DOCSLIB.ORG

Basic Algebra Review

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Basic Algebra Review SC_03213974739_rp08.qxd 1/15/08 9:15 AM Page 1 Algebra Review Exponents (continued) Polynomials (continued) Factoring (continued) Rational Expressions Rational Expressions Equations of Lines Quotient Rules FOIL Expansion for Multiplying Two To find the value(s) for which a rational (continued) TwoVariables (continued) Factoring Trinomials, Z Binomials expression is undefined, set the denominator If a 0, Leading Term Z x2 Intercepts 0 equal to 0 and solve the resulting equation. SIMPLIFYING COMPLEX FRACTIONS i. Zero exponent: a = 1 i. Multiply the first terms. 2 + + Z To find the x-intercept, let y = 0. To factor ax bx c, a 1: Lowest Terms Numbers Linear Equations ii. Multiply the outer terms. Method 1 To find the y-intercept, let x = 0. -n = 1 By Grouping To write a rational expression in lowest terms: ii. Negative exponents: a n iii. Multiply the inner terms. i. Simplify the numerator and denominator FRACTIONS Definition of Subtraction Properties a i. Find m and n such that i. Factor the numerator and denominator. Slope iv. Multiply the last terms. = + = separately. m mn ac and m n b. Suppose (x , y ) and (x , y ) are two differ- Addition and Subtraction x - y = x + -y i. Addition: The same quantity may be a - ii. Divide out common factors. ii. Divide by multiplying the simplified 1 1 2 2 1 2 iii. Quotient rule: = m n v. Collect like terms. Z added to (or subtracted from) each side of n a ii. Then numerator by the reciprocal of the ent points on a line. If x1 x2, then the i. To add or subtract fractions with the same Subtracting Real Numbers a 2 ϩ ϩ ϭ 2 ϩ ϩ ϩ an equality without changing the solution. ax bx c ax mx nx c. OPERATIONS ON RATIONAL simplified denominator. slope is denominator, add or subtract the numera- i. Change the subtraction symbol to the SPECIAL PRODUCTS - tors and keep the same denominator. ii. Multiplication: Each side of an equality iv. Negative to positive: iii. Group the first two terms and the last EXPRESSIONS rise y2 y1 addition symbol. Square of a Binomial two terms. Method 2 m = = . may be multiplied (or divided) by the -m n run x - x ii. To add or subtract fractions with differ- ii. Change the sign of the number being a b 2 2 2 Multiplying Rational Expressions i. Multiply the numerator and denomina- 2 1 same nonzero number without chang- = , a Z 0, b Z 0 x + y = x + 2xy + y iv. Follow the steps for factoring by grouping. ent denominators, find the LCD and -n m 1 2 tor of the complex fraction by the LCD subtracted. ing the solution. b a - 2 = 2 - + 2 i. Multiply numerators and multiply The slope of a vertical line is undefined. write each fraction with this LCD. Then - x y x 2xy y By Trial and Error of all the denominators in the complex iii. Add using the rules for adding real m m 1 2 denominators. follow the procedure in step i. Solving Linear Equalities a b fraction. The slope of a horizontal line is 0. numbers. = , a Z 0, b Z 0 i. Factor a as pq and c as mn. ii. Factor numerators and denominators. a b a b Parallel lines have the same slope. Multiplication i. Simplify each side separately. b a Product of the Sum and Difference of ii. For each such factorization, form the ii. Write in lowest terms. Multiplying Real Numbers Two Terms + + iii. Write expression in lowest terms. Multiply numerators and multiply ii. Isolate the variable term on one side. Scientific Notation product px m qx n and Perpendicular lines have slopes that are 2 2 1 21 2 i. Multiply the absolute value of the two x + y x - y = x - y expand using FOIL. Dividing Rational Expressions SOLVING EQUATIONS WITH negative reciprocals of each other. denominators. iii. Isolate the variable. A number written in scientific notation is in 1 21 2 numbers. * n i. Multiply the first rational expression by RATIONAL EXPRESSIONS Division the form a 10 , where a has one digit in iii. Stop when the expansion matches the EQUATIONS OF LINES ii. If the two numbers have the same sign, front of the decimal point and that digit is Dividing a Polynomial by a original trinomial. the reciprocal of the second rational i. Find the LCD of all denominators in the Multiply the first fraction by the reciprocal APPLICATIONS the product is positve. If the two num- nonzero. To write a number in scientific Monomial expression. equation. Slope–intercept form: y = mx + b, of the second fraction. bers have different signs, the product