Multiplying and Dividing Rational Expressions
|
|
- Florence Flowers
- 5 years ago
- Views:
Transcription
1 Multiplying and Dividing Rational Expressions
2 Warm Up Simplify each expression. Assume all variables are nonzero. 1. x 5 x 2 3. x 6 x 2 x 7 Factor each expression. 2. y 3 y 3 y 6 x 4 4. y 2 1 y 5 y 3 5. x 2 2x 8 (x 4)(x + 2) 6. x 2 5x x(x 5) 7. x 5 9x 3 x 3 (x 3)(x + 3)
3 Objectives Simplify rational expressions. Multiply and divide rational expressions.
4 rational expression Vocabulary
5 A rational expression is a quotient of two polynomials. 5 x Other examples of rational expressions include the following:
6 Because rational expressions are ratios of polynomials, you can simplify them the same way as you simplify fractions. Recall that to write a fraction in simplest form, you can divide out common factors in the numerator and denominator. Caution! When identifying values for which a rational expression is undefined, identify the values of the variable that make the original denominator equal to 0.
7 Example 1A: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined. 10x 8 6x x = Quotient of Powers Property 36 3 x4 The expression is undefined at x = 0 because this value of x makes 6x 4 equal 0.
8 Example 1B: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined. x 2 + x 2 x 2 + 2x 3 (x + 2)(x 1) (x 1)(x + 3) = (x + 2) (x + 3) Factor; then divide out common factors. The expression is undefined at x = 1 and x = 3 because these values of x make the factors (x 1) and (x + 3) equal 0.
9 Example 1B Continued Check Substitute x = 1 and x = 3 into the original expression. (1) 2 + (1) 2 (1) 2 + 2(1) 3 = 0 0 ( 3) 2 + ( 3) 2 ( 3) 2 + 2( 3) 3 = 4 0 Both values of x result in division by 0, which is undefined.
10 Check It Out! Example 1b Simplify. Identify any x-values for which the expression is undefined. 3x + 4 3x 2 + x 4 (3x + 4) (3x + 4)(x 1) = 1 (x 1) Factor; then divide out common factors. The expression is undefined at x = 1 and x = 4 3 because these values of x make the factors (x 1) and (3x + 4) equal 0.
11 Check It Out! Example 1b Continued Check Substitute x = 1 and x = the original expression. 4 3 into 3(1) + 4 3(1) 2 + (1) 4 = 7 0 Both values of x result in division by 0, which is undefined.
12 Check It Out! Example 1c Simplify. Identify any x-values for which the expression is undefined. 6x 2 + 7x + 2 6x 2 5x 6 (2x + 1)(3x + 2) (3x + 2)(2x 3) = (2x + 1) (2x 3) Factor; then divide out common factors. The expression is undefined at x = 2 and x = because these values of x make the factors (3x + 2) and (2x 3) equal 0.
13 Check It Out! Example 1c Continued Check Substitute x = 3 and x = 2 into the original expression. 2 3 Both values of x result in division by 0, which is undefined.
14 Example 2: Simplifying by Factoring by 1 Simplify 4x x 2. Identify any x values x 2 2x 8 for which the expression is undefined. 1(x 2 4x) x 2 2x 8 1(x)(x 4) (x 4)(x + 2) Factor out 1 in the numerator so that x 2 is positive, and reorder the terms. Factor the numerator and denominator. Divide out common factors. x (x + 2 ) Simplify. The expression is undefined at x = 2 and x = 4.
15 Example 2 Continued Check The calculator screens suggest that 4x x 2 = x except when x = 2 x 2 2x 8 (x + 2) or x = 4.
16 Check It Out! Example 2a Simplify 10 2x. Identify any x values x 5 for which the expression is undefined. 1(2x 10) x 5 1(2)(x 5) (x 5) Factor out 1 in the numerator so that x is positive, and reorder the terms. Factor the numerator and denominator. Divide out common factors. 2 1 Simplify. The expression is undefined at x = 5.
