vortices are situated on two perpendicular lines is derived. The method is extended to vortex systems where all the vortices have the same magnitude but may be of either sign. Derivations of the equation of Tkachenko for completely stationary configurations and its extension to translating relative equilibria are given.

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