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Solving Solvable Quintics
mathematics of computation volume 57,number 195 july 1991, pages 387-401 SOLVINGSOLVABLE QUINTICS D. S. DUMMIT Abstract. Let f{x) = x 5 +px 3 +qx 2 +rx + s be an irreducible polynomial of degree 5 with rational coefficients. An explicit resolvent sextic is constructed which has a rational root if and only if f(x) is solvable by radicals (i.e., when its Galois group is contained in the Frobenius group F20 of order 20 in the symmetric group S5). When f(x) is solvable by radicals, formulas for the roots are given in terms of p, q, r, s which produce the roots in a cyclic order. 1. Introduction It is well known that an irreducible quintic with coefficients in the rational numbers Q is solvable by radicals if and only if its Galois group is contained in the Frobenius group F20 of order 20, i.e., if and only if the Galois group is isomorphic to F20 , to the dihedral group DXQof order 10, or to the cyclic group Z/5Z. (More generally, for any prime p, it is easy to see that a solvable subgroup of the symmetric group S whose order is divisible by p is contained in the normalizer of a Sylow p-subgroup of S , cf. [1].) The purpose here is to give a criterion for the solvability of such a general quintic in terms of the existence of a rational root of an explicit associated resolvent sextic polynomial, and when this is the case, to give formulas for the roots analogous to Cardano's formulas for the general cubic and quartic polynomials (cf. -
A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables
mathematics Article A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables Chukwuma Ogbonnaya 1,2,* , Chamil Abeykoon 3 , Adel Nasser 1 and Ali Turan 4 1 Department of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Manchester M13 9PL, UK; [email protected] 2 Faculty of Engineering and Technology, Alex Ekwueme Federal University, Ndufu Alike Ikwo, Abakaliki PMB 1010, Nigeria 3 Aerospace Research Institute and Northwest Composites Centre, School of Materials, The University of Manchester, Manchester M13 9PL, UK; [email protected] 4 Independent Researcher, Manchester M22 4ES, Lancashire, UK; [email protected] * Correspondence: [email protected]; Tel.: +44-(0)74-3850-3799 Abstract: A system of transcendental equations (SoTE) is a set of simultaneous equations containing at least a transcendental function. Solutions involving transcendental equations are often problematic, particularly in the form of a system of equations. This challenge has limited the number of equations, with inter-related multi-functions and multi-variables, often included in the mathematical modelling of physical systems during problem formulation. Here, we presented detailed steps for using a code- based modelling approach for solving SoTEs that may be encountered in science and engineering problems. A SoTE comprising six functions, including Sine-Gordon wave functions, was used to illustrate the steps. Parametric studies were performed to visualize how a change in the variables Citation: Ogbonnaya, C.; Abeykoon, affected the superposition of the waves as the independent variable varies from x1 = 1:0.0005:100 to C.; Nasser, A.; Turan, A. -
Math 1232-04F (Survey of Calculus) Dr. J.S. Zheng Chapter R. Functions
Math 1232-04F (Survey of Calculus) Dr. J.S. Zheng Chapter R. Functions, Graphs, and Models R.4 Slope and Linear Functions R.5* Nonlinear Functions and Models R.6 Exponential and Logarithmic Functions R.7* Mathematical Modeling and Curve Fitting • Linear Functions (11) Graph the following equations. Determine if they are functions. (a) y = 2 (b) x = 2 (c) y = 3x (d) y = −2x + 4 (12) Definition. The variable y is directly proportional to x (or varies directly with x) if there is some positive constant m such that y = mx. We call m the constant of proportionality, or variation constant. (13) The weight M of a person's muscles is directly proportional to the person's body weight W . It is known that a person weighing 200 lb has 80 lb of muscle. (a) Find an equation of variation expressing M as a function of W . (b) What is the muscle weight of a person weighing 120 lb? (14) Definition. A linear function is any function that can be written in the form y = mx + b or f(x) = mx + b, called the slope-intercept equation of a line. The constant m is called the slope. The point (0; b) is called the y-intercept. (15) Find the slope and y-intercept of the graph of 3x + 5y − 2 = 0. (16) Find an equation of the line that has slope 4 and passes through the point (−1; 1). (17) Definition. The equation y − y1 = m(x − x1) is called the point-slope equation of a line. The point is (x1; y1), and the slope is m. -
