...What is Algebra? Algebra is a branch of mathematics that uses mathematical statements to describe relationships between things that vary over time. These variables include things like the relationship between supply of an object and its price. When we use a mathematical statement to describe a relationship, we often use letters to represent the quantity that varies, sisnce it is not a fixed amount. These letters and symbols are referred to as variables. (See the Appendix One for a brief review of constants and variables.) The mathematical statements that describe relationships are expressed using algebraic terms, expressions, or equations (mathematical statements containing letters or symbols to represent numbers). Before we use algebra to find information about these kinds of relationships, it is important to first cover some basic terminology. In this unit we will first define terms, expressions, and equations. In the remaining units in this book we will review how to work with algebraic expressions, solve equations, and how to construct algebraic equations that describe a relationship. We will also introduce the notation used in algebra as we move through this unit. History of algebra The history of algebra began in ancient Egypt and Babylon, where people learned to solve linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations such as x2 + y2 = z2, whereby several unknowns are involved. The ancient Babylonians solved arbitrary quadratic...
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...Many people are intimidated and afraid of mathematics and algebra largely due to the fact that upon first glance, certain problems or expressions may seem overwhelmingly large, difficult, or complicated. Along with remembering formulas, this can often times lead to anger, confusion, and frustration. There are several very important key elements and aspects involved within mathematics that helps combat this confusion and frustration and can even help the most intimidated person feel at ease and comfortable with solving these problems. This particular report will demonstrate the importance of understanding certain key mathematical principles and components and show how understanding and utilizing certain mathematical definitions can help limit the amount of confusion and intimidation one may have. These definitions include but are not limited to simplifying, adding like terms, coefficient, distributive property, and removing parenthesis. This report will also demonstrate how much easier and more simplistic mathematics and algebra can be by remembering and utilizing just a few important concepts. Example 1: 2^a(a-5) +4(a-5) This is the first example that will be used. The variable a is used. This particular example has a coefficient of two. Step 1: The distributive property can be utilized to multiply 2a by everything inside of the parenthesis (a-5 in this case) resulting in: 2a^2-10a Step 2: The distributive property is used once again to multiply 4 by everything in the...
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...The relational algebra is a theoretical language with operations that work on one or more relations to define another relation without changing the original relation. Thus, both the operands and the results are relations; hence the output from one operation can become the input to another operation. This allows expressions to be nested in the relational algebra. This property is called closure. Relational algebra is an abstract language, which means that the queries formulated in relational algebra are not intended to be executed on a computer. Relational algebra consists of group of relational operators that can be used to manipulate relations to obtain a desired result. Knowledge about relational algebra allows us to understand query execution and optimization in relational database management system. Role of Relational Algebra in DBMS Knowledge about relational algebra allows us to understand query execution and optimization in relational database management system. The role of relational algebra in DBMS is shown in Fig. 3.1. From the figure it is evident that when a SQL query has to be converted into an executable code, first it has to be parsed to a valid relational algebraic expression, then there should be a proper query execution plan to speed up the data retrieval. The query execution plan is given by query optimizer. Relational Algebra Operations Operations in relational algebra can be broadly classified into set operation and database operations...
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...History of algebra The history of algebra began in ancient Egypt and Babylon, where people learned to solve linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations such as x2 + y2 = z2, whereby several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equations by essentially the same procedures taught today. They also could solve some indeterminate equations. The Alexandrian mathematicians Hero of Alexandria and Diophantus continued the traditions of Egypt and Babylon, but Diophantus's book Arithmetica is on a much higher level and gives many surprising solutions to difficult indeterminate equations. This ancient knowledge of solutions of equations in turn found a home early in the Islamic world, where it was known as the "science of restoration and balancing." (The Arabic word for restoration, al-jabru,is the root of the word algebra.) In the 9th century, the Arab mathematician al-Khwarizmi wrote one of the first Arabic algebras, a systematic exposé of the basic theory of equations, with both examples and proofs. By the end of the 9th century, the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebra and solved such complicated problems as finding x, y, and z such that x + y + z = 10, x2 + y2 = z2, and xz = y2. Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by medieval times Islamic mathematicians were able to talk about arbitrarily...
