Lesson #9 Simplifying Rational Expressions
|
|
- Kristopher Haynes
- 5 years ago
- Views:
Transcription
1 Lesson #9 Simplifying Rational Epressions A.A.6 Perform arithmetic operations with rational epressions and rename to lowest terms Factor the following epressions: A. 7 4 B. y C. y 49y Simplify: 5 5 = = 0 0 Definitions Rational Epression: A fraction that contains variables. In mathematical epressions such as 6 or 5 6 you can use any number for. Mathematically speaking, the epression is defined for all values of. In other words, there are no restrictions on what we can use for. Rational epressions, on the other hand, can have restrictions on what we can plug in for the variable(s). There are certain values of the variables that will make the epression undefined which means it does not have a definite value. When is a rational epression undefined and why? State the restrictions for each rational epression y ~ ~
2 Simplifying a rational epression means cancelling or dividing out any common factors in the numerator and denominator. We ADD like terms. In rational epressions we cancel common factors NOT like terms. Rules for simplifying rational epressions Monomials (One term) ) Since monomials contain multiplication only, and multiplication is the same operation as the division indicated by the fraction bar, you can cancel any common factors. 6 y y 4 Polynomials (More than one term) ) Since polynomials contain addition/subtraction, they are not the same operation as division. Therefore, you must factor polynomials before you can find the common factors to cancel. Parenthesis around polynomials can help you remember this. 6 6 State the restrictions on each rational epression and then simplify y 0 4y y y ~ ~
3 5. 4y 6y 5 6. r 5r 0 4r Continue simplifying these rational epressions. You do not have to find the restricted values ac ab bc b a ab b 0. 9 y 9 6y y ~ ~
4 Lesson #0 Absolute Value Equations A.A. Solve absolute value equations and inequalities involving linear epressions in one variable Absolute value: a number s distance from zero on a number line. If you know that the absolute value of a number is 0 or in other words that the number is 0 units away from zero on the number line, that number could be or. If 0, what could equal? We solve more comple equations in the same way. Eample) 7.. If necessary isolate the absolute value. (SADMEP). Get rid of the absolute value sign using its inverse: set the epression in the absolute value equal to the answer and the (answer).. Solve each resulting equation and write your solution in roster notation. 4. Check your solutions in the original equation. A common mistake is to change the sign in the absolute value. Do not do this. Always change the sign of the answer. If you know that the absolute value of a number is - or in other words that the number is - units away from zero on the number line, that number could be or. Eample : 4 ~ 4 ~
5 Practice. Solve each equation ~ 5 ~
6 Lesson # Multiplying (and Dividing) Rat. Ep. A.A.6 Multiplying Fractions: Perform arithmetic operations with rational epressions and rename to lowest terms Simplify the following problem without cross canceling: = Simplify the following problem by cross canceling first: = Imagine doing the problem, , without cross cancelling. You would have to use the distributive property in both the numerator & denominator. It would take forever. Q: Why can you cross cancel when multiplying rational epressions? A: Rational epressions are division. For eample, 5 0 is the same as 5 0. Since multiplication and division are really the same operation, you can cancel or divide out common factors before you multiply. Steps for Multiplying Rational Epressions 5 4 ) Factor each numerator and denominator 0 5 (State the restricted values if asked to) ) Cancel any factors that appear in one of the numerators and one of the denominators. ) Multiply the remaining terms in the numerator. Multiply the remaining terms in the denominator. 4) Check to see if you can reduce any further. E) 4 ~ 6 ~
7 Division Dividing is the same as. For eample, 5 is the same 5. This eample helps you to remember to,, when dividing with fractions or rational epressions. Note: With division, the restricted values will come from the two original denominators as well as the new denominator on the second fraction after changing the problem to multiplication. ) ) 8 y y 6y 6y 9y ) ) a 5 ( a 5) a 0 4a 0 5) y y 4y y ~ 7 ~
8 Lesson # Adding and Subtracting Rat. Ep. A.A.6 Perform arithmetic operations with rational epressions and rename to lowest terms Adding is really combining things that are the same or : For each problem, simplify either term first if possible. Decide if the terms are like terms. Write no if they are not. If they are like terms write the name of the terms and combine them together y y 5y What are like terms with fractions? 5 = 4 5 = The denominator: The numerator: ~ 8 ~
