Theory of Equations Lesson 9
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1. Make Sense of Problems and Persevere in Solving Them. Mathematically Proficient Students Start by Explaining to Themselves T
1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize —to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize , to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. -
Higher Mathematics
; HIGHER MATHEMATICS Chapter I. THE SOLUTION OF EQUATIONS. By Mansfield Merriman, Professor of Civil Engineering in Lehigh University. Art. 1. Introduction. In this Chapter will be presented a brief outline of methods, not commonly found in text-books, for the solution of an equation containing one unknown quantity. Graphic, numeric, and algebraic solutions will be given by which the real roots of both algebraic and transcendental equations may be ob- tained, together with historical information and theoretic discussions. An algebraic equation is one that involves only the opera- tions of arithmetic. It is to be first freed from radicals so as to make the exponents of the unknown quantity all integers the degree of the equation is then indicated by the highest ex- ponent of the unknown quantity. The algebraic solution of an algebraic equation is the expression of its roots in terms of literal is the coefficients ; this possible, in general, only for linear, quadratic, cubic, and quartic equations, that is, for equations of the first, second, third, and fourth degrees. A numerical equation is an algebraic equation having all its coefficients real numbers, either positive or negative. For the four degrees 2 THE SOLUTION OF EQUATIONS. [CHAP. I. above mentioned the roots of numerical equations may be computed from the formulas for the algebraic solutions, unless they fall under the so-called irreducible case wherein real quantities are expressed in imaginary forms. An algebraic equation of the n th degree may be written with all its terms transposed to the first member, thus: n- 1 2 x" a x a,x"- . -
Section X.56. Insolvability of the Quintic
X.56 Insolvability of the Quintic 1 Section X.56. Insolvability of the Quintic Note. Now is a good time to reread the first set of notes “Why the Hell Am I in This Class?” As we have claimed, there is the quadratic formula to solve all polynomial equations ax2 +bx+c = 0 (in C, say), there is a cubic equation to solve ax3+bx2+cx+d = 0, and there is a quartic equation to solve ax4+bx3+cx2+dx+e = 0. However, there is not a general algebraic equation which solves the quintic ax5 + bx4 + cx3 + dx2 + ex + f = 0. We now have the equipment to establish this “insolvability of the quintic,” as well as a way to classify which polynomial equations can be solved algebraically (that is, using a finite sequence of operations of addition [or subtraction], multiplication [or division], and taking of roots [or raising to whole number powers]) in a field F . Definition 56.1. An extension field K of a field F is an extension of F by radicals if there are elements α1, α2,...,αr ∈ K and positive integers n1,n2,...,nr such that n1 ni K = F (α1, α2,...,αr), where α1 ∈ F and αi ∈ F (α1, α2,...,αi−1) for 1 <i ≤ r. A polynomial f(x) ∈ F [x] is solvable by radicals over F if the splitting field E of f(x) over F is contained in an extension of F by radicals. n1 Note. The idea in this definition is that F (α1) includes the n1th root of α1 ∈ F . -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Theory of Equations
CHAPTER I - THEORY OF EQUATIONS DEFINITION 1 A function defined by n n-1 f ( x ) = p 0 x + p 1 x + ... + p n-1 x + p n where p 0 ≠ 0, n is a positive integer or zero and p i ( i = 0,1,2...,n) are fixed complex numbers, is called a polynomial function of degree n in the indeterminate x . A complex number a is called a zero of a polynomial f ( x ) if f (a ) = 0. Theorem 1 – Fundamental Theorem of Algebra Every polynomial function of degree greater than or equal to 1 has atleast one zero. Definition 2 n n-1 Let f ( x ) = p 0 x + p 1 x + ... + p n-1 x + p n where p 0 ≠ 0, n is a positive integer . Then f ( x ) = 0 is a polynomial equation of nth degree. Definition 3 A complex number a is called a root ( solution) of a polynomial equation f (x) = 0 if f ( a) = 0. THEOREM 2 – DIVISION ALGORITHM FOR POLYNOMIAL FUNCTIONS If f(x) and g( x ) are two polynomial functions with degree of g(x) is greater than or equal to 1 , then there are unique polynomials q ( x) and r (x ) , called respectively quotient and reminder , such that f (x ) = g (x) q(x) + r (x) with the degree of r (x) less than the degree of g (x). Theorem 3- Reminder Theorem If f (x) is a polynomial , then f(a ) is the remainder when f(x) is divided by x – a . Theorem 4 Every polynomial equation of degree n ≥ 1 has exactly n roots. -
A Student's Question: Why the Hell Am I in This Class?
