Finding Zeros of Quartic Polynomials by Using Radicals
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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. -
Positivity Conditions for Cubic, Quartic and Quintic Polynomials Arxiv:2008.10922V10 [Math.GM] 18 Sep 2020
Positivity Conditions for Cubic, Quartic and Quintic Polynomials Liqun Qi,∗ Yisheng Song,y and Xinzhen Zhang,z August 17, 2021 Abstract We present a necessary and sufficient condition for a cubic polynomial to be positive for all positive reals. We identify the set where the cubic polynomial is nonnegative but not all positive for all positive reals, and explicitly give the points where the cubic polynomial attains zero. We then reformulate a necessary and sufficient condition for a quartic polynomial to be nonnegative for all posi- tive reals. From this, we derive a necessary and sufficient condition for a quartic polynomial to be nonnegative and positive for all reals. Our condition explic- itly exhibits the scope and role of some coefficients, and has strong geometrical meaning. In the interior of the nonnegativity region for all reals, there is an ap- pendix curve. The discriminant is zero at the appendix, and positive in the other part of the interior of the nonnegativity region. By using the Sturm sequences, we present a necessary and sufficient condition for a quintic polynomial to be positive and nonnegative for all positive reals. We show that for polynomials of a fixed even degree higher than or equal to four, if they have no real roots, then their discriminants take the same sign, which depends upon that degree only, except on an appendix set of dimension lower by two, where the discriminants attain zero. arXiv:2008.10922v10 [math.GM] 18 Sep 2020 Key words. Cubic polynomials, quartic polynomials, quintic polynomials, the Sturm theorem, discriminant, appendix. AMS subject classifications. -
Quartic Equation of General Form
EqWorld http://eqworld.ipmnet.ru Exact Solutions > Algebraic Equations and Systems of Algebraic Equations > Algebraic Equations > Quartic Equation of General Form 8. ax4 + bx3 + cx2 + dx + e = 0 (a ≠ 0). Quartic equation of general form. 1±. Reduction to an incomplete equation. The quartic equation in question is reduced to an incom- plete equation y4 + py2 + qy + r = 0.(1) with the change of variable b x = y − 4a 2±. Decartes–Euler solution. The roots of the incomplete equation (1) are given by 1 ¡p p p ¢ 1 ¡p p p ¢ y1 = z1 + z2 + z3 , y2 = z1 − z2 − z3 , 2 2 2 1 ¡ p p p ¢ 1 ¡ p p p ¢ ( ) y3 = 2 − z1 + z2 − z3 , y4 = 2 − z1 − z2 + z3 , where z1, z2, z3 are roots of the cubic equation z3 + 2pz2 + (p2 − 4r)z − q2 = 0,(3) which is called the cubic resolvent of equation (1). The signs of the roots in (2) are chosen so that p p p z1 z2 z3 = −q. The roots of the incomplete quartic equation (1) are determined by the roots of the cubic resolvent (3); see the table below. TABLE Relation between the roots of the incomplete quartic equation and the roots of its cubic resolvent Cubic resolvent (3) Quartic equation (1) All roots are real and positive* Four real roots All roots are real, Two pairs of complex conjugate roots one positive and two negative* One roots is positive Two real and two complex conjugate roots and two roots are complex conjugate 2 * By Vieta’s theorem, the product of the roots z1, z2, z3 is equal to q ≥ 0. -
(Trying To) Solve Higher Order Polynomial Equations. Featuring a Recall of Polynomial Long Division
(Trying to) solve Higher order polynomial equations. Featuring a recall of polynomial long division. Some history: The quadratic formula (Dating back to antiquity) allows us to solve any quadratic equation. ax2 + bx + c = 0 What about any cubic equation? ax3 + bx2 + cx + d = 0? In the 1540's Cardano produced a \cubic formula." It is too complicated to actually write down here. See today's extra credit. What about any quartic equation? ax4 + bx3 + cx2 + dx + e = 0? A few decades after Cardano, Ferrari produced a \quartic formula" More complicated still than the cubic formula. In the 1820's Galois (age ∼19) proved that there is no general algebraic formula for the solutions to a degree 5 polynomial. In fact there is no purely algebraic way to solve x5 − x − 1 = 0: Galois died in a duel at age 21. This means that, as disheartening as it may feel, we will never get a formulaic solution to a general polynomial equation. The best we can get is tricks that work sometimes. Think of these tricks as analogous to the strategies you use to factor a degree 2 polynomial. Trick 1 If you find one solution, then you can find a factor and reduce to a simpler polynomial equation. Example. 