9 Day : Monomial Denominators Least Common Monomial Denominators: a. 4, 7 4 b. 5, 6 8 c., 4 6 d., y e., 4 f. y, 7 y, 6 4y g.,, 4 y y z z h., 4 Steps for Adding and Subtracting Rational Epressions 0. State the restricted values.. Find the LCD and write it on the side. 6 y. Goal: Each rational epression must have the LCD. a. Multiply both numerator and denominator by any missing factors. Use the distributive property if necessary. b. CAUTION: DO NOT cancel here. You wanted to get an LCD. Cancelling will get rid of it.. Add the numerators ( ) and keep the common denominator ( ) Multiplying the numerator and denominator by the same value is the same as multiplying the fraction by. Therefore its value has not changed. 4. Reduce the resulting fraction if possible. REMEMBER YOUR PARENTHESIS! ~ 9 ~
10 ) 4 = ) 6 = ) b a a b = y y 4) y = 5) Which epression is equivalent to () a b () a b a b () a b (4) 6) The sum of, 0 5, is () () () 5 (4) 5 5 a b? This is not meant to confuse you. The problem is just telling you what the restricted value is. 7) What is the sum of 7n and 7 n? () n () 0 n () 4 n (4) 58 n ~ 0 ~
11 Day : Binomial Denominators Nothing changes ecept now you have to think about binomial factors. Remember your. The factors of 5 are: The factors of 5 are: The factors of 0 y are: The factors of 0 are: Remember, when finding the LCD, you are simply making sure every factor in both denominators is present. Least Common Binomial Denominators: a. 4, b. 5, c., 6 6 d., 4 e. 4, f. y 7 y 6,, 7 4 y y y y g., 4 6 h. 4, ~ ~
12 Steps for Adding & Subtracting Rational Epressions Nothing has changed! 0. State the restricted values.. Find the LCD and write it on the side. 5. Goal: Each rational epression must have the LCD. a. Multiply both numerator and denominator by any missing factors. Use the distributive property if necessary. b. CAUTION: DO NOT cancel here. You wanted to get an LCD. Cancelling will get rid of it.. Add the numerators ( ) and keep the common denominator ( ) 4. Reduce the resulting fraction if possible. REMEMBER YOUR PARENTHESIS! ) 8 ) = 9 ~ ~
13 ) ( ) = 4) = 5) 6 6 = 4 6) = 4 5 7) ~ ~
14 Lesson # Rational Equations A.A. Solve rational equations and inequalities Good news: With rational equations, we can always get rid of the fractions, turning them into regular quadratic or linear equations. Denominators in rational epressions are really what you are dividing by. To get LCD rid of division in an equation, multiply both sides by the. Cross multiplying is basically the same thing because you are multiplying by both denominators. ) 7 ) 7 ) ) 4 When solving a rational equation, any solution is valid UNLESS it is a restricted value. Find the restrictions for each of the following equations ~ 4 ~
15 Steps for Solving Rational Equations 4 5. State any restrictions for the rational epressions in the equations. These values cannot be solutions to your equations.. Simplify either side of the equation if possible.. Find the LCD of the denominators on both sides of the equation. 4. Multiply both sides of the equation by the common denominator so that each denominator will cancel. 5. Solve the resulting equation. 6. Check your solutions to make sure that they are not restricted values. Solve each of the following equations. Epress your solutions in roster notation. 6 A. 5 0 B. C ~ 5 ~
16 D. 4 4 E. 5 a a a 6 F. a a a b b G. b 5 b 5 ~ 6 ~
17 Lesson #4 Absolute Value Inequalities A.A. Solve absolute value equations and inequalities involving linear epressions in one variable Review: Steps for Solving Quadratic Inequalities Make the inequality an equation and solve it. 5 6 Set up a number line with the two solutions on it. These are called the critical points. Use open or closed circles depending on the type of inequality. Choose numbers as test points: one smaller than the critical points, one between the critical points, and one larger than the critical points. = = = Check to see if each test point works in the original inequality. Shade the regions on the number line where the test point made the inequality true. Write your solution in either interval notation or set builder notation. When solving absolute value inequalities, most of the steps are the same as quadratic inequalities. These steps that are the same are bolded. Solve the absolute value inequality as if it were an equation. Steps for Solving Absolute Value Inequalities 7 ~ 7 ~
