Lesson 2 - Mini-Lesson. Section 2.1 Properties of Exponents
|
|
- Anis Gilbert
- 5 years ago
- Views:
Transcription
1 Lesson - Mini-Lesson Section.1 Properties of Exponents What is an exponent? An exponent is a number in the superscript location and identifies the number of times the base number is to be multiplied times itself. For example: 3 = In this situation, the exponent 3 is attached to the base. Raising to the 3 rd power indicates that we are to multiply the base times itself a total of 3 times. Here is how we would perform that multiplication: 3 = = 4 = 8 Notes about this property: Zero Exponent Property For any nonzero real number a, a 0 = is undefined a is the base of the exponential expression Problem 1 MEDIA EXAMPLE Zero Exponent Property Simplify the following expressions using the Zero Exponent Property. 5 0 = b) 5 0 = c) 5x 0 = d) 5x 0 = 53
2 Problem WORKED EXAMPLE Zero Exponent Property Simplify the following expressions using the Zero Exponent Property. 0 = 1 b) 0 = ( 1) 0 = ( 1) ( 1) = 1 c) x 0 = ( 1) = Note:x 0 = 1;assumethatx 0 d) ( x) 0 = 1 Note:Thebaseis x;assumethatx 0 Negative Exponent Property For any real number a 0,b 0 : a m = 1 1 a = a m am m a b m = b a m Problem 3 MEDIA EXAMPLE Negative Exponent Property Rewrite each of the following with only positive exponents. Variables represent nonzero quantities. 1 x 3 = b) = x 1 1 c) x = d) 3 = e) 16 = f) 3x 4 = 54
3 Problem 4 WORKED EXAMPLE Negative Exponent Property Rewrite each of the following with only positive exponents. Variables represent nonzero quantities. y 4 = 1 y 4 b) 1 x = x c) 1 1x5 = 5 4x 4 Note:Thecoefficient4doesnotmove d) 4 1 = 1 41 = x5 4 = 1 4 = 1 e) x 5 = x 5 Note:Thecoefficientdoesnotmove. f) = = 3 3 = = 9 3 = 7 Problem 5 You Try Zero Exponent Property and Negative Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. 7a 0 = b) 7a 0 = c) 5a 3 = d) 7 a 1 = e) 7a 1 = f) 7 a 1 = 55
4 Multiplication Property: For any real number a, am a n = a m+n Problem 6 MEDIA EXAMPLE Multiplication Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. n 3 n 9 = b) b5 b 4 b = c) 5x 7x 3 = d) 5x + 7x 3 = Problem 7 WORKED EXAMPLE Multiplication Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. q 7 q 5 = q 7+5 = q 1 b) a 4 a 5 b = a 4+( 5 ) b 8 = a 5 b 3 = a b c) = ( 4 7) ( x) ( y 5 y ) 3 = ( 4 7) ( x) ( y ) 5 + ( 3) = ( 8) ( x) ( y ) 4 y 5 7xy 3 d) 4 y5 +7 y 3 = completelysimplified = 8xy 56
5 Raising an Exponent to an Exponent Property: For any real number a, n = a mn am Problem 8 MEDIA EXAMPLE Raising an Exponent to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. x 3 5 = b) x x 3 = Problem 9 WORKED EXAMPLE Raising an Exponent to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. x ( x ) 3 4 = x x 1 = x +1 = x 14 b) 4x x 3 3 = 4x x 3 i 3 = 4x x = 4x +( ) = 4x 0 = 4 1 = 4 57
6 Product and Quotient to an Exponent Property: For any real number a, PRODUCT: n =a n b n n a = an QUOTIENT: ab b b n, b 0 Problem 10 MEDIA EXAMPLE Product to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. 5x = b) 5 8ab = c) ( 5n4 3n ) 3 = Problem 11 MEDIA EXAMPLE Quotient to an Exponent Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. 5 7 = b) x 5 y 3 4 = Problem 1 WORKED EXAMPLE Product to an Exponent Property and Quotient to an Exponent Property Simplify the following expression. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. 1 1a b 4 4 a = 1 = 1 a 1 4 b 8 4 ( b ) 4 = a b 8 58
7 The Division Property: For any nonzero real number a, a m a = n a(m n) Problem 13 MEDIA EXAMPLE The Division Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. x 4 x = 4a 10 b 5 0 b) 6ab = Problem 14 WORKED EXAMPLE The Division Property Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. y 1 y 6 = y1 6 1 b) y 6 y6 1 = 1 y 1 = y 6 = y 6 1 = 1 y 6 c) 3a 1 b 5 9ab = 1a 1 b 5 = a 1 b 5 6 3ab 6 3a 1 b = a 1 1 b 5 6 = a b 1 = a b = 1 1 3a b Note:Thisproblemshowshorizontalformatwhichisalsoan acceptablewaytowriteyoursolutions. 59