is i. Assign a variable to the unknown Remainder Theorem notation, move the decimal point to follow Divide each term of the polynomial by the ii. Multiply numerators and multiply where m is the slope, and 0, b is the negative. quantity in the problem. If the polynomial P(x) is divided by x – a, then ii. Multiply each side of the equation by 1 2 the first nonzero digit. If the decimal point monomial: denominators. the LCD. y-intercept. ORDER OF OPERATIONS ii. Write an equation involving the unknown. has been moved n places to the left, the the remainder is equal to P(a). x = # 1 Z + iii. Factor numerators and denominators. Simplify within parentheses, brackets, or Definition of Division: x , y 0 exponent on 10 is n. If the decimal point p q p q iii. Solve the resulting equation. x y y y iii. Solve the equation. = + Factor Theorem Intercept form: + = 1, absolute value bars or above and below has been moved n places to the right, the r r r iv. Write expression in lowest terms. iv. Check that the resulting solutions satisfy a b Division by 0 is undefined. For a polynomial P(x) and number a, fraction bars first, in the following order. FORMULAS exponent on 10 is –n. Finding the Least Common the original equation. Dividing a Polynomial by a if P(a) = 0, then x – a is a factor of P(x). where a, 0 is the x-intercept, and 0, b is i. Apply all exponents. Dividing Real Numbers Denominator (LCD) 1 2 1 2 i. To find the value of one of the variables in Polynomial the y-intercept. i. Divide the absolute value of a formula, given values for the others, sub- SPECIAL FACTORIZATIONS ii. Perform any multiplications or divisions i. Factor each denominator into prime Equations of Lines - = - the numbers. stitute the known values into the formula. Polynomials Use long division or synthetic division. Point–slope form: y y1 m x x1 , from left to right. Difference of Squares factors. TwoVariables 1 2 iii. Perform any additions or subtractions ii. If the signs are the same, the answer is ii. To solve a formula for one of the vari- A polynomial is an algebraic expression 2 2 where m is the slope and x1, y1 is any Graphing Simple Polynomials x - y = x + y x - y ii. List each different factor the greatest 1 2 from left to right. positive. If the signs are different, the ables, isolate that variable by treating made up of a term or a finite sum of terms 1 21 2 number of times it appears in any one An ordered pair is a solution of an equation point on the line. answer is negative. with real or complex coefficients and whole the other variables as constants (num- i. Determine several points (ordered pairs) Perfect Square Trinomials denominator. if it satisfies the equation. Standard form: Ax + By = C bers) and using the steps for solving number exponents. satisfying the polynomial equation. 2 + + 2 = + 2 VARIABLES, EXPRESSIONS, x 2xy y x y If the value of either variable in an equation PROPERTIES OF REAL NUMBERS equations. 1 2 iii. Multiply the factors from step ii. Vertical line: x = a AND EQUATIONS The degree of a term is the sum of the ii. Plot the points. x2 - 2xy + y2 = x - y 2 is given, the value of the other variable can Commutative Properties exponents on the variables. The degree of a 1 2 Writing a Rational Expression with a be found by substitution. Horizontal line: y = b An expression containing a variable is evalu- iii. Connect the points with a smooth Difference of Cubes ated by substituting a given number for the + = + polynomial is the highest degree amongst all of curve. Specified Denominator a b b a Exponents 3 - 3 = - 2 + + 2 variable. its terms. x y x y x xy y i. Factor both denominators. GRAPHING LINEAR EQUATIONS ab = ba 1 21 2 Values for a variable that make an equation For any integers m and n, the following rules A monomial is a polynomial with only one Sum of Cubes ii. Determine what factors the given To graph a linear equation: Systems of Linear Equations hold: term. Factoring true are solutions of the equation. Associative Properties x3 + y3 = x + y x2 - xy + y2 denominator must be multiplied by to 1 21 2 i. Find at least two ordered pairs that satisfy TWO VARIABLES a + b + c = a + b + c Product Rule A binomial is a polynomial with exactly two Finding the Greatest Common Factor equal the one given. the equation. REAL NUMBERS AND 1 2 1 2 m # n m+n terms. An ordered pair is a solution of a system if it ab c = a bc a a = a (GCF) SOLVING QUADRATIC EQUATIONS iii. Multiply the rational expression by that ii.