The Diamond Method of Factoring a Quadratic Equation
The Diamond Method of Factoring a Quadratic Equation Important: Remember that the first step in any factoring is to look at each term and factor out the greatest common factor. For example: 3x2 + 6x + 12 = 3(x2 + 2x + 4) AND 5x2 + 10x = 5x(x + 2) If the leading coefficient is negative, always factor out the negative. For example: -2x2 - x + 1 = -1(2x2 + x - 1) = -(2x2 + x - 1) Using the Diamond Method: Example 1 2 Factor 2x + 11x + 15 using the Diamond Method. +30 Step 1: Multiply the coefficient of the x2 term (+2) and the constant (+15) and place this product (+30) in the top quarter of a large “X.” Step 2: Place the coefficient of the middle term in the bottom quarter of the +11 “X.” (+11) Step 3: List all factors of the number in the top quarter of the “X.” +30 (+1)(+30) (-1)(-30) (+2)(+15) (-2)(-15) (+3)(+10) (-3)(-10) (+5)(+6) (-5)(-6) +30 Step 4: Identify the two factors whose sum gives the number in the bottom quarter of the “x.” (5 ∙ 6 = 30 and 5 + 6 = 11) and place these factors in +5 +6 the left and right quarters of the “X” (order is not important). +11 Step 5: Break the middle term of the original trinomial into the sum of two terms formed using the right and left quarters of the “X.” That is, write the first term of the original equation, 2x2 , then write 11x as + 5x + 6x (the num bers from the “X”), and finally write the last term of the original equation, +15 , to get the following 4-term polynomial: 2x2 + 11x + 15 = 2x2 + 5x + 6x + 15 Step 6: Factor by Grouping: Group the first two terms together and the last two terms together. -
Solving Cubic Polynomials
Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial monic. a 2. Then, given x2 + a x + a , substitute x = y − 1 to obtain an equation without the linear term. 1 0 2 (This is the \depressed" equation.) 3. Solve then for y as a square root. (Remember to use both signs of the square root.) a 4. Once this is done, recover x using the fact that x = y − 1 . 2 For example, let's solve 2x2 + 7x − 15 = 0: First, we divide both sides by 2 to create an equation with leading term equal to one: 7 15 x2 + x − = 0: 2 2 a 7 Then replace x by x = y − 1 = y − to obtain: 2 4 169 y2 = 16 Solve for y: 13 13 y = or − 4 4 Then, solving back for x, we have 3 x = or − 5: 2 This method is equivalent to \completing the square" and is the steps taken in developing the much- memorized quadratic formula. For example, if the original equation is our \high school quadratic" ax2 + bx + c = 0 then the first step creates the equation b c x2 + x + = 0: a a b We then write x = y − and obtain, after simplifying, 2a b2 − 4ac y2 − = 0 4a2 so that p b2 − 4ac y = ± 2a and so p b b2 − 4ac x = − ± : 2a 2a 1 The solutions to this quadratic depend heavily on the value of b2 − 4ac. -
Robust Algorithms for Object Localization
Robust Algorithms for Ob ject Lo calizati on y Dinesh Mano cha Aaron S. Wallack Department of Computer Science Computer Science Division University of North Carolina University of California Chap el Hill, NC 27599-3175 Berkeley, CA 94720 mano [email protected] wallack@rob otics.eecs.b erkeley.edu Abstract Ob ject lo calization using sensed data features and corresp onding mo del features is a fun- damental problem in machine vision. We reformulate ob ject lo calization as a least squares problem: the optimal p ose estimate minimizes the squared error discrepancy b etween the sensed and predicted data. The resulting problem is non-linear and previous attempts to estimate the optimal p ose using lo cal metho ds such as gradient descent su er from lo cal minima and, at times, return incorrect results. In this pap er, we describ e an exact, accurate and ecient algorithm based on resultants, linear algebra, and numerical analysis, for solv- ing the nonlinear least squares problem asso ciated with lo calizing two-dimensional ob jects given two-dimensional data. This work is aimed at tasks where the sensor features and the mo del features are of di erenttyp es and where either the sensor features or mo del features are p oints. It is applicable to lo calizing mo deled ob jects from image data, and estimates the p ose using all of the pixels in the detected edges. The algorithm's running time dep ends mainly on the typ e of non-p oint features, and it also dep ends to a small extent on the number of features. -
Casus Irreducibilis and Maple