17 Check It Out! Example 2a Continued Check The calculator screens suggest that 10 2x x 5 = 2 except when x = 5.
18 Check It Out! Example 2b Simplify x 2 + 3x. Identify any x values 2x 2 7x + 3 for which the expression is undefined. 1(x 2 3x) 2x 2 7x + 3 1(x)(x 3) (x 3)(2x 1) Factor out 1 in the numerator so that x is positive, and reorder the terms. Factor the numerator and denominator. Divide out common factors. x 2x 1 Simplify. The expression is undefined at x = 3 and x =. 1 2
19 Check It Out! Example 2b Continued Check The calculator screens suggest that x 2 + 3x = x except when x = 2x 2 7x + 3 2x 1 and x =
20 You can multiply rational expressions the same way that you multiply fractions.
21 Example 3: Multiplying Rational Expressions Multiply. Assume that all expressions are defined. A. 3x 5 y 3 2x 3 y 10x3 y 4 7 9x 2 y 5 3x 5 y 3 10x3 y 4 2x 3 y 7 9x 2 y 5 5x 3 3y B. x 3 4x + 20 x + 5 x 2 9 x 3 4(x + 5) x + 5 (x 3)(x + 3) 1 4(x + 3)
22 Check It Out! Example 3 Multiply. Assume that all expressions are defined. A. 3 x 15 x 15 x x 2x x 7 2 x 4 20 x 4 2 B. 10x 40 x 2 6x + 8 x + 3 5x (x 4) (x 4)(x 2) x + 3 5(x + 3) 2x (x 2)
23 You can also divide rational expressions. Recall that to divide by a fraction, you multiply by its reciprocal = = 2 3
24 Example 4A: Dividing Rational Expressions Divide. Assume that all expressions are defined. 5x 4 8x 2 y y 5 5x 4 8x 2 y 8y Rewrite as multiplication by the reciprocal. 5x 4 8x 2 y 8y x 2 y
25 Example 4B: Dividing Rational Expressions Divide. Assume that all expressions are defined. x 4 9x 2 x 2 4x + 3 x4 + 2x 3 8x 2 x 2 16 x 4 9x 2 x 2 4x + 3 x 2 16 x 4 + 2x 3 8x 2 x 2 (x 2 9) x 2 4x + 3 x 2 16 x 2 (x 2 + 2x 8) x 2 (x 3)(x + 3) (x 3)(x 1) (x + 3)(x 4) (x 1)(x 2) (x + 4)(x 4) x 2 (x 2)(x + 4) Rewrite as multiplication by the reciprocal.
26 2x 2 7x 4 x 2 9 Check It Out! Example 4b Divide. Assume that all expressions are defined. 2x 2 7x 4 x 2 9 4x 2 1 8x 2 28x +12 8x 2 28x +12 4x 2 1 (2x + 1)(x 4) (x + 3)(x 3) 4(2x2 7x + 3) (2x + 1)(2x 1) (2x + 1)(x 4) (x + 3)(x 3) 4(x 4) (x +3) 4(2x 1)(x 3) (2x + 1)(2x 1)
27 Example 5A: Solving Simple Rational Equations Solve. Check your solution. x 2 25 x 5 = 14 (x + 5)(x 5) = 14 (x 5) x + 5 = 14 x = 9 Note that x 5.
28 Example 5A Continued Check x 2 25 x 5 = 14 (9)
29 Check It Out! Example 5a Solve. Check your solution. x 2 + x 12 x + 4 = 7 (x 3)(x + 4) = 7 (x + 4) x 3 = 7 Note that x 4. x = 4 Because the left side of the original equation is undefined when x = 4, there is no solution.
30 Check It Out! Example 5a Continued Check A graphing calculator shows that 4 is not a solution.
31 Check It Out! Example 5b Solve. Check your solution. 4x 2 9 2x + 3 = 5 (2x + 3)(2x 3) = 5 (2x + 3) 2x 3 = 5 x = 4 Note that x. 3 2
32 Check It Out! Example 5b Continued Check 4x 2 9 2x + 3 = 5 4(4) 2 9 2(4)
Warm Up Simplify each expression. Assume all variables are nonzero.
Warm Up Simplify each expression. Assume all variables are nonzero. 1. x 5 x 2 3. x 6 x 2 x 7 x 4 Factor each expression. 2. y 3 y 3 y 6 4. y 2 1 y 5 y 3 5. x 2 2x 8 (x 4)(x + 2) 6. x 2 5x x(x 5) 7. x
More informationMultiplying and Dividing Rational Expressions
Page 1 of 14 Multiplying and Dividing Rational Expressions Attendance Problems. Simplify each expression. Assume all variables are nonzero. x 6 y 2 1. x 5 x 2 2. y 3 y 3 3. 4. x 2 y 5 Factor each expression.