18 Set up a number line with the two solutions on it. These are called the critical points. Use open or closed circles depending on the type of inequality. = = = Choose numbers as test points: one smaller than the critical points, one between the critical points, and one larger than the critical points. Check to see if each test point works in the original inequality. Shade the regions on the number line where the test point made the inequality true. Write your solution in either interval notation or set builder notation. A) Find and graph the solution set: 5 B) Find and graph the solution set: 4 ~ 8 ~
19 C) *Find and graph the solution set: 8 D) *Find and graph the solution set: Thinking questions:. *Which of the following inequalities has (the null or empty set) as its solution set? a. b. c. d.. *Which of the following inequalities has all real numbers as its solution set? a. b. c. d. ~ 9 ~
20 Lesson #5 Simplifying Comple Fractions A.A.7 Simplify comple fractional epressions Simplify the following rational epressions: 0 = = 4 5 Comple fractions - within. For eample, 6 8 is a comple fraction. Comple fractions are not in simplest form because there are still operations to perform. How to simplify comple fractions Epress in simplest form: a. Perform the addition/subtraction in the numerator if the comple fraction. b. Perform the addition/subtraction in the denominator if the comple fraction. (This can be done at the same time as step a). 6 8 = c. Once there is only one fraction in the numerator and one fraction in the denominator, perform the remaining division: keep, change, flip, and multiply. d. Reduce the resulting fraction if possible. ) 4 ~ 0 ~
21 ) ) z z 5 5 4) r m r m 5) 6) 7 ~ ~
22 Lesson #6 Solving Rational Inequalities A.A. Solve rational equations and inequalities Solving rational inequalities will be similar to solving quadratic and absolute value inequalities. The main difference is that the critical values will include the solutions to an equation AND the values where the rational epression(s) are undefined. Find the value(s) where the rational epression(s) are undefined. Any undefined values will be critical points Solve the rational inequality as if it were an equation. Set up a number line with any solutions or restricted values on it. These are called the critical points. Use open or closed circles depending on the type of inequality. = = = Choose numbers in each interval created the critical points. (With rational inequalities the number of critical points can vary.) Test those numbers in the original inequality. Shade the regions on the number line where the value works in the inequality. Write your solution in either interval notation or set builder notation. ~ ~
23 . Solve each inequality and epress your solution in either set builder or interval notation ~ ~
Section 5.1 Model Inverse and Joint Variation
108 Section 5.1 Model Inverse and Joint Variation Remember a Direct Variation Equation y k has a y-intercept of (0, 0). Different Types of Variation Relationship Equation a) y varies directly with. y k
More informationDay 3: Section P-6 Rational Expressions; Section P-7 Equations. Rational Expressions
1 Day : Section P-6 Rational Epressions; Section P-7 Equations Rational Epressions A rational epression (Fractions) is the quotient of two polynomials. The set of real numbers for which an algebraic epression
More informationLESSON #34 - FINDING RESTRICTED VALUES AND SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II
LESSON #4 - FINDING RESTRICTED VALUES AND SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II A rational epression is a fraction that contains variables. A variable is very useful in mathematics. In
More informationReview of Rational Expressions and Equations
Page 1 of 14 Review of Rational Epressions and Equations A rational epression is an epression containing fractions where the numerator and/or denominator may contain algebraic terms 1 Simplify 6 14 Identification/Analysis
More informationReteach Variation Functions
8-1 Variation Functions The variable y varies directly as the variable if y k for some constant k. To solve direct variation problems: k is called the constant of variation. Use the known and y values
More informationACCUPLACER MATH 0310
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to
More informationAdding and Subtracting Rational Expressions
Adding and Subtracting Rational Epressions As a review, adding and subtracting fractions requires the fractions to have the same denominator. If they already have the same denominator, combine the numerators
More informationACCUPLACER MATH 0311 OR MATH 0120
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises
More information6.4 Division of Polynomials. (Long Division and Synthetic Division)