2x2 − x2 − 1 = 0 has x = 1 as a solution. This means that x − 1 MUST divide 2x3 − x2 − 1. Use polynomial long division to write 2x3 − x2 − 1 as (x − 1) · (something). Now find the remaining two roots 1 2 For you: Find all of the solutions to x3 + x2 + x + 1 = 0 given that x = −1 is a solution to this equation. -
Low-Degree Polynomial Roots
Low-Degree Polynomial Roots David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Created: July 15, 1999 Last Modified: September 10, 2019 Contents 1 Introduction 3 2 Discriminants 3 3 Preprocessing the Polynomials5 4 Quadratic Polynomials 6 4.1 A Floating-Point Implementation..................................6 4.2 A Mixed-Type Implementation...................................7 5 Cubic Polynomials 8 5.1 Real Roots of Multiplicity Larger Than One............................8 5.2 One Simple Real Root........................................9 5.3 Three Simple Real Roots......................................9 5.4 A Mixed-Type Implementation................................... 10 6 Quartic Polynomials 12 6.1 Processing the Root Zero...................................... 14 6.2 The Biquadratic Case........................................ 14 6.3 Multiplicity Vector (3; 1; 0; 0).................................... 15 6.4 Multiplicity Vector (2; 2; 0; 0).................................... 15 6.5 Multiplicity Vector (2; 1; 1; 0).................................... 15 6.6 Multiplicity Vector (1; 1; 1; 1).................................... 16 1 6.7 A Mixed-Type Implementation................................... 17 2 1 Introduction Consider a polynomial of degree d of the form d X i p(y) = piy (1) i=0 where the pi are real numbers and where pd 6= 0. A root of the polynomial is a number r, real or non-real (complex-valued with nonzero imaginary part) such that p(r) = 0. The polynomial can be factored as p(y) = (y − r)mf(y), where m is a positive integer and f(r) 6= 0. -
The Conditions for Multiple Roots in Cubic and Quartic Equations
Fort Hays State University FHSU Scholars Repository Master's Theses Graduate School Summer 1953 The Conditions For Multiple Roots In Cubic and Quartic Equations Laurence Dryden Fort Hays Kansas State College Follow this and additional works at: https://scholars.fhsu.edu/theses Part of the Algebraic Geometry Commons Recommended Citation Dryden, Laurence, "The Conditions For Multiple Roots In Cubic and Quartic Equations" (1953). Master's Theses. 508. https://scholars.fhsu.edu/theses/508 This Thesis is brought to you for free and open access by the Graduate School at FHSU Scholars Repository. It has been accepted for inclusion in Master's Theses by an authorized administrator of FHSU Scholars Repository. THE CONDITIONS FOR MULTIPLE ROOTS IN CUBI C AND QUARTIC EQUATIONS being A Thesis Presented to the Graduate F aculty of the Fort Hays Kansas State College in Partial Fulfillment of the Requireme nts for t h e De gree Master of Science by Laurence A. Dryden, B.S. in Education Ohio State University Approved~,R;I;_#~ Major Departme Date 7- '-?.. - ..,-.3 ACKNOWLEDGlvIENT The writer of this thesis wishes to acknowledge all sources of information used in its preparation and is especially indebted to the Head of the Mathematics Department, Professor Emmet C. Stopher, for his assistance, criticisms, and advice. \ TABLE OF CONTENTS CHAPTER PAGE INTRODUCTION 1 I. DISCRil'HNANT 1. Definition of Discriminant ••• . 2 2. Definition of Resultant 2 3. Relation between Discriminant and Resultant 3 4. Definition of Sylvester's Determinant D(f,g) •••• 3 S. Relation between Sylvester's Determinant and Resultant 6. Relation between Discriminant and Sylvester's Determinant 7 7. -