More informationSection 2.3 Rational Numbers. A rational number is a number that may be written in the form a b. for any integer a and any nonzero integer b.
Section 2.3 Rational Numbers A rational number is a number that may be written in the form a b for any integer a and any nonzero integer b. Why is division by zero undefined? For example, we know that
More information1-3 Multiplying and Dividing Real Numbers
Multiplying and Dividing 1-3 Multiplying and Dividing Real Numbers Real Numbers Warm Up Lesson Presentation Lesson Quiz 1 2 pts Bell Quiz 1-3 Add or Subtract 1. 3 8 2 pts 2. - 8 + 12 2 pts 3. 4 (-4) 2
More informationWorking with Rational Expressions
Working with Rational Expressions Return to Table of Contents 4 Goals and Objectives Students will simplify rational expressions, as well as be able to add, subtract, multiply, and divide rational expressions.
More informationAlgebra II Chapter 6: Rational Exponents and Radical Functions
Algebra II Chapter 6: Rational Exponents and Radical Functions Chapter 6 Lesson 1 Evaluate nth Roots and Use Rational Exponents Vocabulary 1 Example 1: Find nth Roots Note: and Example 2: Evaluate Expressions
More informationAlgebra 1 Review. Properties of Real Numbers. Algebraic Expressions
Algebra 1 Review Properties of Real Numbers Algebraic Expressions Real Numbers Natural Numbers: 1, 2, 3, 4,.. Numbers used for counting Whole Numbers: 0, 1, 2, 3, 4,.. Natural Numbers and 0 Integers:,
More informationRational number operations can often be simplified by converting mixed numbers to improper fractions Add EXAMPLE:
Rational number operations can often be simplified by converting mixed numbers to improper fractions Add ( 2) EXAMPLE: 2 Multiply 1 Negative fractions can be written with the negative number in the numerator
More informationA. Incorrect! To simplify this expression you need to find the product of 7 and 4, not the sum.
Problem Solving Drill 05: Exponents and Radicals Question No. 1 of 10 Question 1. Simplify: 7u v 4u 3 v 6 Question #01 (A) 11u 5 v 7 (B) 8u 6 v 6 (C) 8u 5 v 7 (D) 8u 3 v 9 To simplify this expression you
More informationAlgebra II Chapter 8 Part 2: Rational Functions
Algebra II Chapter 8 Part 2: Rational Functions Chapter 8 Lesson 4 Multiply and Divide Rational Functions Vocabulary Words to Review: Reciprocal The rules of fractions DO NOT change! *When adding and subtracting,
More informationRules of Exponents Part 1[Algebra 1](In Class Version).notebook. August 22, 2017 WARM UP. Simplify using order of operations. SOLUTION.
WARM UP Simplify using order of operations. Aug 22 3:22 PM 1 Aug 22 4:09 PM 2 WARM UP a) The equation 3(4x) = (4x)3 illustrates which property? b) Which property of real numbers is illustrated by the equation
More informationFinal Exam: Precalculus
Final Exam: Precalculus Apr. 17, 2018 ANSWERS Without Notes or Calculators Version A 1. Consider the unit circle: a. Angle in degrees: What is the angle in radians? What are the coordinates? b. Coordinates:
More informationFundamentals. Copyright Cengage Learning. All rights reserved.
Fundamentals Copyright Cengage Learning. All rights reserved. 1.4 Rational Expressions Copyright Cengage Learning. All rights reserved. Objectives The Domain of an Algebraic Expression Simplifying Rational
More informationMath-3 Lesson 3-6 Analyze Rational functions The Oblique Asymptote
Math- Lesson - Analyze Rational functions The Oblique Asymptote Quiz: a What is the domain? b Where are the holes? c What is the vertical asymptote? y 4 8 8 a -, b = c = - Last time Zeroes of the numerator
More informationLimits and an Introduction to Calculus. Copyright Cengage Learning. All rights reserved.
Limits and an Introduction to Calculus 12 Copyright Cengage Learning. All rights reserved. 12.2 TECHNIQUES FOR EVALUATING LIMITS Copyright Cengage Learning. All rights reserved. What You Should Learn Use
More information3.3 Division of Fractions and of Mixed Numbers
CCBC Math 0 Division of Fractions and of Mixed Numbers Section.. Division of Fractions and of Mixed Numbers Introduction: http://youtu.be/fsdtivjjq What does it mean to divide? The basic division questions
More informationLesson 1: Arithmetic Review
In this lesson we step back and review several key arithmetic topics that are extremely relevant to this course. Before we work with algebraic expressions and equations, it is important to have a good
More informationAdding and subtracting rational expressions is quite similar to adding and subtracting rational numbers (fractions).