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...Algebra 2 Lesson 5-5 Example 1 Equation with Rational Roots Solve 2x2 – 36x + 162 = 32 by using the Square Root Property. 2x2 – 36x + 162 = 32 Original equation 2(x2 – 18x + 81) = 2(16) Factor out the GCF. x2 – 18x + 81 = 16 Divide each side by 2. (x – 9)2 = 16 Factor the perfect trinomial square. x – 9 = Square Root Property x – 9 = ±4 = 4 x = 9 ± 4 Add 9 to each side. x = 9 + 4 or x = 9 – 4 Write as two equations. x = 13 x = 5 Solve each equation. The solution set is {5, 13}. You can check this result by using factoring to solve the original equation. Example 2 Equation with Irrational Roots Solve x2 + 10x + 25 = 108 by using the Square Root Property. x2 + 10x + 25 = 108 Original equation (x + 5)2 = 108 Factor the perfect square trinomial. x + 5 = Square Root Property x = –5 ±6 Add –5 to each side; = 6 x = –5 + 6 or x = –5 – 6 Write as two equations. x ≈ 5.4 x ≈ –15.4 Use a calculator. The exact solutions of this equation are –5 – 6 and –5 + 6. The approximate solutions are –15.4 and 5.4. Check these results by finding and graphing the related quadratic function. x2 + 10x + 25 = 108 Original equation x2 + 10x – 83 = 0 Subtract 108 from each side. y = x2 + 10x – 83 Related quadratic function. CHECK Use the ZERO function of a graphing calculator. The approximate zeros of the...
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...Name: ______________________ Class: _________________ Date: _________ ID: A Solving Real-World problems with System of Linear Equations ____ 1 Mr. Frankel bought 7 tickets to a puppet show and spent $43. He bought a combination of child tickets for $4 each and adult tickets for $9 each. Which system of equations below will determine the number of adult tickets, a, and the number of child tickets, c, he bought? A. a = c - 9 9a + 4c = 43 B. 9a + 4c = 43 a +c=7 C. a + c = 301 a +c=7 D. 4a + 4c = 50 a +c=7 2 Tyrone is packaging a mix of bluegrass seed and drought-resistant seed for people buying grass seed for their lawns. The bluegrass seed costs him $2 per pound while the drought-resistant grass seed costs him $3 per pound. a. Write an equation showing that Tyrone spent $68 altogether for the two types of grass seed. b. Write an equation showing that Tyrone bought a total of 25 lb of the two types of grass seed. c. Solve the system of equations to find out how many pounds of each type of grass seed Tyrone bought. Mr. Jarvis invested a total of $9,112 in two savings accounts. One account earns 7.5% simple interest per year and the other earns 8.5% simple interest per year. Last year, the two investments earned a total of $884.88 in interest. Write a system of equations that could be used to determine the amount Mr. Jarvis initially invested in each account. Let x represent the amount invested at 7.5% and let y represent the amount invested at 8.5%. A. x + y = 9, 112 0.075x +...
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...MAPÚA INSTITUTE OF TECHNOLOGY Department of Mathematics COURSE SYLLABUS 1. Course Code: Math 10-3 2. Course Title: Algebra 3. Pre-requisite: none 4. Co-requisite: none 5. Credit: 3 units 6. Course Description: This course covers discussions on a wide range of topics necessary to meet the demands of college mathematics. The course discussion starts with an introductory set theories then progresses to cover the following topics: the real number system, algebraic expressions, rational expressions, rational exponents and radicals, linear and quadratic equations and their applications, inequalities, and ratio, proportion and variations. 7. Student Outcomes and Relationship to Program Educational Objectives Student Outcomes Program Educational Objectives 1 2 (a) an ability to apply knowledge of mathematics, science, and engineering √ (b) an ability to design and conduct experiments, as well as to analyze and interpret from data √ (c) an ability to design a system, component, or process to meet desired needs √ (d) an ability to function on multidisciplinary teams √ √ (e) an ability to identify, formulate, and solve engineering problems √ (f) an understanding of professional and ethical responsibility √ (g) an ability to communicate effectively √ √ (h) the broad education necessary to understand the impact of engineering solutions in the global and societal context √ √ (i) a recognition of the need for, and an ability to engage...
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... | | |Axia College | | |MAT/117 Version 7 | | |Algebra 1B | Copyright © 2010, 2009, 2007 by University of Phoenix. All rights reserved. Course Description This course explores advanced algebra concepts and assists in building the algebraic and problem-solving skills developed in Algebra 1A. Students solve polynomials, quadratic equations, rational equations, and radical equations. These concepts and skills serve as a foundation for subsequent business coursework. Applications to real-world problems are also explored throughout the course. This course is the second half of the college algebra sequence, which began with MAT/116, Algebra 1A. Policies Faculty and students/learners will be held responsible for understanding and adhering to all policies contained within the following two documents: • University policies: You must be logged into the student website to view this document. • Instructor policies: This document is posted in the Course Materials forum. University policies are subject to change. Be sure to read the policies...