Why Are We In This Class? 1 A Student’s Question: Why The Hell Am I In This Class? Note. Mathematics of the current era consists of the broad areas of (1) geometry, (2) analysis, (3) discrete math, and (4) algebra. This is an oversimplification; this is not a complete list and these areas are not disjoint. You are familiar with geometry from your high school experience, and analysis is basically the study of calculus in a rigorous/axiomatic way. You have probably encountered some of the topics from discrete math (graphs, networks, Latin squares, finite geometries, number theory). However, surprisingly, this is likely your first encounter with areas of algebra (in the modern sense). Modern algebra is roughly 100–200 years old, with most of the ideas originally developed in the nineteenth century and brought to rigorous completion in the twentieth century. Modern algebra is the study of groups, rings, and fields. However, these ideas grow out of the classical ideas of algebra from your previous experience (primarily, polynomial equations). The purpose of this presentation is to link the topics of classical algebra to the topics of modern algebra. Babylonian Mathematics The ancient city of Babylon was located in the southern part of Mesopotamia, about 50 miles south of present day Baghdad, Iraq. Clay tablets containing a type of writing called “cuneiform” survive from Babylonian times, and some of them reflect that the Babylonians had a sophisticated knowledge of certain mathematical ideas, some geometric and some arithmetic. [Bardi, page 28] Why Are We In This Class? 2 The best known surviving tablet with mathematical content is known as Plimp- ton 322. -
Unit 3 Cubic and Biquadratic Equations
UNIT 3 CUBIC AND BIQUADRATIC EQUATIONS Structure 3.1 Introduction Objectives 3.2 Let Us Recall Linear Equations Quadratic Equations 3.3 Cubic Equations Cardano's Solution Roots And Their Relation With Coefficients 3.4 Biquadratic Equations Ferrari's Solution Descartes' Solution Roots And Their Relation With Coeftlcients 3.5 Summary - 3.1 INTRODUCTION In this unit we will look at an aspect of algebra that has exercised the minds of several ~nntllematicians through the ages. We are talking about the solution of polynomial equations over It. The ancient Hindu, Arabic and Babylonian mathematicians had discovered methods ni solving linear and quadratic equations. The ancient Babylonians and Greeks had also discovered methods of solviilg some cubic equations. But, as we have said in Unit 2, they had llol thought of complex iiumbers. So, for them, a lot of quadratic and cubic equations had no solutioit$. hl the 16th century various Italian mathematicians were looking into the geometrical prob- lein of trisectiiig an angle by straight edge and compass. In the process they discovered a inethod for solviilg the general cubic equation. This method was divulged by Girolanlo Cardano, and hence, is named after him. This is the same Cardano who was the first to iiilroduce coinplex numbers into algebra. Cardano also publicised a method developed by his contemporary, Ferrari, for solving quartic equations. Lake, in the 17th century, the French mathematician Descartes developed another method for solving 4th degree equrtioiis. In this unit we will acquaint you with the solutions due to Cardano, Ferrari and Descartes. Rut first we will quickly cover methods for solving linear and quadratic equations. -
ON the CASUS IRREDUCIBILIS of SOLVING the CUBIC EQUATION Jay Villanueva Florida Memorial University Miami, FL 33055 Jvillanu@Fmu