7.2: Adding and Subtracting Rational Expressions, Simplifying Complex Fractions Adding and subtracting rational expressions is quite similar to adding and subtracting rational numbers (fractions). Adding
More information(-,+) (+,+) Plotting Points
Algebra Basics +y (-,+) (+,+) -x +x (-,-) (+,-) Plotting Points -y Commutative Property of Addition/Multiplication * You can commute or move the terms * This only applies to addition and multiplication
More information6.3. Complex Fractions
6. Comple Fractions 1. Simplify comple fractions by simplifying the numerator and denominator (Method 1).. Simplify comple fractions by multiplying by a common denominator (Method ).. Compare the two methods
More information1-3 Square Roots. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
1-3 Square Roots Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Round to the nearest tenth. 1. 3.14 3.1 2. 1.97 2.0 Find each square root. 3. 4 4. 25 Write each fraction in simplest form. 5. 6. Simplify.
More informationMultiplying and Dividing Rational Expressions
COMMON CORE Locker LESSON 9. Multiplying and Dividing Rational Expressions Name Class Date 9. Multiplying and Dividing Rational Expressions Essential Question: How can you multiply and divide rational
More informationMini-Lesson 1. Section 1.1: Order of Operations PEMDAS
Name: Date: 1 Section 1.1: Order of Operations PEMDAS If we are working with a mathematical expression that contains more than one operation, then we need to understand how to simplify. The acronym PEMDAS
More informationChapter 9 Review. By Charlie and Amy
Chapter 9 Review By Charlie and Amy 9.1- Inverse and Joint Variation- Explanation There are 3 basic types of variation: direct, indirect, and joint. Direct: y = kx Inverse: y = (k/x) Joint: y=kxz k is
More information4.1: Equivalent and Rational Expressions
Math 3201 Chapter 4: Rational Expressions and Equations 12 Hours 4.1: Equivalent and Rational Expressions A Rational Expression is any expression that can be written as the quotient of two polynomials,
More informationUnit 3: Multiplication and Division Reference Guide pages x 7 = 392 factors: 56, 7 product 392
Lesson 1: Multiplying Integers and Decimals, part 1 factor: any two or more numbers multiplied to form a product 56 x 7 = 392 factors: 56, 7 product 392 Integers: all positive and negative whole numbers
More informationWriting Linear Functions
2- Writing Linear Functions Objectives Use slope-intercept form and point-slope form to write linear functions. Write linear functions to solve problems. Vocabulary Point-slope form Why learn this? When
More informationFinding Asymptotes KEY
Unit: 0 Lesson: 0 Discontinuities Rational functions of the form f ( are undefined at values of that make 0. Wherever a rational function is undefined, a break occurs in its graph. Each such break is called
More information4.3 Rational Thinking
RATIONAL EXPRESSIONS & FUNCTIONS -4.3 4.3 Rational Thinking A Solidify Understanding Task The broad category of functions that contains the function!(#) = & ' is called rational functions. A rational number
More informationCSE 215: Foundations of Computer Science Recitation Exercises Set #4 Stony Brook University. Name: ID#: Section #: Score: / 4
CSE 215: Foundations of Computer Science Recitation Exercises Set #4 Stony Brook University Name: ID#: Section #: Score: / 4 Unit 7: Direct Proof Introduction 1. The statement below is true. Rewrite the
More informationChapter 4 Section 2 Operations on Decimals
Chapter 4 Section 2 Operations on Decimals Addition and subtraction of decimals To add decimals, write the numbers so that the decimal points are on a vertical line. Add as you would with whole numbers.
More informationRadical Expressions LESSON. 36 Unit 1: Relationships between Quantities and Expressions
LESSON 6 Radical Expressions UNDERSTAND You can use the following to simplify radical expressions. Product property of radicals: The square root of a product is equal to the square root of the factors.
