Algebraic Equations Worksheets with Answers
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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"- . -
Algorithmic Factorization of Polynomials Over Number Fields
Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 5-18-2017 Algorithmic Factorization of Polynomials over Number Fields Christian Schulz Rose-Hulman Institute of Technology Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Number Theory Commons, and the Theory and Algorithms Commons Recommended Citation Schulz, Christian, "Algorithmic Factorization of Polynomials over Number Fields" (2017). Mathematical Sciences Technical Reports (MSTR). 163. https://scholar.rose-hulman.edu/math_mstr/163 This Dissertation is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Algorithmic Factorization of Polynomials over Number Fields Christian Schulz May 18, 2017 Abstract The problem of exact polynomial factorization, in other words expressing a poly- nomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used to form the algebraic number field in question, let s be the sum of the squares of these degrees, and let d be the degree of the polynomial to be factored; then the runtime of this algorithm is found to be O(d4sk2 + 2dd3). -
Some Properties of the Discriminant Matrices of a Linear Associative Algebra*
570 R. F. RINEHART [August, SOME PROPERTIES OF THE DISCRIMINANT MATRICES OF A LINEAR ASSOCIATIVE ALGEBRA* BY R. F. RINEHART 1. Introduction. Let A be a linear associative algebra over an algebraic field. Let d, e2, • • • , en be a basis for A and let £»•/*., (hjik = l,2, • • • , n), be the constants of multiplication corre sponding to this basis. The first and second discriminant mat rices of A, relative to this basis, are defined by Ti(A) = \\h(eres[ CrsiCij i, j=l T2(A) = \\h{eres / ,J CrsiC j II i,j=l where ti(eres) and fa{erea) are the first and second traces, respec tively, of eres. The first forms in terms of the constants of multi plication arise from the isomorphism between the first and sec ond matrices of the elements of A and the elements themselves. The second forms result from direct calculation of the traces of R(er)R(es) and S(er)S(es), R{ei) and S(ei) denoting, respectively, the first and second matrices of ei. The last forms of the dis criminant matrices show that each is symmetric. E. Noetherf and C. C. MacDuffeeJ discovered some of the interesting properties of these matrices, and shed new light on the particular case of the discriminant matrix of an algebraic equation. It is the purpose of this paper to develop additional properties of these matrices, and to interpret them in some fa miliar instances. Let A be subjected to a transformation of basis, of matrix M, 7 J rH%j€j 1,2, *0). -
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 . -
A Historical Survey of Methods of Solving Cubic Equations Minna Burgess Connor
University of Richmond UR Scholarship Repository Master's Theses Student Research 7-1-1956 A historical survey of methods of solving cubic equations Minna Burgess Connor Follow this and additional works at: http://scholarship.richmond.edu/masters-theses Recommended Citation Connor, Minna Burgess, "A historical survey of methods of solving cubic equations" (1956). Master's Theses. Paper 114. This Thesis is brought to you for free and open access by the Student Research at UR Scholarship Repository. It has been accepted for inclusion in Master's Theses by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. A HISTORICAL SURVEY OF METHODS OF SOLVING CUBIC E<~UATIONS A Thesis Presented' to the Faculty or the Department of Mathematics University of Richmond In Partial Fulfillment ot the Requirements tor the Degree Master of Science by Minna Burgess Connor August 1956 LIBRARY UNIVERStTY OF RICHMOND VIRGlNIA 23173 - . TABLE Olf CONTENTS CHAPTER PAGE OUTLINE OF HISTORY INTRODUCTION' I. THE BABYLONIANS l) II. THE GREEKS 16 III. THE HINDUS 32 IV. THE CHINESE, lAPANESE AND 31 ARABS v. THE RENAISSANCE 47 VI. THE SEVEW.l'EEl'iTH AND S6 EIGHTEENTH CENTURIES VII. THE NINETEENTH AND 70 TWENTIETH C:BNTURIES VIII• CONCLUSION, BIBLIOGRAPHY 76 AND NOTES OUTLINE OF HISTORY OF SOLUTIONS I. The Babylonians (1800 B. c.) Solutions by use ot. :tables II. The Greeks·. cs·oo ·B.c,. - )00 A~D.) Hippocrates of Chios (~440) Hippias ot Elis (•420) (the quadratrix) Archytas (~400) _ .M~naeobmus J ""375) ,{,conic section~) Archimedes (-240) {conioisections) Nicomedea (-180) (the conchoid) Diophantus ot Alexander (75) (right-angled tr~angle) Pappus (300) · III. -