8 Problem 15 You Try Properties of Exponents Simplify the following expressions. Write your answers with only positive exponents. Variables represent nonzero quantities. Simplify if possible. Show all steps as in the media examples. x 4 = b) x 3 = 3 c) 8g 5g 4 = d) 6 n 3 = e) 3a x 3 y 8 = f) 9xy = 5 60
9 Polynomials and Polynomial Multiplication: Variables raised to exponents and joined by addition form the building blocks, called terms, of algebraic expressions known as polynomials. Some examples are below. Note that the coefficient is the number in front of the variable part of a term. Expression Terms Name Coefficients in Order x 3 1 term:x 3 Monomial 4x 7 terms: 4x, 7 Binomial 4, 7 x + 5 terms: x, 5 Binomial 1, 5 x + 3x 5 3 terms: Trinomial, 3, 5 x, 3x, 5 Polynomial multiplication requires the use of the exponent properties learned previously in this lesson. Some examples are below. Problem 16 Multiply and simplify. MEDIA EXAMPLE Multiplying Polynomials 3x x 4 = b) x x 3 x + 5 = c) 3x x 4 + ( x 4) = Problem 17 Multiply and simplify. x x 4 WORKED EXAMPLE Multiplying Polynomials = x( x) + x( 4) =x 8x b) x( x 4 3x + 6) = ( x) x 4 +( x) ( 3x)+ x = x 5 + ( 3x )+( 6x) = x 5 + 3x 6x ( 6) 61
10 Multiplying Binomials and the FOIL Method: A binomial as a two-term polynomial. When we multiply two binomials, we can use the FOIL method (meaning First, Outside, Inside, Last) to help us keep track of our multiplications. Here is the general form: When we multiply two binomials together, we initially obtain four terms. Usually, two of these will combine resulting in three terms, or, a trinomial. Problem 18 MEDIA EXAMPLE Multiplying Binomials/Higher Order Polynomials Multiply and simplify. x + 3 ( x + 4)= b) d 4 ( 3d + 5)= = d) 3 a c) x x + x 4 = 6
11 Problem 19 WORKED EXAMPLE Multiplying Binomials/Higher Order Polynomials Multiply each set of polynomials below and combine like terms to simplify. ( 3x )( 4x + 3) = ( 3x) ( 4x)+ ( 3x) ( 3)+ ( ) ( 4x)+ = ( 1x )+( 9x)+ ( 8x)+ 6 ( 3) = 1x + 9x 8x 6 = 1x + x 6 b) ( x +1) x + x 5 = ( x) ( x )+( x) ( x)+ ( x) ( 5)+ ( 1) ( x )+( 1) ( x)+ 1 = ( x 3 )+( 1x )+( 5x)+ ( x )+( 1x)+ ( 5) = ( x 3 )+( 1x x )+( 5x +1x)+ ( 5) = ( x 3 )+( x )+( 4x)+ ( 5) = x 3 x 4x 5 ( 5) 63
12 Problem 0 YOU TRY Multiplying Binomials/Higher Order Polynomials Multiply each set of polynomials below and combine like terms to simplify. x 1 ( x + 4) = b) 3x 4 ( 5x + ) = c) x + 5 = d) x 5 = e) x 5 ( x +5) = f) x + ( x ) = g) x 4 ( x + x + 1) = h) x 1 ( x x + 1) = 64
13 Section. Using Properties of Exponents to Evaluate Functions Problem 1 MEDIA EXAMPLE Function Evaluation Given the function, f (x)= x, evaluate each of the following. Show your work. Write final results as ordered pairs. f 5 5 = b) f 6 = c) f 10x + 1 = d) f x 3 = Problem WORKED EXAMPLE Function Evaluation Given f x pairs. = 5 x, evaluate each of the following. Show your work. Write final results as ordered f ( 5) = 5 [Replacexwith 5inf x ( 5) = 5 5 [ ( 5 ) = ( 5) ( 5) = 5] = 1 5 [Reducefraction] 5, 1 5 ] b) = 5 [Replacexwith5xinf x ( 5x) f 5x = 5 5x = 1 5x 1 5x, 5x [( 5x) = ( 5x) ( 5x) = 5x ] [Reducefraction] ] 65
14 Problem 3 You Try Function Evaluation Given f x = 3x, evaluate each of the following. Show your work. Write final results as ordered pairs. f 3 = b) f 5x 3 = c) f x 5 = 66