Load more

Recommended publications
  • LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
    LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students.
    [Show full text]
  • Schaum's Outline of Linear Algebra (4Th Edition)
    SCHAUM’S SCHAUM’S outlines outlines Linear Algebra Fourth Edition Seymour Lipschutz, Ph.D. Temple University Marc Lars Lipson, Ph.D. University of Virginia Schaum’s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2009, 2001, 1991, 1968 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior writ- ten permission of the publisher. ISBN: 978-0-07-154353-8 MHID: 0-07-154353-8 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-154352-1, MHID: 0-07-154352-X. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at [email protected]. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work.
    [Show full text]
  • Problems in Abstract Algebra
    STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth 10.1090/stml/082 STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 00A07, 12-01, 13-01, 15-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-82 Library of Congress Cataloging-in-Publication Data Names: Wadsworth, Adrian R., 1947– Title: Problems in abstract algebra / A. R. Wadsworth. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Student mathematical library; volume 82 | Includes bibliographical references and index. Identifiers: LCCN 2016057500 | ISBN 9781470435837 (alk. paper) Subjects: LCSH: Algebra, Abstract – Textbooks. | AMS: General – General and miscellaneous specific topics – Problem books. msc | Field theory and polyno- mials – Instructional exposition (textbooks, tutorial papers, etc.). msc | Com- mutative algebra – Instructional exposition (textbooks, tutorial papers, etc.). msc | Linear and multilinear algebra; matrix theory – Instructional exposition (textbooks, tutorial papers, etc.). msc | Group theory and generalizations – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA162 .W33 2017 | DDC 512/.02–dc23 LC record available at https://lccn.loc.gov/2016057500 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.
    [Show full text]
  • From Arithmetic to Algebra
    From arithmetic to algebra Slightly edited version of a presentation at the University of Oregon, Eugene, OR February 20, 2009 H. Wu Why can’t our students achieve introductory algebra? This presentation specifically addresses only introductory alge- bra, which refers roughly to what is called Algebra I in the usual curriculum. Its main focus is on all students’ access to the truly basic part of algebra that an average citizen needs in the high- tech age. The content of the traditional Algebra II course is on the whole more technical and is designed for future STEM students. In place of Algebra II, future non-STEM would benefit more from a mathematics-culture course devoted, for example, to an understanding of probability and data, recently solved famous problems in mathematics, and history of mathematics. At least three reasons for students’ failure: (A) Arithmetic is about computation of specific numbers. Algebra is about what is true in general for all numbers, all whole numbers, all integers, etc. Going from the specific to the general is a giant conceptual leap. Students are not prepared by our curriculum for this leap. (B) They don’t get the foundational skills needed for algebra. (C) They are taught incorrect mathematics in algebra classes. Garbage in, garbage out. These are not independent statements. They are inter-related. Consider (A) and (B): The K–3 school math curriculum is mainly exploratory, and will be ignored in this presentation for simplicity. Grades 5–7 directly prepare students for algebra. Will focus on these grades. Here, abstract mathematics appears in the form of fractions, geometry, and especially negative fractions.
