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.

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.

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.

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