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Math 100B Textbook Beginning Algebra MATH 100B Math Study Center BYU-Idaho Preface This math book has been created by the BYU-Idaho Math Study Center for the college student who needs an introduction to Algebra. This book is the product of many years of implementation of an extremely successful Beginning Algebra program and includes perspectives and tips from experienced instructors and tutors. Videos of instruction and solutions can be found at the following url: https://youtu.be/-taqTmqALPg?list=PL_YZ8kB-SJ1v2M-oBKzQWMtKP8UprXu8q The following individuals have assisted in authoring: Rich Llewellyn Daniel Baird Diana Wilson Kelly Wilson Kenna Campbell Robert Christenson Brandon Dunn Hannah Sherman We hope that it will be helpful to you as you take Algebra this semester. The BYU-Idaho Math Study Center Math 100B Table of Contents Chapter 1 – Arithmetic and Variables Section 1.1 – LCM and Factoring…..……………………………………………………………6 Finding factors, Prime Factorization, Least Common Multiples Section 1.2 – Fractions…………………………………………………………………………..14 Addition, Subtraction, Multiplication, Division Section 1.3 – Decimals, Percents………………………………………………………..……….21 Decimals; Conversion between decimals, fractions, and percents; finding percents of totals Section 1.4 – Rounding, Estimation, Exponents, Roots, and Order of Operations…...…………28 Natural number exponents and roots of integers Section 1.5 – Variables and Geometry ……………………………………………………….…39 Introductions to variables and their use in formulas Section 1.6 – Negative and Signed Numbers...…………………………………………………..51 Section 1.7 – Simplifying…………………………………………………………………..……60 Combining; Commutative, Associative, and Distributive Properties; Identities and Inverses Chapter 2 – Linear Equations Section 2.1 – Linear Equations …………………………………………………………………79 Steps 2 and 3, Addition and Multiplication Principles Section 2.2 – Applications of Linear Equations…………………………………………………91 Translation, Shapes, and Formulas Section 2.3 – More Linear Equations…………………………………………………………..100 Step 1, Combining and clearing parentheses Section 2.4 – Applications of Linear Equations………………………………………………..106 Substitution, Consecutive, Percent Word Problems, Forward and Backward Section 2.5 – Linear Equations with Fractions…………………………………………………117 Step 1, Clearing fractions; Summary of 3-step process Section 2.6 – Inequalities……………………………………………………………………….122 Greater than, less than symbols, Solving Inequalities Section 2.7 – Applications of Inequalities……………………………………………………...130 Inequality translation and Word Problems Chapter 3 – Lines Section 3.1 – Graphing…………………………………………………………………………144 Graphing by Pick ‘n Stick, Intercepts Section 3.2 – Slope……………………………………………………………………………..154 Section 3.3 – Graphing with slope, Slope-Intercept……………………………………………161 Section 3.4 – Graphing with slope, Standard……...……………………………………………168 Section 3.5 – Writing Equations………………………………………………………………..178 Find the Equation of a line given a slope and a point or two points Chapter 4 – Exponents and Polynomials Section 4.1 – Laws of Exponents………………………………………………………………193 Mutliplication, Power, Division, Zero rules for exponents; Scientific Notation Arithmetic Section 4.2 – Intro to Polynomials……………………………………………………………..202 Terminology, Addition and Subtraction of Polynomials Section 4.3 – Multiplication of Polynomials…………………………………………………..211 Monomial × Polynomials, Special Cases, Binomial Squared, Binomial × Binomial Section 4.4 – Division of Polynomials…………………………………………………………221 Division of Polynomials by Monomials Chapter 5 – Factoring Section 5.1 – Intro to Factoring, Methods 1 and 2……………………………………………..234 Factoring by pulling out GCF, Grouping with 4 terms Section 5.2 – Factoring Trinomials, Method 3…………………………………………………244 Factoring Trinomials with lead coefficient =1 Section 5.3 – Factoring Trinomials, Method 4…………………………………………………249 Factoring Trinomials with lead coefficient ≠ 1, ac-method Section 5.4 – Factoring Special Cases………………………………………………………….255 Factoring Perfect Squares and Difference of Squares Section 5.5 – Factoring With All Methods……………………………………………………..264 Holistic Approach to determine which method to use Section 5.6 – Solving Polynomial Equations…………………………………………………..268 Zero Multiplication; Solving Polynomial Equations by factoring 5 Chapter 1: ARITHMETIC & VARIABLES OVERVIEW 1.1 LCM and Factoring 1.2 Fractions 1.3 Decimals 1.4 Exponents, Order of Operations, Rounding 1.5 Variables and Formulas 1.6 Negatives 1.7 Laws of Simplifying 6 1.1 Factoring and Finding the LCM (Least Common Multiple) OBJECTIVES • Know how to find the factors of a number • Use factors to find two or more numbers LCM A RULES OF ARITHMETIC AND VARIABLES (Overview) Factoring a number means break that number down into numbers that go into it. Finding the different factors of a number can