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...1.1 EXERCISE SET In Exercises 1–14, write each English phrase as an algebraic expression. Let x represent the number. 1. Five more than a number X+5 3. Four less than a number X-4 5. Four times a number 4X 7. Ten more than twice a number 2X+10 9. The difference of six and half of a number 6 – ½X 11. Two less than the quotient of four and a number 4/X -2 13. The quotient of three and the difference of five and a number = 3/5-X In Exercises 15–26, evaluate each algebraic expression for the given value or values of the variable(s). 15. 7 + 5x, for x = 10 7+5.10 7+50 = 57 17. 6x − y, for x = 3 and y = 8 6.3-8 18-8=10 19. x2 + 3x, for = 1 10/9 21. x2 − 6x + 3, for x = 7 49-42+3 = 10 23. 4 + 5(x − 7)3, for x = 9 4+5 (9.-7 ) ³ 4+5 (2)³ 4+5.8 4+40 = 44 25. x2 − 3(x − y), for x = 8 and y = 2 X²-3X+3Y 8²-3 (8)+3(2) 64-24+6 = 46 expresses the relationship between Fahrenheit temperature, F, and Celsius temperature, C. In Exercises 95–96, use the formula to convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale. 95. 50°F = 10 C = ( 50-32) = 5/9 C = (12 ) 5/9 C+ 10 A football was kicked vertically upward from a height of 4 feet with an initial speed of 60 feet per second. The formula H= 4+60t -16t² describes the ball’s height above the ground, h, in feet, t seconds after it was kicked. Use this formula to solve Exercises 97–98. 97. What was the ball’s height 2 seconds...
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...Algebra I Quarter 3 Exam Name/Student Number:__________________________ Score:_______/________ Directions: For each question show all work that is required to arrive at the solution. Save this document with your answers and submit as an attachment to be graded. Simplify each expression. Use positive exponents. 1. m3n–6p0 2. a 4 b 3 ab 2 3. (x–2y–4x3) –2 4. Write the explicit formula that represents the geometric sequence -2, 8, -32, 128 5. Evaluate the function f (x) 4 • 7x for x 1 and x = 2. Show your work. 6. Simplify the quotient 4.5 x 103 9 x 107 . Write your answer in scientific notation. Show your work. Simplify the expressions. Show your work. 7. 3x(4x4 – 5x) 8. (5x4 – 3x3 + 6x) – ( 3x3 + 11x2 – 8x) 9. (x – 2) (3x-4) 10. (x + 6)2 Factor each expression. Show your work. 11. r2 + 12r + 27 12. g2 – 9 13. 2p3 + 6p2 + 3p + 9 Solve each quadratic equation. Show your work. 14. (2x – 1)(x + 7) = 0 15. x2 + 3x = 10 16. 4x2 = 100 17. Find the roots of the quadratic equation x2 – 8x = 9 by completing the square. Show your work. 18. Use the discriminant to find the number of real solutions of the equation 3x2 – 5x + 4 = 0. Show your work. A water balloon is tossed into the air with an upward velocity of 25 ft/s. Its height h(t) in ft after t seconds is given by the function h(t) = − 16t2 + 25t + 3. Show your work. 19. After how many seconds will the balloon hit the ground? (hint: Use the...
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...Algebra 2 Honors Name ________________________________________ Test #1 1st 9-weeks September 2, 2011 SHOW ALL WORK to ensure maximum credit. Each question is worth 10 points for a total of 100 points possible. Extra credit is awarded for dressing up. 1. Write the solutions represented below in interval notation. A.) [pic] B.) [pic] 2. Use the tax formula [pic] A.) Solve for I. B.) What is the income, I, when the Tax value, T, is $184? 3. The M&M’s company makes individual bags of M&M’s for sale. In production, the company allows between 20 and 26 m&m’s, including 20 and 26. Write an absolute value inequality describing the acceptable number of m&m’s in each bag. EXPLAIN your reasoning. 4. Solve and graph the solution. [pic] 5. Solve and graph the solution. [pic] 6. Solve. [pic] 7. Solve. [pic] 8. True or False. If false, EXPLAIN why it is false. A.) An absolute value equation always has two solutions. B.) 3 is a solution to the absolute value inequality [pic] C.) 8 is a solution to the compound inequality x < 10 AND x > 0. 9. Solve for w. [pic] 10. You plant a 1.5 foot tall sawtooth oak that grows 3.5 feet per year. You want to know how many years it would take for the tree to outgrow your 20 foot roof. A.) Write an inequality that defines x as the number of years of growth. B.) Determine the number of years, to nearest hundredth, it...