    [Show full text]
  • Choosing Your Math Course
    CHOOSING YOUR MATH COURSE When choosing your first math class, it’s important to consider your current skills, the topics covered, and course expectations. It’s also helpful to think about how each course aligns with both your interests and your career and transfer goals. If you have questions, talk with your advisor. The courses on page 1 are developmental-level and the courses on page 2 are college-level. • Developmental math courses (Foundations of Math and Math & Algebra for College) offer the chance to develop concepts and skills you may have forgotten or never had the chance to learn. They focus less on lecture and more on practicing concepts individually and in groups. Your homework will emphasize developing and practicing new skills. • College-level math courses build on foundational skills taught in developmental math courses. Homework, quizzes, and exams will often include word problems that require multiple steps and incorporate a variety of math skills. You should expect three to four exams per semester, which cover a variety of concepts, and shorter weekly quizzes which focus on one or two concepts. You may be asked to complete a final project and submit a paper using course concepts, research, and writing skills. Course Your Skills Readiness Topics Covered & Course Expectations Foundations of 1. All multiplication facts (through tens, preferably twelves) should be memorized. In Foundations of Mathematics, you will: Mathematics • learn to use frections, decimals, percentages, whole numbers, & 2. Whole number addition, subtraction, multiplication, division (without a calculator): integers to solve problems • interpret information that is communicated in a graph, chart & table Math & Algebra 1.
    [Show full text]
  • Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students
    EDUCATOR’S PRACTICE GUIDE A set of recommendations to address challenges in classrooms and schools WHAT WORKS CLEARINGHOUSE™ Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students NCEE 2015-4010 U.S. DEPARTMENT OF EDUCATION About this practice guide The Institute of Education Sciences (IES) publishes practice guides in education to provide edu- cators with the best available evidence and expertise on current challenges in education. The What Works Clearinghouse (WWC) develops practice guides in conjunction with an expert panel, combining the panel’s expertise with the findings of existing rigorous research to produce spe- cific recommendations for addressing these challenges. The WWC and the panel rate the strength of the research evidence supporting each of their recommendations. See Appendix A for a full description of practice guides. The goal of this practice guide is to offer educators specific, evidence-based recommendations that address the challenges of teaching algebra to students in grades 6 through 12. This guide synthesizes the best available research and shares practices that are supported by evidence. It is intended to be practical and easy for teachers to use. The guide includes many examples in each recommendation to demonstrate the concepts discussed. Practice guides published by IES are available on the What Works Clearinghouse website at http://whatworks.ed.gov. How to use this guide This guide provides educators with instructional recommendations that can be implemented in conjunction with existing standards or curricula and does not recommend a particular curriculum. Teachers can use the guide when planning instruction to prepare students for future mathemat- ics and post-secondary success.
    [Show full text]
  • Taming the Unknown. a History of Algebra from Antiquity to the Early Twentieth Century, by Victor J
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 4, October 2015, Pages 725–731 S 0273-0979(2015)01491-6 Article electronically published on March 23, 2015 Taming the unknown. A history of algebra from antiquity to the early twentieth century, by Victor J. Katz and Karen Hunger Parshall, Princeton University Press, Princeton and Oxford, 2014, xvi+485 pp., ISBN 978-0-691-14905-9, US $49.50 1. Algebra now Algebra has been a part of mathematics since ancient times, but only in the 20th century did it come to play a crucial role in other areas of mathematics. Algebraic number theory, algebraic geometry, and algebraic topology now cast a big shadow on their parent disciplines of number theory, geometry, and topology. And algebraic topology gave birth to category theory, the algebraic view of mathematics that is now a dominant way of thinking in almost all areas. Even analysis, in some ways the antithesis of algebra, has been affected. As long ago as 1882, algebra captured some important territory from analysis when Dedekind and Weber absorbed the Abel and Riemann–Roch theorems into their theory of algebraic function fields, now part of the foundations of algebraic geometry. Not all mathematicians are happy with these developments. In a speech in 2000, entitled Mathematics in the 20th Century, Michael Atiyah said: Algebra is the offer made by the devil to the mathematician. The devil says “I will give you this powerful machine, and it will answer any question you like. All you need to do is give me your soul; give up geometry and you will have this marvellous machine.” Atiyah (2001), p.