ON THE CASUS IRREDUCIBILIS OF SOLVING THE CUBIC EQUATION Jay Villanueva Florida Memorial University Miami, FL 33055 [email protected] I. Introduction II. Cardan’s formulas A. 1 real, 2 complex roots B. Multiple roots C. 3 real roots – the casus irreducibilis III. Examples IV. Significance V. Conclusion ******* I. Introduction We often need to solve equations as teachers and researchers in mathematics. The linear and quadratic equations are easy. There are formulas for the cubic and quartic equations, though less familiar. There are no general methods to solve the quintic and other higher order equations. When we deal with the cubic equation one surprising result is that often we have to express the roots of the equation in terms of complex numbers although the roots are real. For example, the equation – 4 = 0 has all roots real, yet when we use the formula we get . This root is really 4, for, as Bombelli noted in 1550, and , and therefore This is one example of the casus irreducibilis on solving the cubic equation with three real roots. 205 II. Cardan’s formulas The quadratic equation with real coefficients, has the solutions (1) . The discriminant < 0, two complex roots (2) ∆ real double root > 0, two real roots. The (monic) cubic equation can be reduced by the transformation to the form where (3) Using the abbreviations (4) and , we get Cardans’ formulas (1545): (5) The complete solutions of the cubic are: (6) The roots are characterized by the discriminant (7) < 0, one real, two complex roots = 0, multiple roots > 0, three real roots. -
Algebraic Solution of Nonlinear Equation Systems in REDUCE
Algebraic Solution of Nonlinear Equation Systems in REDUCE Herbert Melenk∗ A modern computer algebra system (CAS) is primarily a workbench offering an extensive set of computational tools from various field of mathematics and engineering. As most of these tools have an advanced and complicated theoretical background they often are hard for an unexperienced user to apply successfully. The idea of a SOLV E facility in a CAS is to find solutions for mathematical problems by applying the available tools in an automatic and invisible manner. In REDUCE the functionality of SOLVE has grown over the last years to an important extent. In the circle of REDUCE developers the Konrad–Zuse–Zentrum Berlin (ZIB) has been engaged in the field of solving nonlinear algebraic equation systems, a typical task for a CAS. The algebraic kernel for such computations is Buchberger’s algorithm for com- puting Gr¨obner bases and related techniques, published in 1966, first introduced in REDUCE 3.3 (1988) and substantially revised and extended in several steps for RE- DUCE 3.4 (1991). This version also organized for the first time the automatical invo- cation of Gr¨obner bases for the solution of pure polynomial systems in the context of REDUCE’s SOLVE command. In the meantime the range of automatically soluble system has been enlarged substan- tially. This report describes the algorithms and techniques which have been implemented in this context. Parts of the new features have been incorporated in REDUCE 3.4.1 (in July 1992), the rest will become available in the following release. E-mail address: [email protected] 1 Some of the developments have been encouraged to an important extent by colleagues who use these modules for their research, especially Hubert Caprasse (Li`ege) [4] and Jarmo Hietarinta (Turku) [10]. -
MA3D5 Galois Theory
MA3D5 Galois theory Miles Reid Jan{Mar 2004 printed Jan 2014 Contents 1 The theory of equations 3 1.1 Primitive question . 3 1.2 Quadratic equations . 3 1.3 The remainder theorem . 4 1.4 Relation between coefficients and roots . 5 1.5 Complex roots of 1 . 7 1.6 Cubic equations . 9 1.7 Quartic equations . 10 1.8 The quintic is insoluble . 11 1.9 Prerequisites and books . 13 Exercises to Chapter 1 . 14 2 Rings and fields 18 2.1 Definitions and elementary properties . 18 2.2 Factorisation in Z ......................... 21 2.3 Factorisation in k[x] ....................... 23 2.4 Factorisation in Z[x], Eisenstein's criterion . 28 Exercises to Chapter 2 . 32 3 Basic properties of field extensions 35 3.1 Degree of extension . 35 3.2 Applications to ruler-and-compass constructions . 40 3.3 Normal extensions . 46 3.4 Application to finite fields . 51 1 3.5 Separable extensions . 53 Exercises to Chapter 3 . 56 4 Galois theory 60 4.1 Counting field homomorphisms . 60 4.2 Fixed subfields, Galois extensions . 64 4.3 The Galois correspondences and the Main Theorem . 68 4.4 Soluble groups . 73 4.5 Solving equations by radicals . 76 Exercises to Chapter 4 . 80 5 Additional material 84 5.1 Substantial examples with complicated Gal(L=k) . 84 5.2 The primitive element theorem . 84 5.3 The regular element theorem . 84 5.4 Artin{Schreier extensions . 84 5.5 Algebraic closure . 85 5.6 Transcendence degree . 85 5.7 Rings of invariants and quotients in algebraic geometry . -