6.4 Division of Polynomials (Long Division and Synthetic Division) When we combine fractions that have a common denominator, we just add or subtract the numerators and then keep the common denominator
More informationDefine a rational expression: a quotient of two polynomials. ..( 3 10) (3 2) Rational expressions have the same properties as rational numbers:
1 UNIT 7 RATIONAL EXPRESSIONS & EQUATIONS Simplifying Rational Epressions Define a rational epression: a quotient of two polynomials. A rational epression always indicates division EX: 10 means..( 10)
More informationA2T. Rational Expressions/Equations. Name: Teacher: Pd:
AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 1-3 HW: Pages -3 in Packet o Day : SWBAT: Factor using the Greatest
More informationChapter 7 Rational Expressions, Equations, and Functions
Chapter 7 Rational Expressions, Equations, and Functions Section 7.1: Simplifying, Multiplying, and Dividing Rational Expressions and Functions Section 7.2: Adding and Subtracting Rational Expressions
More informationComplex fraction: - a fraction which has rational expressions in the numerator and/or denominator
Comple fraction: - a fraction which has rational epressions in the numerator and/or denominator o 2 2 4 y 2 + y 2 y 2 2 Steps for Simplifying Comple Fractions. simplify the numerator and/or the denominator
More informationLesson #33 Solving Incomplete Quadratics
Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique
More information7.1 Rational Expressions and Their Simplification
7.1 Rational Epressions and Their Simplification Learning Objectives: 1. Find numbers for which a rational epression is undefined.. Simplify rational epressions. Eamples of rational epressions: 3 and 1
More information4.3 Division of Polynomials
4.3 Division of Polynomials Learning Objectives Divide a polynomials by a monomial. Divide a polynomial by a binomial. Rewrite and graph rational functions. Introduction A rational epression is formed
More information8.3 Zero, Negative, and Fractional Exponents
www.ck2.org Chapter 8. Eponents and Polynomials 8.3 Zero, Negative, and Fractional Eponents Learning Objectives Simplify epressions with zero eponents. Simplify epressions with negative eponents. Simplify
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationCOUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra
COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed
More informationSection 4.3: Quadratic Formula
Objective: Solve quadratic equations using the quadratic formula. In this section we will develop a formula to solve any quadratic equation ab c 0 where a b and c are real numbers and a 0. Solve for this
More informationSelf-Directed Course: Transitional Math Module 4: Algebra
Lesson #1: Solving for the Unknown with no Coefficients During this unit, we will be dealing with several terms: Variable a letter that is used to represent an unknown number Coefficient a number placed
More informationAlgebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.
C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each
More informationMulti-Step Equations and Inequalities
Multi-Step Equations and Inequalities Syllabus Objective (1.13): The student will combine like terms in an epression when simplifying variable epressions. Term: the parts of an epression that are either
More informationAlgebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?
Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using
More information1. Write three things you already know about expressions. Share your work with a classmate. Did your classmate understand what you wrote?
LESSON 1: RATIONAL EXPONENTS 1. Write three things you already know about epressions. Share your work with a classmate. Did your classmate understand what you wrote?. Write your wonderings about working
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections.6 and.) 8. Equivalent Inequalities Definition 8. Two inequalities are equivalent
More informationEby, MATH 0310 Spring 2017 Page 53. Parentheses are IMPORTANT!! Exponents only change what they! So if a is not inside parentheses, then it
Eby, MATH 010 Spring 017 Page 5 5.1 Eponents Parentheses are IMPORTANT!! Eponents only change what they! So if a is not inside parentheses, then it get raised to the power! Eample 1 4 b) 4 c) 4 ( ) d)
More informationChapter 7 Class Notes. Intermediate Algebra, MAT1033C. SI Leader Joe Brownlee. Palm Beach State College
Chapter 7 Class Notes Intermediate Algebra, MAT033C Palm Beach State College Class Notes 7. Professor Burkett 7. Rational Expressions and Functions; Multiplying and Dividing Chapter 7 takes factoring to
More informationQUADRATIC EQUATIONS. + 6 = 0 This is a quadratic equation written in standard form. x x = 0 (standard form with c=0). 2 = 9