    [Show full text]
  • Linear Algebra Methods in Combinatorics
    Linear Algebra methods in Combinatorics Dani¨elKroes Jacob Naranjo Jiaxi Nie [email protected] [email protected] [email protected] Jason O'Neill Nicholas Sieger Sam Sprio Emily Zhu [email protected] [email protected] [email protected] [email protected] Fall 2019 Quarter Abstract Linear algebra tools have been used to solve many problems in extremal combinatorics. The far reaching nature of the subject matter has resulted in a book [4] written by Babai and Frankl. Many of the proofs in the area are short, elegant and straightforward and as a result fit perfectly in a graduate student seminar. We will start with a few results which are fundamental in extremal set theory including the oddtown/eventown problem and the Frankl-Wilson Theorem. The seminar will culminate with some big results in the past few years including the method of slice rank to solve capset problem and the recently solved sensitivity conjecture. 1 Introduction These notes are comprised from an eight lecture series for graduate students in combinatorics at UCSD during the Fall 2019 Quarter. The organization of these expository notes is as follows. Each section corresponds to a fifty minute lecture given as part of the seminar. We shall first establish some common notation. 1.1 Notation Throughout these notes, we let [n] := f1; 2; : : : ; ng and consider 2[n] := fA ⊂ [n]g: Moreover, we are interested in the collection of size k subsets of an n-element set which we denote [n] as k := fA ⊂ [n] : jAj = kg: [n] [n] We denote subfamilies of k and 2 by calligraphic letters, sets by capital letters and elements by lower case letters.
    [Show full text]
  • Elementary Algebra > Syllabus
    San Antonio College · - · MATH-Mathematics Elementary Algebra MATH-0410 Summer - 8 Week Session Summer 2018 Section 026.14294 4-4-1 Credits 06/04/2018 to 07/26/2018 Modified 05/31/2018 Contact Information Professor Roland Treviño McCreless Hall, MCCH 126C [email protected] (Canvas email preferred) Math Department – MCCH 221 – 210.486.0270 Administrative Services Specialist - Patricia Gonzalez – [email protected] Academic Unit Assistant - Cynthia Morton – [email protected] Program Coordinator - Paula McKenna – [email protected] Department Chair – Dr. Said Fariabi – [email protected] Materials PREREQUISITES, CO-REQUISITES and OTHER REQUIREMENTS: Prerequisite(s):TSI score MATH Numeracy 310-335 with ABE 3 -6. Course placement advisement is available in the Mathematics/Computer Science Office located at MCCH 221. TEXTBOOKS (including ISBN#) and REQUIRED MATERIALS/RECOMMENDED READINGS: The student should go to www.connectmath.com to register for the online math program that accompanies this course. Register as a new student even if you have used this program in the past. You will be asked for the following Course Code: E4AFH-YHTVQ You already purchased access to this program at registration. There is no need for any more purchases beyond this. COURSE CONTENT: Topics include those listed below. Please see the Methods of Measurement section below to see other requirements such as exams. Chapter 1: Whole Numbers 1.2 – 1.6 (REVIEW ONLY) 1.7 Exponents, Algebraic Expressions, and the Order of Operations Chapter 2: Integers and Algebraic
    [Show full text]
  • Mathematics.Pdf
    Algebra Algebra II w/ Trigonometry AP Calculus AB AP Calculus BC AP Statistics Calculus III College Algebra Geometry Linear Algebra Pre-Calculus Pre-Calculus Career Internship Program Mathematics 101 Mr. Collin Voigt, Division Chair TEL: SC: (708) 579-6580, NC: (708) 579-6410 FAX: (708) 579-6038 EMAIL: [email protected] Mr. Joseph Barker, Assistant Division Chair Ms. Annette Orrico, Assistant Division Chair TEL: SC (708) 579-6581, NC (708) 579-6412 TEL: SC: (708) 579-6583, NC: (708) 579-6411 FAX: (708) 579-6038 FAX: (708) 579-6038 EMAIL: [email protected] EMAIL: [email protected] Mathematics Department Philosophy Th e mathematics curriculum has been developed to help students value mathematics, become confi dent in their abilities to do mathematics, become mathematical problem solvers, and to communicate and reason mathemati- cally. Students, as a result of the high school mathematics experiences, should be able