501 Algebra Questions 2Nd Edition
501 Algebra Questions 501 Algebra Questions 2nd Edition ® NEW YORK Copyright © 2006 LearningExpress, LLC. All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by LearningExpress, LLC, New York. Library of Congress Cataloging-in-Publication Data: 501 algebra questions.—2nd ed. p. cm. Rev. ed. of: 501 algebra questions / [William Recco]. 1st ed. © 2002. ISBN 1-57685-552-X 1. Algebra—Problems, exercises, etc. I. Recco, William. 501 algebra questions. II. LearningExpress (Organization). III. Title: Five hundred one algebra questions. IV. Title: Five hundred and one algebra questions. QA157.A15 2006 512—dc22 2006040834 Printed in the United States of America 98765432 1 Second Edition ISBN 1-57685-552-X For more information or to place an order, contact LearningExpress at: 55 Broadway 8th Floor New York, NY 10006 Or visit us at: www.learnatest.com The LearningExpress Skill Builder in Focus Writing Team is comprised of experts in test preparation, as well as educators and teachers who specialize in language arts and math. LearningExpress Skill Builder in Focus Writing Team Brigit Dermott Freelance Writer English Tutor, New York Cares New York, New York Sandy Gade Project Editor LearningExpress New York, New York Kerry McLean Project Editor Math Tutor Shirley, New York William Recco Middle School Math Teacher, Grade 8 New York Shoreham/Wading River School District Math Tutor St. James, New York Colleen Schultz Middle School Math Teacher, Grade 8 Vestal Central School District Math Tutor -
The Evolution of Equation-Solving: Linear, Quadratic, and Cubic
California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2006 The evolution of equation-solving: Linear, quadratic, and cubic Annabelle Louise Porter Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Porter, Annabelle Louise, "The evolution of equation-solving: Linear, quadratic, and cubic" (2006). Theses Digitization Project. 3069. https://scholarworks.lib.csusb.edu/etd-project/3069 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. THE EVOLUTION OF EQUATION-SOLVING LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degre Master of Arts in Teaching: Mathematics by Annabelle Louise Porter June 2006 THE EVOLUTION OF EQUATION-SOLVING: LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino by Annabelle Louise Porter June 2006 Approved by: Shawnee McMurran, Committee Chair Date Laura Wallace, Committee Member , (Committee Member Peter Williams, Chair Davida Fischman Department of Mathematics MAT Coordinator Department of Mathematics ABSTRACT Algebra and algebraic thinking have been cornerstones of problem solving in many different cultures over time. Since ancient times, algebra has been used and developed in cultures around the world, and has undergone quite a bit of transformation. This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work. -
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]. -
Interactive Mathematics Program Curriculum Framework