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. -
Cardano's Method for Roots of Cubic Equations
Lecture 23 : Solutions of Cubic and Quartic Equations Objectives (1) Cardano's method for roots of cubic equations. (2) Lagrange's method for roots of quartic equations. (3) Ferrari's method for roots of quartic equations. Keywords and Phrases : Cubic equations, quartic equations. In this section we present algorithms for finding roots of cubic and quartic polynomials over any field F of characteristic different from 2 and 3: This is to make sure that irreducible cubics and quartics are separable. Cubic polynomials Cardano published Tartaglia's method to find roots of cubic polynomials in 1545. This is known as Cardano's method. We may assume that the given cubic is of the form f(x) = x3 + px + q since a general cubic can be transformed into this form without changing its splitting field. One begins by introducing two unknowns u and v: Put x = u + v into f(x) = 0 to get u3 + v3 + 3u2v + 3uv2 + p(u + v) + q = u3 + v3 + q + (3uv + p)(u + v) = 0: We set u3 + v3 + q = 0 and 3uv + p = 0: Hence v = −p=3u: Put this into the first equation to get u6 + qu3 − p3=27 = 0: This is a quadratic equation in u3: Put D = −(4p3 +27q2): By the quadratic formula we get −q ± pq2 + (4p3=27) q u3 = = − ± p−D=108: 2 2 Set A = −q=2 + p−D=108 and B = −q=2 − p−D=108: By symmetry of u and v; we set u3 = A and v3 = B: Let ! be a primitive cube root of unity. Then p p p p p p u = 3 A; ! 3 A; !2 3 A; and v = 3 B; ! 3 B; !2 3 B: 97 98 p p We must choose cube roots of A and B in such a way that 3 A 3 B = −p=3: Having chosen these we see that the three roots of f(x) are p p p p p p 3 A + 3 B; ! 3 A + !2 3 B; !2 3 A + ! 3 B: Example 23.1. -
Reflection Theorems for Number Rings
Reflection theorems for number rings Evan O’Dorney July 13, 2021 Contents I Introduction 4 1 Introduction 4 1.1 Historicalbackground . ............ 4 1.2 Methods......................................... ....... 5 1.3 Results......................................... ........ 6 1.4 Outlineofthepaper ............................... .......... 7 1.5 Acknowledgements ................................ .......... 7 2 Examples for the lay reader 7 2.1 Reflectionforquadraticequations. ................ 7 2.2 Reflectionforcubicequations . .............. 9 2.3 Reflection for 2 n n boxes.................................... 11 2.4 Reflectionforquarticequations× × . ............... 12 3 Notation 15 II Galois cohomology 15 4 Étale algebras and their Galois groups 16 4.1 Étalealgebras................................... .......... 16 4.2 TheGaloisgroupofanétalealgebra . .............. 17 4.3 Resolvents...................................... ......... 17 4.4 Subextensionsandautomorphisms . .............. 18 4.5 Torsors ......................................... ....... 18 arXiv:2107.04727v1 [math.NT] 10 Jul 2021 4.6 AfreshlookatGaloiscohomology . ............. 20 5 Extensions of Kummer theory to explicitize Galois cohomology 23 5.1 TheTatepairingandtheHilbertsymbol. ............... 27 6 Rings over a Dedekind domain 29 6.1 Indicesoflattices............................... ............ 29 6.2 Discriminants................................... .......... 30 6.3 Quadraticrings.................................. .......... 32 6.4 Cubicrings ..................................... -
View This Volume's Front and Back Matter
i i “IrvingBook” — 2013/5/22 — 15:39 — page i — #1 i i 10.1090/clrm/043 Beyond the Quadratic Formula i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page ii — #2 i i c 2013 by the Mathematical Association of America, Inc. Library of Congress Catalog Card Number 2013940989 Print edition ISBN 978-0-88385-783-0 Electronic edition ISBN 978-1-61444-112-0 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page iii — #3 i i Beyond the Quadratic Formula Ron Irving University of Washington Published and Distributed by The Mathematical Association of America i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page iv — #4 i i Council on Publications and Communications Frank Farris, Chair Committee on Books Gerald M. Bryce, Chair Classroom Resource