More informationChapter 0: Algebra II Review
Chapter 0: Algebra II Review Topic 1: Simplifying Polynomials & Exponential Expressions p. 2 - Homework: Worksheet Topic 2: Radical Expressions p. 32 - Homework: p. 45 #33-74 Even Topic 3: Factoring All
More informationSolving Algebraic Equations
Lesson 4. Solving Algebraic Equations 3 3 3 3 3 8 8 4 Add 3 to both sides. Divide both sides by. 4 gives the solution of the equation 3. Check: Substitute 4 for x into the original equation. 3 4 3 When
More informationSection 5.4 Properties of Rational Functions
Rational Function A rational function is a function of the form R(xx) = P(xx), where P(xx)and Q(xx) are polynomial Q(xx) functions and Q(xx) 0. Domain is the set of all real numbers xx except the value(s)
More informationStudent Exploration: General Form of a Rational Function
Name: Date: Student Eploration: General Form of a Rational Function Vocabulary: asymptote, degree of a polynomial, discontinuity, rational function, root Prior Knowledge Questions (Do these BEFORE using
More informationNote-Taking Guides. How to use these documents for success
1 Note-Taking Guides How to use these documents for success Print all the pages for the module. Open the first lesson on the computer. Fill in the guide as you read. Do the practice problems on notebook
More informationMath Sections 4.4 and 4.5 Rational Functions. 1) A rational function is a quotient of polynomial functions:
1) A rational function is a quotient of polynomial functions: 2) Explain how you find the domain of a rational function: a) Write a rational function with domain x 3 b) Write a rational function with domain
More information, 1 1, A complex fraction is a quotient of rational expressions (including their sums) that result
RT. Complex Fractions Wen working wit algebraic expressions, sometimes we come across needing to simplify expressions like tese: xx 9 xx +, xx + xx + xx, yy xx + xx + +, aa Simplifying Complex Fractions
More informationSlide 1 / 180. Radicals and Rational Exponents
Slide 1 / 180 Radicals and Rational Exponents Slide 2 / 180 Roots and Radicals Table of Contents: Square Roots Intro to Cube Roots n th Roots Irrational Roots Rational Exponents Operations with Radicals
More informationObjectives Graph and Analyze Rational Functions Find the Domain, Asymptotes, Holes, and Intercepts of a Rational Function
SECTIONS 3.5: Rational Functions Objectives Graph and Analyze Rational Functions Find the Domain, Asymptotes, Holes, and Intercepts of a Rational Function I. Rational Functions A rational function is a
More informationMultiply the dividend by the reciprocal of the divisor.
Domain Lesson 6 Complex Fractions Common Core Standards: 7.RP., 7.RP. Getting the Idea To divide fractions, first find the reciprocal of the divisor. Then multiply the dividend by the reciprocal of the
More informationDomain: The domain of f is all real numbers except those values for which Q(x) =0.
Math 1330 Section.3.3: Rational Functions Definition: A rational function is a function that can be written in the form P() f(), where f and g are polynomials. Q() The domain of the rational function such
More information11.2 Techniques for Evaluating Limits
11.2 Techniques for Evaluating Limits Copyright Cengage Learning. All rights reserved. What You Should Learn Use the dividing out technique to evaluate limits of functions. Use the rationalizing technique
More informationRational Functions HONORS PRECALCULUS :: MR. VELAZQUEZ
Rational Functions HONORS PRECALCULUS :: MR. VELAZQUEZ Definition of Rational Functions Rational Functions are defined as the quotient of two polynomial functions. This means any rational function can
More informationSection 2-7. Graphs of Rational Functions
Section 2-7 Graphs of Rational Functions Section 2-7 rational functions and domain transforming the reciprocal function finding horizontal and vertical asymptotes graphing a rational function analyzing
More informationADDING AND SUBTRACTING RATIONAL EXPRESSIONS
ADDING AND SUBTRACTING RATIONAL EXPRESSIONS To Add or Subtract Two Fractions, 0, 0 Example 1 a) Add b) Subtract a) b) The same principles apply when adding or subtracting rational expressions containing
More information2-9 Operations with Complex Numbers
2-9 Operations with Complex Numbers Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Express each number in terms of i. 1. 9i 2. Find each complex conjugate. 3. 4. Find each product. 5. 6. Objective
More informationStudy Guide For use with pages
. GOAL For use with pages Write fractions as decimals and vice versa. VOCABULARY A rational number is a number that can be written as a quotient of two integers. In a terminating decimal, the division
More informationRational Functions. By: Kaushik Sriram, Roshan Kuntamukkala, and Sheshanth Vijayakumar
Rational Functions By: Kaushik Sriram, Roshan Kuntamukkala, and Sheshanth Vijayakumar What are Rational Functions? Dictionary Definition: In mathematics, a rational function is any function which can be
More information1. To add (or subtract) fractions, the denominators must be equal! a. Build each fraction (if needed) so that both denominators are equal.