48 Casus irreducibilis and Maple Rudolf V´yborn´y Abstract We give a proof that there is no formula which uses only addition, multiplication and extraction of real roots on the coefficients of an irreducible cubic equation with three real roots that would provide a solution. 1 Introduction The Cardano formulae for the roots of a cubic equation with real coefficients and three real roots give the solution in a rather complicated form involving complex numbers. Any effort to simplify it is doomed to failure; trying to get rid of complex numbers leads back to the original equation. For this reason, this case of a cubic is called casus irreducibilis: the irreducible case. The usual proof uses the Galois theory [3]. Here we give a fairly simple proof which perhaps is not quite elementary but should be accessible to undergraduates. It is well known that a convenient solution for a cubic with real roots is in terms of trigono- metric functions. In the last section we handle the irreducible case in Maple and obtain the trigonometric solution. 2 Prerequisites By N, Q, R and C we denote the natural numbers, the rationals, the reals and the com- plex numbers, respectively. If F is a field then F [X] denotes the ring of polynomials with coefficients in F . If F ⊂ C is a field, a ∈ C but a∈ / F then there exists a smallest field of complex numbers which contains both F and a, we denote it by F (a). Obviously it is the intersection of all fields which contain F as well as a. -
On Popularization of Scientific Education in Italy Between 12Th and 16Th Century
PROBLEMS OF EDUCATION IN THE 21st CENTURY Volume 57, 2013 90 ON POPULARIZATION OF SCIENTIFIC EDUCATION IN ITALY BETWEEN 12TH AND 16TH CENTURY Raffaele Pisano University of Lille1, France E–mail: [email protected] Paolo Bussotti University of West Bohemia, Czech Republic E–mail: [email protected] Abstract Mathematics education is also a social phenomenon because it is influenced both by the needs of the labour market and by the basic knowledge of mathematics necessary for every person to be able to face some operations indispensable in the social and economic daily life. Therefore the way in which mathe- matics education is framed changes according to modifications of the social environment and know–how. For example, until the end of the 20th century, in the Italian faculties of engineering the teaching of math- ematical analysis was profound: there were two complex examinations in which the theory was as impor- tant as the ability in solving exercises. Now the situation is different. In some universities there is only a proof of mathematical analysis; in others there are two proves, but they are sixth–month and not annual proves. The theoretical requirements have been drastically reduced and the exercises themselves are often far easier than those proposed in the recent past. With some modifications, the situation is similar for the teaching of other modern mathematical disciplines: many operations needing of calculations and math- ematical reasoning are developed by the computers or other intelligent machines and hence an engineer needs less theoretical mathematics than in the past. The problem has historical roots. In this research an analysis of the phenomenon of “scientific education” (teaching geometry, arithmetic, mathematics only) with respect the methods used from the late Middle Ages by “maestri d’abaco” to the Renaissance hu- manists, and with respect to mathematics education nowadays is discussed. -
The Fast Quartic Solver Peter Strobach AST-Consulting Inc., Bahnsteig 6, 94133 Röhrnbach, Germany Article Info a B S T R a C T
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 234 (2010) 3007–3024 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam The fast quartic solver Peter Strobach AST-Consulting Inc., Bahnsteig 6, 94133 Röhrnbach, Germany article info a b s t r a c t Article history: A fast and highly accurate algorithm for solving quartic equations is introduced. This Received 27 February 2010 new algorithm is more than six times as fast and several times more accurate than the Received in revised form 14 April 2010 quasi-standard Companion matrix eigenvalue quartic solver. Moreover, the method is exceptionally robust in cases of extreme root spread. The new algorithm is based on MSC: a factorization of the quartic in two quadratics, which are solved in closed form. The 65E05 performance key at this point is a fixed-point iteration based fitting algorithm for backward Keywords: optimization of the underlying quartic-to-quadratic polynomial decomposition. Detailed Quartic function experimental results confirm our claims. Polynomial factorization ' 2010 Elsevier B.V. All rights reserved. Polynomial rooting Companion matrix Eigenvalues 1. Introduction Consider the practical rooting of quartic polynomials or functions of the type f .x/ D x4 C ax3 C bx2 C cx C d D x2 C αx C β x2 C γ x C δ D .x − x1/.x − x2/.x − x3/.x − x4/ : (1) A quartic solver is an algorithm that relates a set of real or complex conjugate or mixed real/complex conjugate roots fx1; x2; x3; x4g to a given set of real quartic coefficients fa; b; c; dg. -