QUADRATIC EQUATIONS A quadratic equation is always written in the form of: a + b + c = where a The form a + b + c = is called the standard form of a quadratic equation. Eamples: 5 + 6 = This is a quadratic
More informationINTRODUCTION TO RATIONAL FUNCTIONS COMMON CORE ALGEBRA II
Name: Date: INTRODUCTION TO RATIONAL FUNCTIONS COMMON CORE ALGEBRA II Rational functions are simply the ratio of polynomial functions. They take on more interesting properties and have more interesting
More informationMath 154 :: Elementary Algebra
Math 4 :: Elementary Algebra Section. Additive Property of Equality Section. Multiplicative Property of Equality Section.3 Linear Equations in One-Variable Section.4 Linear Equations in One-Variable with
More informationBasic Property: of Rational Expressions. Multiplication and Division of Rational Expressions. The Domain of a Rational Function: P Q WARNING:
Basic roperties of Rational Epressions A rational epression is any epression of the form Q where and Q are polynomials and Q 0. In the following properties, no denominator is allowed to be zero. The Domain
More informationProblem 1 Oh Snap... Look at the Denominator on that Rational
Problem Oh Snap... Look at the Denominator on that Rational Previously, you learned that dividing polynomials was just like dividing integers. Well, performing operations on rational epressions involving
More informationINTRODUCTION TO RATIONAL EXPRESSIONS EXAMPLE:
INTRODUCTION TO RATIONAL EXPRESSIONS EXAMPLE: You decide to open a small business making gluten-free cakes. Your start-up costs were $, 000. In addition, it costs $ 0 to produce each cake. What is the
More informationRational Expressions & Equations
Chapter 9 Rational Epressions & Equations Sec. 1 Simplifying Rational Epressions We simply rational epressions the same way we simplified fractions. When we first began to simplify fractions, we factored
More informationAlgebra 1: Hutschenreuter Chapter 11 Note Packet Ratio and Proportion
Algebra 1: Hutschenreuter Chapter 11 Note Packet Name 11.1 Ratio and Proportion Proportion: an equation that states that two ratios are equal a c = b 0, d 0 a is to b as c is to d b d Etremes: a and d
More informationPractical Algebra. A Step-by-step Approach. Brought to you by Softmath, producers of Algebrator Software
Practical Algebra A Step-by-step Approach Brought to you by Softmath, producers of Algebrator Software 2 Algebra e-book Table of Contents Chapter 1 Algebraic expressions 5 1 Collecting... like terms 5
More informationRational Expressions
CHAPTER 6 Rational Epressions 6. Rational Functions and Multiplying and Dividing Rational Epressions 6. Adding and Subtracting Rational Epressions 6.3 Simplifying Comple Fractions 6. Dividing Polynomials:
More informationAlgebra, Part I. x m x = n xm i x n = x m n = 1
Lesson 7 Algebra, Part I Rules and Definitions Rules Additive property of equality: If a, b, and c represent real numbers, and if a=b, then a + c = b + c. Also, c + a = c + b Multiplicative property of
More informationMath 1320, Section 10 Quiz IV Solutions 20 Points
Math 1320, Section 10 Quiz IV Solutions 20 Points Please answer each question. To receive full credit you must show all work and give answers in simplest form. Cell phones and graphing calculators are
More informationUniversity of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra
University of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra Table of Contents Chapter The Algebra of Polynomials Chapter Factoring 7 Chapter 3 Fractions Chapter 4 Eponents and Radicals
More informationSec. 1 Simplifying Rational Expressions: +
Chapter 9 Rational Epressions Sec. Simplifying Rational Epressions: + The procedure used to add and subtract rational epressions in algebra is the same used in adding and subtracting fractions in 5 th
More informationAlgebra 2 Chapter 9 Page 1
Section 9.1A Introduction to Rational Functions Work Together How many pounds of peanuts do you think and average person consumed last year? Us the table at the right. What was the average peanut consumption
More informationDr. Relja Vulanovic Professor of Mathematics Kent State University at Stark c 2008
MATH-LITERACY MANUAL Dr. Relja Vulanovic Professor of Mathematics Kent State University at Stark c 2008 2 Algebraic Epressions 2.1 Terms and Factors 29 2.2 Types of Algebraic Epressions 32 2.3 Transforming
More information5.1 Simplifying Rational Expressions
5. Simplifying Rational Expressions Now that we have mastered the process of factoring, in this chapter, we will have to use a great deal of the factoring concepts that we just learned. We begin with the
More informationCHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section Multiplying and Dividing Rational Expressions
Name Objectives: Period CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section 8.3 - Multiplying and Dividing Rational Expressions Multiply and divide rational expressions. Simplify rational expressions,
More informationf(x) = 2x 2 + 2x - 4
4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms
More informationOBJECTIVES UNIT 1. Lesson 1.0
OBJECTIVES UNIT 1 Lesson 1.0 1. Define "set," "element," "finite set," and "infinite set," "empty set," and "null set" and give two examples of each term. 2. Define "subset," "universal set," and "disjoint
More information4.2 Reducing Rational Functions
Section. Reducing Rational Functions 1. Reducing Rational Functions The goal of this section is to review how to reduce a rational epression to lowest terms. Let s begin with a most important piece of
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More informationMath 1 Variable Manipulation Part 4 Student
Math 1 Variable Manipulation Part 4 Student 1 SOLVING AN EQUATION THAT INCLUDES ABSOLUTE VALUE SIGNS To solve an equation that includes absolute value signs, think about the two different cases-one where
More informationSECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION
2.25 SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION PART A: LONG DIVISION Ancient Example with Integers 2 4 9 8 1 In general: dividend, f divisor, d We can say: 9 4 = 2 + 1 4 By multiplying both sides
More informationAlgebra. Robert Taggart
Algebra Robert Taggart Table of Contents To the Student.............................................. v Unit 1: Algebra Basics Lesson 1: Negative and Positive Numbers....................... Lesson 2: Operations
More informationBishop Kelley High School Summer Math Program Course: Algebra 2 A
06 07 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 6 pages of this packet provide eamples as to how to work some of the problems
More informationRational Functions. A rational function is a function that is a ratio of 2 polynomials (in reduced form), e.g.
Rational Functions A rational function is a function that is a ratio of polynomials (in reduced form), e.g. f() = p( ) q( ) where p() and q() are polynomials The function is defined when the denominator
More information4.4 Rational Expressions
4.4 Rational Epressions Learning Objectives Simplify rational epressions. Find ecluded values of rational epressions. Simplify rational models of real-world situations. Introduction A rational epression
More informationPre-Algebra 8 Notes Unit 02B: Linear Equations in One Variable Multi-Step Equations
Pre-Algebra 8 Notes Unit 02B: Linear Equations in One Variable Multi-Step Equations Solving Two-Step Equations The general strategy for solving a multi-step equation in one variable is to rewrite the equation
More informationIntermediate Algebra. Gregg Waterman Oregon Institute of Technology
Intermediate Algebra Gregg Waterman Oregon Institute of Technology c August 2013 Gregg Waterman This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
More informationLESSON 8.1 RATIONAL EXPRESSIONS I
LESSON 8. RATIONAL EXPRESSIONS I LESSON 8. RATIONAL EXPRESSIONS I 7 OVERVIEW Here is what you'll learn in this lesson: Multiplying and Dividing a. Determining when a rational expression is undefined Almost
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types
More informationTABLE OF CONTENTS. Introduction to Finish Line Indiana Math 10. UNIT 1: Number Sense, Expressions, and Computation. Real Numbers
TABLE OF CONTENTS Introduction to Finish Line Indiana Math 10 UNIT 1: Number Sense, Expressions, and Computation LESSON 1 8.NS.1, 8.NS.2, A1.RNE.1, A1.RNE.2 LESSON 2 8.NS.3, 8.NS.4 LESSON 3 A1.RNE.3 LESSON
More informationChapter 9 Notes SN AA U2C9
Chapter 9 Notes SN AA U2C9 Name Period Section 2-3: Direct Variation Section 9-1: Inverse Variation Two variables x and y show direct variation if y = kx for some nonzero constant k. Another kind of variation
More information4.5 Multiplication and Division of Rational Expressions
.5. Multiplication and Division of Rational Epressions www.ck2.org.5 Multiplication and Division of Rational Epressions Learning Objectives Multiply rational epressions involving monomials. Multiply rational
More informationElementary Algebra Study Guide Some Basic Facts This section will cover the following topics
Elementary Algebra Study Guide Some Basic Facts This section will cover the following topics Notation Order of Operations Notation Math is a language of its own. It has vocabulary and punctuation (notation)
More informationBishop Kelley High School Summer Math Program Course: Algebra 2 A