MAT000- Fractions Purpose One of the areas most frustrating for teachers and students alike is the study of fractions, specifically operations with fractions. Year after year, students learn and forget
More informationLesson 1: Arithmetic Review
Lesson 1: Arithmetic Review Topics and Objectives: Order of Operations Fractions o Improper fractions and mixed numbers o Equivalent fractions o Fractions in simplest form o One and zero Operations on
More information6.3 ADDING and SUBTRACTING Rational Expressions REVIEW. When you ADD rational numbers (fractions): 1) Write each number with common denominator
6.3 ADDING and SUBTRACTING Rational REVIEW When you ADD rational numbers (fractions): 1) Write each number with common denominator 4 5 + 10 12 = 6.3 ADDING and SUBTRACTING Rational 4 5 + 10 12 = REVIEW
More informationWalt Whitman High School SUMMER REVIEW PACKET. For students entering AP CALCULUS BC
Walt Whitman High School SUMMER REVIEW PACKET For students entering AP CALCULUS BC Name: 1. This packet is to be handed in to your Calculus teacher on the first day of the school year.. All work must be
More informationRational functions, like rational numbers, will involve a fraction. We will discuss rational functions in the form:
Name: Date: Period: Chapter 2: Polynomial and Rational Functions Topic 6: Rational Functions & Their Graphs Rational functions, like rational numbers, will involve a fraction. We will discuss rational
More informationGraphing Rational Functions
Graphing Rational Functions Return to Table of Contents 109 Vocabulary Review x-intercept: The point where a graph intersects with the x-axis and the y-value is zero. y-intercept: The point where a graph
More informationAP Calculus Summer Review Packet
AP Calculus Summer Review Packet Name: Date began: Completed: **A Formula Sheet has been stapled to the back for your convenience!** Email anytime with questions: danna.seigle@henry.k1.ga.us Complex Fractions
More information1-6 Order of Operations
1-6 Order of Operations Warm Up Lesson Presentation Lesson Quiz 2 pts 3 pts Bell Quiz 1-6 Find each square root. 1. 25 Write all classifications that apply to each real number. 3. -55 5 pts possible Questions
More informationRational and Irrational Numbers
LESSON. Rational and Irrational Numbers.NS. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion;... lso.ns.2,.ee.2? ESSENTIL QUESTION
More information50 MATHCOUNTS LECTURES (6) OPERATIONS WITH DECIMALS
BASIC KNOWLEDGE 1. Decimal representation: A decimal is used to represent a portion of whole. It contains three parts: an integer (which indicates the number of wholes), a decimal point (which separates
More information1. Answer: x or x. Explanation Set up the two equations, then solve each equation. x. Check
Thinkwell s Placement Test 5 Answer Key If you answered 7 or more Test 5 questions correctly, we recommend Thinkwell's Algebra. If you answered fewer than 7 Test 5 questions correctly, we recommend Thinkwell's
More informationELEMENTARY NUMBER THEORY AND METHODS OF PROOF
CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF Copyright Cengage Learning. All rights reserved. SECTION 4.2 Direct Proof and Counterexample II: Rational Numbers Copyright Cengage Learning. All
More informationSNAP Centre Workshop. Graphing Lines
SNAP Centre Workshop Graphing Lines 45 Graphing a Line Using Test Values A simple way to linear equation involves finding test values, plotting the points on a coordinate plane, and connecting the points.
More informationReteaching. Comparing and Ordering Integers
- Comparing and Ordering Integers The numbers and - are opposites. The numbers 7 and -7 are opposites. Integers are the set of positive whole numbers, their opposites, and zero. 7 6 4 0 negative zero You
More informationName: Rational Functions 2.1H. Set Topic: Simplifying rational expressions & operations on rational expressions
Name: Rational Functions 2.1H Ready, Set, Go! Ready Topic: Polynomial division Use division to determine if the given linear term is a factor of the polynomial. If it is a linear factor, then find the
More informationCollege Prep Algebra II Summer Packet
Name: College Prep Algebra II Summer Packet This packet is an optional review which is highly recommended before entering CP Algebra II. It provides practice for necessary Algebra I topics. Remember: When
More informationPure Math 30: Explained!