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...SCHAUM’S outlines SCHAUM’S 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 written 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 bulksales@mcgraw-hill.com. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies,...
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...A Brief Look at the Origin of Algebra Connie Beach Professor Clifton E. Collins, Sr. Math 105: Introduction to College Mathematics May 22, 2010 Abstract In this paper we look at the history of algebra and some of its different writers. Algebra originated in ancient Egypt and Babylon around 1650 B.C. Diophantus of Alexandria, a Greek mathematician, and Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī, a Persian mathematician from Baghdad, astronomer and geographer, shared the credit of being the founders of algebra. Diophantus, who is known as the “father of algebra”, carried on the work of the ancient Egyptians and Babylonians, but the word Algebra actually came from the word al-jabr, which is from al-Khwārizmī’s work, Kitab al-Jabr wa-l-Muqabala. The algebraic notation had gone through 3 stages: rhetorical (or verbal), stage, syncopated (use of abbreviated words) stage, and symbolism (the use of letters for the unknown) stage. As a matter of fact, the algebra that we know of today began during the 16th century, even though its history shows that it began almost 4000 years ago. A Brief Look at the Origin of Algebra I have always had a love for math. My favorite math class was Algebra; in fact, I had taken Algebra I, II, III, and IV all through high school, and aced every class. I can just look at a problem and know the answer. Then, I returned to college after 30 years, and took an Intro to College Math class. I wasn’t sure if I still remembered what I had learned...
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...Algebra I Suggested Teaching Strategies The curriculum guide is a set of suggested teaching strategies designed to be only a starting point for innovative teaching. The teaching strategies are optional, not mandatory. A teaching strategy in this guide could be a task, activity, or suggested method that is part of an instructional unit. It should not be considered sufficient to teach the competency and the associated objective(s); the teaching strategy could be one small component of the unit. There may not be enough instructional time to utilize every strategy in the curriculum guide. The 2007 Mississippi Mathematics Framework Revised includes the Depthof-Knowledge (DOK) level for each objective. As closely as possible, each strategy addresses the DOK level specified for that objective or a higher level. Suggestions or techniques for increasing the level of thinking may be included in the strategy(ies). In addition, the process strands (problem solving, communication, connections, reasoning and proof, and representation) are included in the strategies. The purpose of the suggested teaching strategies is to assist school districts and teachers in the development of possible methods of organizing the competencies and objectives to be taught. Since the competencies and objectives require multiple assessment methods, some assessment ideas may be included in the strategy. October 2007 2007 Mississippi Mathematics Framework Revised Strategies Comp. 1 2 Obj. a g ...
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...Math 202 - Assignment 6 Authors: Yusuf Goren, Miguel-Angel Manrique and Rory Laster Exercise 14.8.1. Proof. The discriminant of x4 + 1 is D = 256 = 28 . We have x4 + 1 ≡ (x + 1)4 (mod 2). Let p be an odd prime (so p D), and suppose the irreducible factors of x4 + 1 have degrees n1 , n2 , . . . , nk . By Corollary 41, the Galois group of x4 + 1 contains an element with cycle structure (n1 , n2 , . . . , nk ). Since the Galois group of x4 + 1 over Q is the Klein 4-group, in which every element has order dividing 2, it follows that each ni = 1 or 2. This gives the possibilities (1, 1, 1, 1), (1, 1, 2), (2, 2). However, D is a square and so the Galois group in contained in A4 ; in particular it contains no transpositions, so (1, 1, 2) is ruled out. This leaves the possibilities (1, 1, 1, 1), and (2, 2), which correspond to the factorization into 4 linear factors or 2 quadratic factors, respectively. Exercise 14.8.3. Proof. The polynomial f (x) = x5 + 20x + 16 is irreducible mod 3 and hence must be irreducible. The Galois group is therefore a transitive subgroup of S5 . The discriminant of f (x) is 216 56 and hence a square; therefore the Galois group is a subgroup of A5 . Modulo 7, we have factorization into irreducibles as f (x) ≡ (x + 2)(x + 3)(x3 + 2x2 + 5x + 5) (mod 7). Therefore the Galois group contains a 3 cycle. From the table on page 643, we see that the Galois group must be isomorphic to A5 . Exercise 14.8.6. Proof. By Eisenstein at 3, we see that f (x) is irreducible,...
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