to model problems with the appropriate operations and equations, apply a variety of approaches and techniques to solve problems, understand the underlying mathematical features of problems, see the applicability of mathematical ideas to common and complex problems, use logical reasoning to present an argument, and employ technology to explore mathematical ideas and solve problems. Students who successfully completed Algebra (Accel) in Grade 7th or 8th grade will receive one unit of high school credit on a pass/fail basis. Th e high school credit will be awarded aft er successful completion of one year of mathematics while enrolled in high school. Mathematics 102 Mathematics Department Standards The LTHS Mathematics Department has adopted the following eight principles in conjunction with both the Illinois State Standards and the Common Core State Standards.
    [Show full text]
  • Abstract Algebra Paul Garrett
    - Abstract Algebra Paul Garrett ii I covered this material in a two-semester graduate course in abstract algebra in 2004-05, rethinking the material from scratch, ignoring traditional prejudices. I wrote proofs which are natural outcomes of the viewpoint. A viewpoint is good if taking it up means that there is less to remember. Robustness, as opposed to fragility, is a desirable feature of an argument. It is burdensome to be clever. Since it is non-trivial to arrive at a viewpoint that allows proofs to seem easy, such a viewpoint is revisionist. However, this is a good revisionism, as opposed to much worse, destructive revisionisms which are nevertheless popular, most notably the misguided impulse to logical perfection [sic]. Logical streamlining is not the same as optimizing for performance. The worked examples are meant to be model solutions for many of the standard traditional exercises. I no longer believe that everyone is obliged to redo everything themselves. Hopefully it is possible to learn from others’ efforts. Paul Garrett June, 2007, Minneapolis Garrett: Abstract Algebra iii Introduction Abstract Algebra is not a conceptually well-defined body of material, but a conventional name that refers roughly to one of the several lists of things that mathematicians need to know to be competent, effective, and sensible. This material fits a two-semester beginning graduate course in abstract algebra. It is a how-to manual, not a monument to traditional icons. Rather than an encyclopedic reference, it tells a story, with plot-lines and character development propelling it forward. The main novelty is that most of the standard exercises in abstract algebra are given here as worked examples.
    [Show full text]
  • Elementary Algebra
    ELEMENTARY ALGEBRA 10 Overview The Elementary Algebra section of ACCUPLACER contains 12 multiple choice Algebra questions that are similar to material seen in a Pre-Algebra or Algebra I pre-college course. A calculator is provided by the computer on questions where its use would be beneficial. On other questions, solving the problem using scratch paper may be necessary. Expect to see the following concepts covered on this portion of the test: Operations with integers and rational numbers, computation with integers and negative rationals, absolute values, and ordering. Operations with algebraic expressions that must be solved using simple formulas and expressions, adding and subtracting monomials and polynomials, multiplying and dividing monomials and polynomials, positive rational roots and exponents, simplifying algebraic fractions, and factoring. Operations that require solving equations, inequalities, and word problems, solving linear equations and inequalities, using factoring to solve quadratic equations, solving word problems and written phrases using algebraic concepts, and geometric reasoning and graphing. Testing Tips Use resources provided such as scratch paper or the calculator to solve the problem. DO NOT attempt to only solve problems in your head. Start the solving process by writing down the formula or mathematic rule associated with solving the particular problem. Put your answer back into the original problem to confirm that your answer is correct. Make an educated guess if you are unsure of the answer. 11 Algebra
    [Show full text]