15 Section.3 Combining Functions Basic Mathematical Operations The basic mathematical operations are: addition, subtraction, multiplication, and division. When working with function notation, these operations will look like this: Addition Subtraction Multiplication Division f ( x ) g( x) f ( x f g( x) g( x) g x f ( x )+ g( x) 0 Problem 4 WORKED EXAMPLE Adding and Subtracting Functions Given f x = x +3x 5 and g x = x +5x +1, determine each of the following. f ( x)+ g( x) = ( x +3x 5)+ ( x +5x +1) = x + 3x 5 x + 5x + 1 = x x + 3x + 5x = x + 8x 4 b) f ( x) g( x) = ( x + 3x 5) ( x + 5x + 1) = x + 3x 5 + x 5x 1 = x + x + 3x 5x 5 1 = 3x x 6 Problem 5 MEDIA EXAMPLE Adding and Subtracting Functions Given f x = 3x + x 1 and g x = x x + 5, determine each of the following. f x + g( x) = b) f x g( x) = Problem 6 YOU TRY Adding and Subtracting Functions Given f x = x + 4 and g x = x 3x + 1, determine each of the following. f x + g( x) = b) f x g( x) = 67
16 Function Multiplication and the Multiplication Property of Exponents When multiplying functions, you will often need to work with exponents. Try to recognize the examples below as being similar to ones completed earlier in the lesson. Problem 7 WORKED EXAMPLE Function Multiplication For each set of functions below, show all work to determine f x g x. Givenf x f ( x) g x = 8x 4 and g( x) = 5x 3, = ( 8x )( 4 5x ) 3 = ( 8 5) ( x 4 x ) 3 = 40x 7 b) Givenf x f ( x) g x = 3x + and g( x) = x 5, = ( 3x +) ( x 5) = ( 3x) ( x)+ ( 3x) ( 5)+ ( ) x = 6x 15x + 4x 10 = 6x 11x 10 + ( 5) Problem 8 MEDIA EXAMPLE Function Multiplication f ( x ) g x Given f x = 3x + and g x = x +3x 1, show all work to determine. Problem 9 YOU TRY Function Multiplication For each of the following, show all work to determine f x g( x). f x = 3x and g( x) = 3x + b) f x = x and g( x) = x 3 4x +5 68
17 Function Division and the Division Property of Exponents When dividing functions, you will also need to work with exponents. Try to recognize the examples below as being similar to ones completed earlier in the lesson. Problem 30 WORKED EXAMPLE Function Division f x For each of the following, determine g x f ( x ) = 15x 15 and g( x) = 3x 9 = 15x f x g x Problem 31 3x 9 15 = 5x 15 9 = 5x 6. Use only positive exponents in your final answer. b) f x MEDIA EXAMPLE Function Division f x For each of the following, determine g x f x = 10x 4 + 3x and g x = 4x 5 and g( x) = x 8 f ( g( x) = 4x 5 x 8 = x 5 8 = x 3 = x 3. Use only positive exponents in your final answer. = x b) f x = 1x 5 + 8x + 5 and g( x) = 4x Problem 3 YOU TRY Function Division f x For each of the following, determine g x f x = 5x 5 4x 7 and g x. Use only positive exponents in your final answer. = 5x 4 b) f x = 0x 6 16x and g( x) = 4x 3 69
18 Working with Functions in Different Forms: Tables and Graphs Functions can be presented in multiple ways including: equations, data sets, graphs and applications. If you understand function notation, then the process for working with functions is the same no matter how the information if presented. Problem 33 MEDIA EXAMPLE Functions in Table Form Functions f (x) and g(x) are defined in the tables below. Find a e below using the tables. f x x g x x f 1 = b) g 9 = c) f 0 + g( 0) = d) g 5 f ( 8) = e) f 0 g( 3) = Problem 34 YOU TRY Functions in Table Form Given the following two tables, complete the third table. Show work in the table cell for each column. The first one is done for you. Show your work the same way as the sample. f x x g x x x f ( 0)+ g( 0) f ( x )+ g( x) = = 10 70
19 Problem 35 YOU TRY Functions in Graph Form Use the graph to determine each of the following. Assume integer answers. g 4 = b) f = c) g 0 = d) If f x = 0, x = e) If g x = 0, x = f) f 1 + g( 1) = g) g 6 f ( 7) = h) f 1 g( ) = i) g 0 f ( 1) = = k) f 1 j) g 6 = g 6 f 1 71