Materials Editorial Board Gerald M. Bryce, Editor Michael Bardzell Jennifer Bergner Diane L. Herrmann Paul R. Klingsberg Mary Morley Philip P. Mummert Mark Parker Barbara E. Reynolds Susan G. Staples Philip D. Straffin Cynthia J Woodburn i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page v — #5 i i CLASSROOM RESOURCE MATERIALS Classroom Resource Materials is intended to provide supplementary class- room material for students—laboratory exercises, projects, historical in- formation, textbooks with unusual approaches for presenting mathematical ideas, career information, etc. 101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Beyond the Quadratic Formula, Ronald S. -
Cubic and Quartic Equations (Pdf File)
CUBIC AND QUARTIC EQUATIONS From the point of view of medieval mathematicians, there are actually 13 different types of cubic equations rather than just one. Basically this is because they not merely did not admit imaginary or complex numbers, but only considered positive real num- bers, so also did not admit negative numbers or zero. Thus to them ¢¡¤£¥ §¦©¨ was a different type of equation from ¢¡ ¦ £ ¨ . Now we can write a general cubic equation ¡ £££¦ ¦ (in which § or the equation is not genuinely cubic) in the form ¡ £¥ £¥£¥ ¦! "¦# after dividing by a constant. The case where ¦! leads to an inadmissible root and anyway is easily soluble, so we get different cases as $ $ $ ¦ % '& ¦ % (& ¦ % leading to 18 cases. However, the cases $ )¦ )¦!'&* % )¦ )¦!'&* are not taken seriously as cubic equations (and indeed the first has no real positive solution), while evidently the cases $ $ $ '&* (&+ $ $ )¦!'&* (&+ $ $ '&*,¦ (&+ have no real positive solution. This leaves 13 cases to be considered. The case of the cosa and the cube, in modern notation the case ¡-£¥.¦ where and are positive, was solved by Scipione del Ferro (1465–1626) early in the six- teenth century. He taught his method to his pupil Antonio Maria Fior (Florido) (dates unknown), who had a contest with Nicolo` Tartaglia (1500–1557) which resulted in the latter’s discovery of the method for solving this particular type. Girolamo Cardano (Jerome Cardan) (1501–1576) persuaded Tartaglia to tell him the solution, first in a cryptic verse and then with a full explanation, after swearing he would keep the solu- tion secret. But, after he had found the solution in the posthumous papers of del Ferro, Cardan felt free to publish, which he did in his Ars Magna (1545). -
ON the COBLE QUARTIC 1. Introduction in This Paper We
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Archivio della ricerca- Università di Roma La Sapienza ON THE COBLE QUARTIC SAMUEL GRUSHEVSKY AND RICCARDO SALVATI MANNI Abstract. We review and extend the known constructions relat- ing Kummer threefolds, G¨opel systems, theta constants and their derivatives, and the GIT quotient for 7 points in P2 to obtain an explicit expression for the Coble quartic. The Coble quartic was recently determined completely in [RSSS12], where (Theorem 7.1a) it was computed completely explicitly, as a polynomial with 372060 monomials of bidegree (28; 4) in theta constants of the second or- der and theta functions of the second order, respectively. Our expression is in terms of products of theta constants with charac- teristics corresponding to G¨opel systems, and is a polynomial with 134 terms. Our approach uses the relationship of G¨opel systems and Jacobian determinants of theta functions, and highlights the geometry and combinatorics of syzygetic octets of characteristics, and the corresponding representations of Sp(6; F2). In genus 2, we similarly obtain a short explicit equation for the universal Kummer surface, and relating modular forms of level two to binary invariants of six points on P1. 1. Introduction In this paper we provide an explicit description of the Coble quartic. This is a quartic equation in 8 variables whose eight partial derivatives defines the Kummer variety of a smooth plane quartic. This sixfold is particularly interesting as it is the moduli space of semistable vector bundles of rank 2 with trivial determinant on the genus three curve [?] or [?].