www.puremath30.com 30 part i: stretches about other lines Stretches about other lines: Stretches about lines other than the x & y axis are frequently required. Example 1: Stretch the graph horizontally
More informationMath Circle Beginners Group October 18, 2015 Solutions
Math Circle Beginners Group October 18, 2015 Solutions Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that
More informationSummer Assignment Glossary
Algebra 1.1 Summer Assignment Name: Date: Hour: Directions: Show all work for full credit using a pencil. Circle your final answer. This assignment is due the first day of school. Use the summer assignment
More informationIntegers and Rational Numbers
A A Family Letter: Integers Dear Family, The student will be learning about integers and how these numbers relate to the coordinate plane. The set of integers includes the set of whole numbers (0, 1,,,...)
More informationNAME UNIT 4 ALGEBRA II. NOTES PACKET ON RADICALS, RATIONALS d COMPLEX NUMBERS
NAME UNIT 4 ALGEBRA II NOTES PACKET ON RADICALS, RATIONALS d COMPLEX NUMBERS Properties for Algebra II Name: PROPERTIES OF EQUALITY EXAMPLE/MEANING Reflexive a - a Any quantity is equal to itself. Symmetric
More informationSummer Review for Students Entering Pre-Calculus with Trigonometry. TI-84 Plus Graphing Calculator is required for this course.
1. Using Function Notation and Identifying Domain and Range 2. Multiplying Polynomials and Solving Quadratics 3. Solving with Trig Ratios and Pythagorean Theorem 4. Multiplying and Dividing Rational Expressions
More information3.1 Dividing a Whole into Fractional Parts. 3.1 Dividing a Set into Fractional Parts. 3.2 Identifying Parts of Wholes.
. Dividing a Whole into Fractional Parts Fraction: represents a part of a whole object or unit Numerator: (top number) represents number of parts of the whole Denominator: (bottom number) represents how
More informationLearning Packet. Lesson 6 Exponents and Rational Functions THIS BOX FOR INSTRUCTOR GRADING USE ONLY
Learning Packet Student Name Due Date Class Time/Day Submission Date THIS BOX FOR INSTRUCTOR GRADING USE ONLY Mini-Lesson is complete and information presented is as found on media links (0 5 pts) Comments:
More informationGRAPHING RATIONAL FUNCTIONS DAY 2 & 3. Unit 12
1 GRAPHING RATIONAL FUNCTIONS DAY 2 & 3 Unit 12 2 Warm up! Analyze the graph Domain: Range: Even/Odd Symmetry: End behavior: Increasing: Decreasing: Intercepts: Vertical Asymptotes: Horizontal Asymptotes:
More informationMath 083 Final Exam Practice
Math 083 Final Exam Practice Name: 1. Simplify the expression. Remember, negative exponents give reciprocals.. Combine the expressions. 3. Write the expression in simplified form. (Assume the variables
More information6.2 Adding and Subtracting Rational Expressions
8 CHAPTER 6 Rational Epressions Simplify. Assume that no denominator is 0. 99. p - - p 00. + q n q n + n + k - 9 0. n 0. - 6 + k Perform the indicated operation. Write all answers in lowest terms. 0. 0.
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 24 - Study Guide - Chapter 1 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Give one number between -8 and 8 that is a negative real
More informationChapter 1 Section 1 Solving Linear Equations in One Variable
Chapter Section Solving Linear Equations in One Variable A linear equation in one variable is an equation which can be written in the form: ax + b = c for a, b, and c real numbers with a 0. Linear equations
More informationCW Middle School. Math RtI 7 A. 4 Pro cient I can add and subtract positive fractions with unlike denominators and simplify the result.
1. Foundations (14.29%) 1.1 I can add and subtract positive fractions with unlike denominators and simplify the result. 4 Pro cient I can add and subtract positive fractions with unlike denominators and
More informationSummer Review for Students Entering Pre-Calculus with Trigonometry. TI-84 Plus Graphing Calculator is required for this course.
