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Examples and applications
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
In many applications one needs to find roots of polynomial equations or to locate them approximately. In this section we will assume that polynomial means polynomial with complex coefficients in a complex variable z. The fundamental theorem of algebra (Theorem 5.5 of Chapter 4) guarantees that a non-constant polynomial has (possibly complex) roots. For polynomials of degree two, the quadratic formula (Exercise 1.1 of Chapter 2) provides formulas for these roots in terms of the coefficients. For polynomials of degree three and four, rather complicated formulas exist for the roots in terms of the coefficients, but for polynomials of degree at least five, it can be proved that no such formula can exist. One must therefore resort to numerical methods. We briefly consider several such methods in this section.
Introduction
Published in Roberts Charles, Elementary Differential Equations, 2018
The fundamental theorem of algebra is an existence theorem, since it states that there exist n roots to a polynomial equation of degree n. Of course, the set of roots of a polynomial equation is unique. So the solution of the equation 2x4-3x3-13x2+37x-15=0 $ 2x^{4} - 3x^{3} - 13x^{2} + 37x - 15 = 0 $ is a set of four complex numbers and that set is unique. Can you solve this equation?
Elements of Higher-Order Linear Equations
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
Now, recalling the Fundamental Theorem of Algebra, we know an nth degree polynomial has n roots if we allow for the possibility of complex roots (and we count repeated roots, too). There are three cases to consider based on these n roots. We may have roots that are real and distinct, roots that are repeated, or roots that are complex. In general, we may have a combination of all three. We will consider each of these cases.
Multidimensional realisation theory and polynomial system solving
Published in International Journal of Control, 2018
Philippe Dreesen, Kim Batselier, Bart De Moor
In the one-dimensional case, the Fundamental Theorem of Algebra states that a univariate degree d polynomial f(x) has exactly d roots in the field of complex numbers. When several of these roots coincide, we say that they occur with multiplicity. This happens if f(x) has a horizontal tangent at the position of a multiple root. The multidimensional counterpart of the Fundamental Theorem of Algebra is called Bezout's theorem (see Cox et al. 2005, p. 97 and Shafarevich 2013, p. 246). This theorem states that a set of Equations (18) that describes a zero-dimensional solution set has exactly m = ∏idi solutions in the projective space, counted with multiplicity.
Textbook accounts of the rules of indices with rational exponents
Published in International Journal of Mathematical Education in Science and Technology, 2019
I have chosen not to look at algebra texts prior to 1800 because what is now elementary algebra was then the subject of mathematical research. In 1799 Gauss published a criticism of d'Alembert's proof of the Fundamental Theorem of Algebra (FTA), and provided a proof of his own. Regardless of whether contemporary mathematicians accept this as the first satisfactory proof, between 1759 and 1800, the FTA was an active topic of mathematics research by some of the most pre-eminent mathematicians, see [15]. Indeed, after 1800 mathematical research continues to develop what are now considered elementary topics. The quadratic formula is well known, and Cardano's method leads to the formula (2) for a root of the cubic in terms of the coefficients. The search for general methods for solving classes of equations was still an area of active research prior to this period. The work of Évariste Galois, who died in 1832, is now used to show that there is no corresponding formula for equations of degree 5 and higher. The first algebra book in my corpus by Euler [16] does not envisage this negative result. § 780. This is the greatest length to which we have yet arrived in the resolution of algebraic equations. All the pains that have been taken in order to resolve equations of the fifth degree, and those of higher dimensions, in the same manner, or, at least, to reduce them to inferior degrees, have been unsuccessful: so that we cannot give any general rules for finding the roots of equations, which exceed the fourth degree [16]. It is interesting to speculate if Euler suspected the impossibility of the solution of the quintic by radicals, but he provides no hint to the reader here. Wessel gave the first geometric interpretation of complex numbers in 1797 and Argand in 1806, and these ideas were extended and developed by Gauss and others. The period (1814–1851) saw rapid and profound research in complex analysis [17]. Prior to 1800 complex numbers were not well understood or even accepted. As late as 1872 Dedekind [18, p. 22] remarked that had not been proved rigorously. He was criticizing a reliance on formal rules of calculation, rather than using a definition of real number. The formal algebraic rules, and the justifications which Dedekind criticizes, are precisely those I consider in this paper and these rules remain as a core component of practical elementary algebraic manipulation.