Introduction to Modern Algebra David Joyce Clark University Version 0.0.6‚ 3 Oct 2008 1 Copyright (C) 2008. 1 ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write this book. I’m sorry that he did not live to see it finished. Contents 1 Introduction 1.1 Structures in Modern Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Operations on sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Fields . . . . .
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10. Find the GCF of 12x3‚ 14x6‚ and 18x2 11. Find the GCF of 17x5‚ x2133‚ and 4x9 12. Find the GCF of 18x6y3‚ 9x10y2‚ and 3x6y4 3. Factoring Monomials ****To factor a polynomial means to write the polynomial as a product of prime polynomials. 1. Find the GCF of each of the terms 2. Factor out the GCF from each of the terms 1. 6a3 + 15a = 2. 32b2 + 12b = 3. 12a5b2 + 16a4b = 4. 9x2 + 18y4 = 5. 7x2 – 15y = 6. y4 – 3y2 – 2y = 7
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Therefore all eigenvectors are actually the roots of the monic polynomial det(xI −T ) in K. This polynomial is called the characteristic polynomial of T and is denoted by cT (x). Since the degree of cT (x) is n‚ the dimension of V‚ T cannot have more than n eigenvalues counted with multiplicities. If A ∈ K n×n ‚ then A can be regarded as a linear mapping from K n to itself‚ and so the polynomial cA (x) = det(xIn − A) is the characteristic polynomial of the matrix A‚ and its roots in K are the eigenvalues
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CONTENTS INTRODUCTION 3 DESCRIPTION 5 UNIT CREDIT 6 TIME ALLOTMENT 6 EXPECTANCIES 7 SCOPE AND SEQUENCE 8 SUGGESTED STRATEGIES AND MATERIALS 9 GRADING SYSTEM 10 LEARNING COMPETENCIES 11 SAMPLE LESSON PLANS 30 INTRODUCTION This Handbook aims to provide the general public – parents‚ students‚ researchers‚ and other stakeholders – an overview of the Mathematics program at the secondary level. Those in education‚ however‚ may use it as a
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205 How could you use Descartes’ rule and the Fundamental Theorem of Algebra to predict the number of complex roots to a polynomial as well as find the number of possible positive and negative real roots to a polynomial? | Descartes rule is really helpful because it eliminates the long list of possible rational roots and you can tell how many positives or negatives roots you will have. Fundamental Theorem of Algebra finds the maximum number of zeros which includes real and complex numbers
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Support Vector Machine Luyang Liu 152003815 I. Introduction Support vector machines (SVMs) are a basic machine learning method for supervised learning models with associated learning algorithms that analyze data and recognize patterns‚ used for classification and regression analysis. Introduced by Vladimir Vapnik and his colleagues‚ SVMs are a relatively new learning method used for binary classification. The basic idea is to find a hyperplane which separates the d-dimensional data perfectly into
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Chapter 1 THE PROBLEM AND ITS BACKGROUND Introduction Mathematics is a very important subject. The effect of Mathematics in our daily struggles in life is very evident. Everything involves Mathematics that is why Mathematics is very important on the part of the students to study. Over the years‚ schools have always been concerned with the development of effective learning experience for the learners. It is generally accepted that the quality of education students get largely dependent upon
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INTRODUCTION TO THE THEORY OF COMPUTATION‚ SECOND EDITION MICHAEL SIPSER MassachusettsInstitute of Technology THOMSON COURSE TECHNOLOGY Australia * Canada * Mexico * Singapore * Spain * United Kingdom * United States THOIVISON COURSE TECHNOLOGY Introduction to the Theory of Computation‚ Second Edition by Michael Sipser Senior Product Manager: Alyssa Pratt Executive Editor: Mac Mendelsohn Associate Production Manager: Aimee Poirier Senior Marketing Manager: Karen Seitz COPYRIGHT
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1) Evaluate each algebraic expression‚ given that x= -1‚ y=3‚ z=2‚ a =1/2‚ b= -2/3. a) b) c) 2) Determine the degree of each of the following polynomials. a) b) c) 3) Remove the symbols of grouping and simplify the resulting expressions by combining like terms. a) (x + 3y – z) – (2y – x +3z) + (4z – 3x +2y) b) c) 3 – {2x – [1 –(x +y)] + [x – 2y]} 4) Add the algebraic expressions in each of the following groups. a) b) 5) Subtract the algebraic expressions
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-12) 44() – 2n = -20 12m – 6n = -60 – 2n = -20 9m + 6n = -48-2n = -20 + 21m = -108(-2n = ) 2121 m = n= -2/7 5. The polynomial x3 + mx2 +nx – 3 leaves a remainder of 21 when divided by x – 2 and a remainder of 3 when divided by x + 1. Calculate the remainder when the polynomial is divided by x – 1. x3 + mx2 + nx – 3 = 21x3 + mx2 + nx – 3 =3 23 + (m2)2 + n(2) – 3 = 21(-1)3 + m(-1)2 + n(-1) -3 = 3 4m + 2n = 16-1 + m – n -3 = 3 m – n
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