then help us in algebra, adding and subtracting fractions, and many other things. Finding the LCM is seeing the multiples of two numbers meet. This can be especially helpful for fractions. Throughout this chapter, you will be able to review all of arithmetic and the concepts and rules of what variables are and how they work. CHAPTER ONE TOPICS Find Least Common LCM AND FACTORING Find Factors Multiples (LCM) FRACTIONS Addition/Subtraction Multiplication Division DECIMALS AND Addition/Subtraction Multiplication Division PERCENTS Change to decimals Change to fractions ROUNDING, ESTIMATION, Nearest place value; EXPONENTS, ORDER OF Round and then compute OPERATIONS VARIABLES AND Replace numbers and FORMULAS make formulas NEGATIVES Addition/Subtraction Multiplication Division Laws of Simplifying B FACTORING DEFINITIONS & BASICS PRODUCT: Multiplication of two numbers will get you a product PRIME NUMBER: When a natural number has only two factors, the number 1 and itself. Example: The numbers 3, 5, 7, 11 are prime Example: The number 1 is not a prime number because it does not have two different factors. Section 1.1 7 ONE: If two numbers only have the number one as a common factor that means these numbers are “relatively prime”. Example: The numbers 7 and 12 are relatively prime because the only factor they have in common 1. FACTOR: The numbers that are multiplied together to get the product are factors. Example: The factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 because these are the only natural numbers that can multiply to get a product of 42. LCM: Least Common Multiple. You can find this when the multiples of two or more numbers meet up at a common number. Example: The multiples of 8 are 8, 16, 24, 32, 40, 48, and so on. The multiples of 12 are 12, 24, 36, 48, 60, and so on. To find the LCM you find the smallest multiple these two numbers have in common. 8 and 12 both have 24, 48 in common but we are looking for the least common multiple, which means the smallest number. Therefore, the LCM is 24 because it is the smallest common multiple between 8 and 12. I. Find All Factors Factoring is a useful and necessary skill when adding and subtracting fractions and will be a very helpful skill to have in algebra. Find Factors 1. Start with 1 and move up finding numbers that are factors. 2. List the numbers you have found. These are all the factors EXAMPLES For factoring number, we simply write down all the numbers that go into it. Number to be Factored Factors 12 1, 2, 3, 4, 6, 12 20 1, 2, 4, 5, 10, 20 27 1, 3, 9, 27 Section 1.1 8 EXAMPLES Find all factors of 48 1 Step 1: Find all factors that multiply to be the product 48, starting with the number 1 and 48 and moving up the 1 Ɛ 48 Ɣ 48 2 Ɛ 24 Ɣ 48 number line. 3 Ɛ 16 Ɣ 48 4 Ɛ 12 Ɣ 48 Note: In the last box we see 8x6=48. The 8 has already 6 Ɛ 8 Ɣ 48 8 Ɛ 6 Ɣ 48 been used in the factors, so we know that all of the factors have been found. To make them a little easier to see we can Step 3: Now we will list all the numbers we used up put them in numerical order from smallest until we saw the repeated number and these will be our to largest. factors. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 1, 48, 2, 24, 3, 16, 4, 12, 6, 8 II. Find Prime Factorization of a Number Once again, factor each number individually just like we did in the previous example. Prime Factorize 1. Find a factor, break the number up. 2. Repeat until all factors are prime. EXAMPLES Find the prime factorization of 60 and 72 2 60 72 Step 1: Find the factors of 60 until the factors are all prime.. 10 6 2 36 5 2 2 3 2 18 Step 2: Repeat step one for 72 6 3 2 3 ̏̉ Ɣ ̋ Ɛ ̋ Ɛ ̌ Ɛ ̎ ̐̋ Ɣ ̋ Ɛ ̋ Ɛ ̋ Ɛ ̌ Ɛ ̌ Section 1.1 9 Find LCM (Least Common Multiple) EXAMPLE 3 Multiples of 6 Multiples of 4 4 and 6 have the numbers 6 4 12 , 24, 36, 48 . 12 8 as common multiples. 18 12 24 16 The smallest of these is: 30 20 12 36 24 42 28 Thus 12 is the LCM (Least Common Multiple) 48 32 of 4 and 6. 54 36 60 40 Find the LCM (Observation) 1. Write out multiples for two or more numbers 2. Find what the two lists have in common. 3. The smallest multiple is the LCM Section 1.1 10 EXAMPLE Find the LCM of 4 and 5 4 4 5 Step 1: Write out the multiples for 4 and 5 8 10 Step 2: The first number that both multiples hit is 20 12 15 16 20 Step 3: The LCM of 4 and 5 is 20. 20 25 24 30 28 35 32 40 We can also find the LCM of numbers by using prime factorization EXAMPLE 5 Prime factorization of 4 Prime factorization of 6 4 = 2 × 2 6 = 2 × 3 2 × 2 × 3 Prime factorization of the smallest number that both will go in to 2 × 2 × 3 = 12 Thus 12 is the LCM of 4 and 6 Section 1.1 11 Find the LCM (Prime Factorization) 1. Prime factorize 2. Write out the smallest number that they all can go into. 3. Multiply it out = LCM EXAMPLE 6 Find the LCM of 40 and 36.
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