Factoring Polynomials
EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: — Set the factors of a polynomial equation (as opposed to an expression) equal to zero in order to solve for a variable: Example: To solve ,; , The flowchart below illustrates a sequence of steps for factoring polynomials. First, always factor out the Greatest Common Factor (GCF), if one exists. Is the equation a Binomial or a Trinomial? 1 Prime polynomials cannot be factored Yes No using integers alone. The Sum of Squares and the Four or more quadratic factors Special Cases? terms of the Sum and Difference of Binomial Trinomial Squares are (two terms) (three terms) Factor by Grouping: always Prime. 1. Group the terms with common factors and factor 1. Difference of Squares: out the GCF from each Perfe ct Square grouping. 1 , 3 Trinomial: 2. Sum of Squares: 1. 2. Continue factoring—by looking for Special Cases, 1 , 2 2. 3. Difference of Cubes: Grouping, etc.—until the 3 equation is in simplest form FYI: A Sum of Squares can 1 , 2 (or all factors are Prime). 4. Sum of Cubes: be factored using imaginary numbers if you rewrite it as a Difference of Squares: — 2 Use S.O.A.P to No Special √1 √1 Cases remember the signs for the factors of the 4 Completing the Square and the Quadratic Formula Sum and Difference Choose: of Cubes: are primarily methods for solving equations rather 1. Factor by Grouping than simply factoring expressions. -
Cracking the Cubic: Cardano, Controversy, and Creasing Alissa S
Cracking the Cubic: Cardano, Controversy, and Creasing Alissa S. Crans Loyola Marymount University MAA MD-DC-VA Spring Meeting Stevenson University April 14, 2012 These images are from the Wikipedia articles on Niccolò Fontana Tartaglia and Gerolamo Cardano. Both images belong to the public domain. 1 Quadratic Equation A brief history... • 400 BC Babylonians • 300 BC Euclid 323 - 283 BC This image is from the website entry for Euclid from the MacTutor History of Mathematics. It belongs to the public domain. 3 Quadratic Equation • 600 AD Brahmagupta To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. Brahmasphutasiddhanta Colebook translation, 1817, pg 346 598 - 668 BC This image is from the website entry for Brahmagupta from the The Story of Mathematics. It belongs to the public domain. 4 Quadratic Equation • 600 AD Brahmagupta To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. Brahmasphutasiddhanta Colebook translation, 1817, pg 346 598 - 668 BC ax2 + bx = c This image is from the website entry for Brahmagupta from the The Story of Mathematics. It belongs to the public domain. 4 Quadratic Equation • 600 AD Brahmagupta To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. -
Math: Honors Geometry UNIT/Weeks (Not Timeline/Topics Essential Questions Consecutive)
Math: Honors Geometry UNIT/Weeks (not Timeline/Topics Essential Questions consecutive) Reasoning and Proof How can you make a Patterns and Inductive Reasoning conjecture and prove that it is Conditional Statements 2 true? Biconditionals Deductive Reasoning Reasoning in Algebra and Geometry Proving Angles Congruent Congruent Triangles How do you identify Congruent Figures corresponding parts of Triangle Congruence by SSS and SAS congruent triangles? Triangle Congruence by ASA and AAS How do you show that two 3 Using Corresponding Parts of Congruent triangles are congruent? Triangles How can you tell whether a Isosceles and Equilateral Triangles triangle is isosceles or Congruence in Right Triangles equilateral? Congruence in Overlapping Triangles Relationships Within Triangles Mid segments of Triangles How do you use coordinate Perpendicular and Angle Bisectors geometry to find relationships Bisectors in Triangles within triangles? 3 Medians and Altitudes How do you solve problems Indirect Proof that involve measurements of Inequalities in One Triangle triangles? Inequalities in Two Triangles How do you write indirect proofs? Polygons and Quadrilaterals How can you find the sum of The Polygon Angle-Sum Theorems the measures of polygon Properties of Parallelograms angles? Proving that a Quadrilateral is a Parallelogram How can you classify 3.2 Properties of Rhombuses, Rectangles and quadrilaterals? Squares How can you use coordinate Conditions for Rhombuses, Rectangles and geometry to prove general Squares