015 016 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 16 pages of this packet provide eamples as to how to work some of the problems
More information5.2 Polynomial Operations
5.2 Polynomial Operations At times we ll need to perform operations with polynomials. At this level we ll just be adding, subtracting, or multiplying polynomials. Dividing polynomials will happen in future
More informationMath-1010 Lesson 4-2. Add and Subtract Rational Expressions
Math-00 Lesson - Add and Subtract Rational Epressions What are like terms? Like variables: Like powers: y y Multiples of the same variable same base and same eponent. Like radicals: same radicand and same
More informationUnit 9 Study Sheet Rational Expressions and Types of Equations
Algebraic Fractions: Unit 9 Study Sheet Rational Expressions and Types of Equations Simplifying Algebraic Fractions: To simplify an algebraic fraction means to reduce it to lowest terms. This is done by
More informationAP Calculus AB Summer Assignment
Name: AP Calculus AB Summer Assignment Due Date: The beginning of class on the last class day of the first week of school. The purpose of this assignment is to have you practice the mathematical skills
More informationBasic ALGEBRA 2 SUMMER PACKET
Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout
More informationIntermediate Algebra 100A Final Exam Review Fall 2007
1 Basic Concepts 1. Sets and Other Basic Concepts Words/Concepts to Know: roster form, set builder notation, union, intersection, real numbers, natural numbers, whole numbers, integers, rational numbers,
More informationCh. 12 Rational Functions
Ch. 12 Rational Functions 12.1 Finding the Domains of Rational F(n) & Reducing Rational Expressions Outline Review Rational Numbers { a / b a and b are integers, b 0} Multiplying a rational number by a
More informationTEKS: 2A.10F. Terms. Functions Equations Inequalities Linear Domain Factor
POLYNOMIALS UNIT TEKS: A.10F Terms: Functions Equations Inequalities Linear Domain Factor Polynomials Monomial, Like Terms, binomials, leading coefficient, degree of polynomial, standard form, terms, Parent
More informationAlgebra Final Exam Review Packet
Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:
More informationMath 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have
Math 10 Final Eam Review 1. 4 5 6 5 4 4 4 7 5 Worked out solutions. In this problem, we are subtracting one polynomial from another. When adding or subtracting polynomials, we combine like terms. Remember
More informationPolynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.
Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10
More informationFundamental Theorem of Algebra (NEW): A polynomial function of degree n > 0 has n complex zeros. Some of these zeros may be repeated.
.5 and.6 Comple Numbers, Comple Zeros and the Fundamental Theorem of Algebra Pre Calculus.5 COMPLEX NUMBERS 1. Understand that - 1 is an imaginary number denoted by the letter i.. Evaluate the square root
More informationFactor each expression. Remember, always find the GCF first. Then if applicable use the x-box method and also look for difference of squares.
NOTES 11: RATIONAL EXPRESSIONS AND EQUATIONS Name: Date: Period: Mrs. Nguyen s Initial: LESSON 11.1 SIMPLIFYING RATIONAL EXPRESSIONS Lesson Preview Review Factoring Skills and Simplifying Fractions Factor
More informationIntroduction. So, why did I even bother to write this?
Introduction This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The review contains the occasional
More informationSolving Equations. Solving Equations - decimal coefficients and constants. 2) Solve for x: 3(3x 6) = 3(x -2) 1) Solve for x: 5 x 2 28 x
Level C Review Packet This packet briefly reviews the topics covered on the Level A Math Skills Assessment. If you need additional study resources and/or assistance with any of the topics below, please
More informationUnit 2 Notes Packet on Quadratic Functions and Factoring
Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a
More informationAlgebra I Unit Report Summary
Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02
More informationR3.6 Solving Linear Inequalities. 3) Solve: 2(x 4) - 3 > 3x ) Solve: 3(x 2) > 7-4x. R8.7 Rational Exponents
Level D Review Packet - MMT This packet briefly reviews the topics covered on the Level D Math Skills Assessment. If you need additional study resources and/or assistance with any of the topics below,
More informationFinite Mathematics : A Business Approach
Finite Mathematics : A Business Approach Dr. Brian Travers and Prof. James Lampes Second Edition Cover Art by Stephanie Oxenford Additional Editing by John Gambino Contents What You Should Already Know
More informationReview for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.
LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in
More informationMA Lesson 25 Section 2.6
MA 1500 Lesson 5 Section.6 I The Domain of a Function Remember that the domain is the set of x s in a function, or the set of first things. For many functions, such as f ( x, x could be replaced with any
More informationMA 180 Lecture. Chapter 0. College Algebra and Calculus by Larson/Hodgkins. Fundamental Concepts of Algebra
0.) Real Numbers: Order and Absolute Value Definitions: Set: is a collection of objections in mathematics Real Numbers: set of numbers used in arithmetic MA 80 Lecture Chapter 0 College Algebra and Calculus
More informationPolynomials and Polynomial Functions
Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial
More informationP.1 Prerequisite skills Basic Algebra Skills
P.1 Prerequisite skills Basic Algebra Skills Topics: Evaluate an algebraic expression for given values of variables Combine like terms/simplify algebraic expressions Solve equations for a specified variable
More informationFigure 1.1: Positioning numbers on the number line.
1B. INEQUALITIES 13 1b Inequalities In Chapter 1, we introduced the natural numbers N = {1, 2, 3,...}, the whole numbers W = {0, 1, 2, 3,...}, and the integers Z = {..., 3, 2, 1, 0, 1, 2, 3,...}. Later
More informationMultiplying a Polynomial by a Monomial
Lesson -3 Multiplying a Polynomial by a Monomial Lesson -3 BIG IDEA To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial and add the products. In earlier chapters,
More informationWhat makes f '(x) undefined? (set the denominator = 0)
Chapter 3A Review 1. Find all critical numbers for the function ** Critical numbers find the first derivative and then find what makes f '(x) = 0 or undefined Q: What is the domain of this function (especially
More informationSection 3.6 Complex Zeros
04 Chapter Section 6 Complex Zeros When finding the zeros of polynomials, at some point you're faced with the problem x = While there are clearly no real numbers that are solutions to this equation, leaving
More information1 a) Remember, the negative in the front and the negative in the exponent have nothing to do w/ 1 each other. Answer: 3/ 2 3/ 4. 8x y.
AP Calculus Summer Packer Key a) Remember, the negative in the front and the negative in the eponent have nothing to do w/ each other. Answer: b) Answer: c) Answer: ( ) 4 5 = 5 or 0 /. 9 8 d) The 6,, and
More informationJune If you want, you may scan your assignment and convert it to a.pdf file and it to me.
Summer Assignment Pre-Calculus Honors June 2016 Dear Student: This assignment is a mandatory part of the Pre-Calculus Honors course. Students who do not complete the assignment will be placed in the regular
More informationEvaluate algebraic expressions for given values of the variables.
Algebra I Unit Lesson Title Lesson Objectives 1 FOUNDATIONS OF ALGEBRA Variables and Expressions Exponents and Order of Operations Identify a variable expression and its components: variable, coefficient,
More informationAlgebra I. abscissa the distance along the horizontal axis in a coordinate graph; graphs the domain.
Algebra I abscissa the distance along the horizontal axis in a coordinate graph; graphs the domain. absolute value the numerical [value] when direction or sign is not considered. (two words) additive inverse
More informationSeptember 12, Math Analysis Ch 1 Review Solutions. #1. 8x + 10 = 4x 30 4x 4x 4x + 10 = x = x = 10.
#1. 8x + 10 = 4x 30 4x 4x 4x + 10 = 30 10 10 4x = 40 4 4 x = 10 Sep 5 7:00 AM 1 #. 4 3(x + ) = 5x 7(4 x) 4 3x 6 = 5x 8 + 7x CLT 3x = 1x 8 +3x +3x = 15x 8 +8 +8 6 = 15x 15 15 x = 6 15 Sep 5 7:00 AM #3.
More informationSolutions to the Math 1051 Sample Final Exam (from Spring 2003) Page 1
Solutions to the Math 0 Sample Final Eam (from Spring 00) Page Part : Multiple Choice Questions. Here ou work out the problems and then select the answer that matches our answer. No partial credit is given
More informationAlgebra I Chapter 6: Solving and Graphing Linear Inequalities
Algebra I Chapter 6: Solving and Graphing Linear Inequalities Jun 10 9:21 AM Chapter 6 Lesson 1 Solve Inequalities Using Addition and Subtraction Vocabulary Words to Review: Inequality Solution of an Inequality
More information