Summer Review for Students Entering Pre-Calculus with Trigonometry 1. Using Function Notation and Identifying Domain and Range 2. Multiplying Polynomials and Solving Quadratics 3. Solving with Trig Ratios
More informationFinal Exam MAT 100 JS 2018
Final Exam MAT 100 JS 2018 Miles College T Dabit MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Tell which set or sets the number belongs to: natural
More information3.6-Rational Functions & Their Graphs
.6-Rational Functions & Their Graphs What is a Rational Function? A rational function is a function that is the ratio of two polynomial functions. This definition is similar to a rational number which
More information( 3) ( 4 ) 1. Exponents and Radicals ( ) ( xy) 1. MATH 102 College Algebra. still holds when m = n, we are led to the result
Exponents and Radicals ZERO & NEGATIVE EXPONENTS If we assume that the relation still holds when m = n, we are led to the result m m a m n 0 a = a = a. Consequently, = 1, a 0 n n a a a 0 = 1, a 0. Then
More information10-2 Circles. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra2 2
10-2 Circles Warm Up Lesson Presentation Lesson Quiz Holt Algebra2 2 Warm Up Find the slope of the line that connects each pair of points. 1. (5, 7) and ( 1, 6) 1 6 2. (3, 4) and ( 4, 3) 1 Warm Up Find
More informationSLOPE A MEASURE OF STEEPNESS through 7.1.5
SLOPE A MEASURE OF STEEPNESS 7.1. through 7.1.5 Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in -form, the m is the
More informationRational and Irrational Numbers can be written as 1_ 2.
? L E S S O N 1.1 Rational and Irrational Numbers ESSENTIAL QUESTION 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion;
More informationRational Expressions Sections
Rational Expressions Sections Multiplying / Dividing Let s first review how we multiply and divide fractions. Multiplying / Dividing When multiplying/ dividing, do we have to have a common denominator?
More informationSection Rational Functions and Inequalities. A rational function is a quotient of two polynomials. That is, is a rational function if
Section 6.1 --- Rational Functions and Inequalities A rational function is a quotient of two polynomials. That is, is a rational function if =, where and are polynomials and is not the zero polynomial.
More informationExploring Rational Functions
Name Date Period Exploring Rational Functions Part I - The numerator is a constant and the denominator is a linear factor. 1. The parent function for rational functions is: Graph and analyze this function:
More informationMath 1330 Section : Rational Functions Definition: A rational function is a function that can be written in the form f ( x ), where
2.3: Rational Functions P( x ) Definition: A rational function is a function that can be written in the form f ( x ), where Q( x ) and Q are polynomials, consists of all real numbers x such that You will
More informationor 5.00 or 5.000, and so on You can expand the decimal places of a number that already has digits to the right of the decimal point.
1 LESSON Understanding Rational and Irrational Numbers UNDERSTAND All numbers can be written with a For example, you can rewrite 22 and 5 with decimal points without changing their values. 22 5 22.0 or
More informationExpress Math. Simplifying Expressions Using Distributive Properties. Learning Goals. Key Terms. Essential Ideas
Express Math Simplifying Expressions Using Distributive Properties Learning Goals In this lesson, you will: Write and use the distributive properties. Use distributive properties to simplify expressions.
More information16 Rational Functions Worksheet
16 Rational Functions Worksheet Concepts: The Definition of a Rational Function Identifying Rational Functions Finding the Domain of a Rational Function The Big-Little Principle The Graphs of Rational
More informationPreCalculus 300. Algebra 2 Review
PreCalculus 00 Algebra Review Algebra Review The following topics are a review of some of what you learned last year in Algebra. I will spend some time reviewing them in class. You are responsible for
More informationRational numbers as decimals and as integer fractions
Rational numbers as decimals and as integer fractions Given a rational number expressed as an integer fraction reduced to the lowest terms, the quotient of that fraction will be: an integer, if the denominator
More informationIntegers are whole numbers; they include negative whole numbers and zero. For example -7, 0, 18 are integers, 1.5 is not.
What is an INTEGER/NONINTEGER? Integers are whole numbers; they include negative whole numbers and zero. For example -7, 0, 18 are integers, 1.5 is not. What is a REAL/IMAGINARY number? A real number is
More informationDecimals. Chapter Five
Chapter Five Decimals 5.1 Introductions to Decimals 5.2 Adding & Subtracting Decimals 5.3 Multiplying Decimals & Circumference of a Circle 5.4 Dividing Decimals 5.5 Fractions, Decimals, & Order of Operations
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II. 3 rd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More information2-3 Graphing Rational Functions
2-3 Graphing Rational Functions Factor What are the end behaviors of the Graph? Sketch a graph How to identify the intercepts, asymptotes and end behavior of a rational function. How to sketch the graph
More information