DWCRTTPORS *Algebra, *Tnstr:Uction, Mathematical Concepts, *Mathematicq, Mathematics Education, *Secondary School Mathematics
DOCUMENT RESUME ED 035 974 24 SE 007 912 AUTHOR nixon, Lyle J. TTmLE An Analysis of Certain Aspects of Approximation Theory as Related to Mathematical Instruction in. Algebra. Final Report. TmsTTTUmTON Kansas state Univ., Manhattan. MOONS AGENCY Office of Education (DHEW) , Washington, D.C. Bureau of Research. W17,ATT NO RP-8-F-03n PUP DAV' Aug 6() (1PANm OFG-6-9-008030-0035(097) NOTr 99D. "UPS DRICP ;?)RS price MP-s0.50 HC-4;3.05 DWCRTTPORS *Algebra, *Tnstr:uction, Mathematical Concepts, *Mathematicq, Mathematics Education, *Secondary School mathematics AsSTRACT This study is conc9rned with certain aspects of approximation theory which can he introduced into '.he mathematics curriculum at the secondary school level. The investigation examines existing literature in mathematics which relates to this subject in an effort to determine what is available in the way of mathematical concepts pertinent to this study. As a result of the literature review the material collected has been arranged in a structured mathematical form and existing mathematical theory has been extended to make the material useful to instructional problems in high school algebra. The results of the study are found in the expository material which comprises the ma-ior portion of this report. This material contains ideas of approximation theory which relate to elementary algebra. Also, included is a collection of references for this material. The report concludes that certain techniques of approximating a root of a polynomial by finding roots of a derived polynomial can be presented in a manner which is suitable for courses in elementary algebra. (DI') EDUCATION A WELFARE U.S. -
A Simplified Expression for the Solution of Cubic Polynomial Equations
A simplified expression for the solution of cubic polynomial equations using function evaluation Ababu T. Tiruneh, University of Eswatini, Department of Env. Health Science, Swaziland. Email: [email protected] Abstract This paper presents a simplified method of expressing the solution to cubic equations in terms of function evaluation only. The method eliminates the need to manipulate the orig- inal coefficients of the cubic polynomial and makes the solution free from such coefficients. In addition, the usual substitution needed to reduce the cubics is implicit in that the final solution is expressed in terms only of the function and derivative values of the given cubic polynomial at a single point. The proposed methodology simplifies the solution to cubic equations making them easy to remember and solve. Keywords: Polynomial equations, algebra, cubic equations, solution of equations, cubic polynomials, mathematics 1 Introduction Polynomials of higher degree arise often in problems in science and engineering. According to the fundamental theorem of algebra, a polynomial equation of degree n has at most n distinct solutions [1]. The history of symbolic manipulation for solving polynomial equation notes the work of Luca Pacioli in 1494 [2] in which the basis was laid for solving linear and quadratic equations while he stated the cubic equation as impossible to solve and asking the Italian math- ematical community to take the challenge. Mathematicians seemed to have felt hitting a wall after solving quadratic equations as it took quite a while for the solution to cubic equation to be found [3]. The solution to the cubic in the depressed form x3 +bx+c was discovered by del Ferro initially but he passed it to his student instead of publishing it.