Interactive Mathematics Program Curriculum Framework School: __Delaware STEM Academy________ Curricular Tool: _IMP________ Grade or Course _Year 1 (grade 9) Unit Concepts / Standards Alignment Essential Questions Assessments Big Ideas from IMP Unit One: Patterns Timeline: 6 weeks Interpret expressions that represent a quantity in terms of its Patterns emphasizes extended, open-ended Can students use variables and All assessments are context. CC.A-SSE.1 exploration and the search for patterns. algebraic expressions to listed at the end of the Important mathematics introduced or represent concrete situations, curriculum map. Understand that a function from one set (called the domain) reviewed in Patterns includes In-Out tables, generalize results, and describe to another set (called the range) assigns to each element of functions, variables, positive and negative functions? the domain exactly one element of the range. If f is a function numbers, and basic geometry concepts Can students use different and x is an element of its domain, then f(x) denotes the output related to polygons. Proof, another major representations of functions— of f corresponding to the input x. The graph of f is the graph theme, is developed as part of the larger symbolic, graphical, situational, of the equation y = f(x). CC.F-IF.1 theme of reasoning and explaining. and numerical—and Students’ ability to create and understand understanding the connections Recognize that sequences are functions, sometimes defined proofs will develop over their four years in between these representations? recursively, whose domain is a subset of the integers. For IMP; their work in this unit is an important example, the Fibonacci sequence is defined recursively by start. -
Solving Polynomial Equations
Seminar on Advanced Topics in Mathematics Solving Polynomial Equations 5 December 2006 Dr. Tuen Wai Ng, HKU What do we mean by solving an equation ? Example 1. Solve the equation x2 = 1. x2 = 1 x2 ¡ 1 = 0 (x ¡ 1)(x + 1) = 0 x = 1 or = ¡1 ² Need to check that in fact (1)2 = 1 and (¡1)2 = 1. Exercise. Solve the equation p p x + x ¡ a = 2 where a is a positive real number. What do we mean by solving a polynomial equation ? Meaning I: Solving polynomial equations: ¯nding numbers that make the polynomial take on the value zero when they replace the variable. ² We have discovered that x, which is something we didn't know, turns out to be 1 or ¡1. Example 2. Solve the equation x2 = 5. x2 = 5 2 p x p¡ 5 = 0 (x ¡ 5)(x + 5) = 0p p x = 5 or ¡ 5 p p ² But what is 5 ? Well, 5 is the positive real number that square to 5. ² We have "learned" that the positive solution to the equation x2 = 5 is the positive real number that square to 5 !!! ² So there is a sense of circularity in what we have done here. ² Same thing happens when we say that i is a solution of x2 = ¡1. What are \solved" when we solve these equations ? ² The equations x2 = 5 and x2 = ¡1 draw the attention to an inadequacy in a certain number system (it does not contain a solution to the equation). ² One is therefore driven to extend the number system by introducing, or `adjoining', a solution. -
The Roots of Any Polynomial Equation
The roots of any polynomial equation G.A.Uytdewilligen, Bergen op Zoomstraat 76, 5652 KE Eindhoven. [email protected] Abstract We provide a method for solving the roots of the general polynomial equation n n−1 a ⋅x + a ⋅x + . + a ⋅x + s 0 n n−1 1 (1) To do so, we express x as a powerseries of s, and calculate the first n-1 coefficients. We turn the polynomial equation into a differential equation that has the roots as solutions. Then we express the powerseries’ coefficients in the first n-1 coefficients. Then the variable s is set to a0. A free parameter is added to make the series convergent. © 2004 G.A.Uytdewilligen. All rights reserved. Keywords: Algebraic equation The method The method is based on [1]. Let’s take the first n-1 derivatives of (1) to s. Equate these derivatives to zero. di Then find x ( s ) in terms of x(s) for i from 1 to n-1. Now make a new differential equation dsi n 1 n 2 d − d − m1⋅ x(s) + m2⋅ x(s) + . + m ⋅x(s) + m 0 n−1 n−2 n n+1 ds ds (2) di and fill in our x ( s ) in (2). Multiply by the denominator of the expression. Now we have a dsi polynomial in x(s) of degree higher then n. Using (1) as property, we simplify this polynomial to the degree of n. Set it equal to (1) and solve m1 .. mn+1 in terms of s and a1 .. an Substituting these in (2) gives a differential equation that has the zeros of (1) among its solutions.