20 Section.4 Applications of Function Operations One of the classic applications of function operations is the forming of the profit function, P x by subtracting the cost function, C x, from the revenue function, R x, as shown below. Profit = Revenue Cost Given functions P x = Profit, R x = Revenue, and C x = Cost: P ( x ) = R( x) C( x), Problem 36 MEDIA EXAMPLE Cost, Revenue, Profit A local courier service estimates its monthly operating costs to be $1500 plus $0.85 per delivery. The revenues are $6 for each delivery. Let x = the number of deliveries in a given month. Write a function, C x, to represent the monthly costs for making x deliveries per month. b) Write a function, R x, to represent the revenue for making x deliveries per month. c) Write a function, P x, that represents the monthly profits for making x deliveries per month. d) Determine algebraically the break-even point for the function P x and how many deliveries you must make each month to begin making money. e) Solve the equation P x = 0 graphically to confirm the break-even point. 7
21 Problem 37 YOU TRY Cost, Revenue, Profit Charlie s Chocolate Shoppe sells their chocolates for $1.80 per piece. The fixed costs to run the Chocolate Shoppe total $450 for the week, and Charlie estimates that each chocolate costs about $0.60 to produce. Charlie estimates that he can produce up to 3,000 chocolates in one week. Write a function, C n, to model Charlie s total weekly costs if he makes n chocolates. b) Write a function, R n, to represent the revenue from the sale of n chocolates. c) Write a function, P n, that represents Charlie s profit from selling n chocolates. d) Interpret the meaning of the statement P 300 = 90. Write your answer as a complete sentence. e) Determine the practical domain and practical range for P n, then use that information to define an appropriate viewing window. Sketch the graph from your calculator. Practical Domain: Practical Range: f) How many chocolates must Charlie sell in order to break even? Show complete work. Write your final answer as a complete sentence. Mark the break-even point on the graph above. 73
22
Unit 13: Polynomials and Exponents
Section 13.1: Polynomials Section 13.2: Operations on Polynomials Section 13.3: Properties of Exponents Section 13.4: Multiplication of Polynomials Section 13.5: Applications from Geometry Section 13.6:
More informationAlgebra II Chapter 5: Polynomials and Polynomial Functions Part 1
Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions
More informationLesson 3: Polynomials and Exponents, Part 1
Lesson 2: Introduction to Variables Assessment Lesson 3: Polynomials and Exponents, Part 1 When working with algebraic expressions, variables raised to a power play a major role. In this lesson, we look
More informationLesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o
Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o o ( 1)(9) 3 ( 1) 3 9 1 Evaluate the second expression at the left, if
More informationWhen you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut.
Squaring a Binomial When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Solve. (x 3) 2 Step 1 Square the first term. Rules
More informationChapter 5: Exponents and Polynomials
Chapter 5: Exponents and Polynomials 5.1 Multiplication with Exponents and Scientific Notation 5.2 Division with Exponents 5.3 Operations with Monomials 5.4 Addition and Subtraction of Polynomials 5.5
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 information20A. Build. Build and add. Build a rectangle and find the area (product). l e s s o n p r a c t i c e 1. X X X 2 + 6X X
l e s s o n p r a c t i c e 0A Build.. X X. X 6X 8 3. X 8 Build and add. 4. X 6X 3 3X 7X 9 5. X 8 X 6X 7 6. X 0X 7 X 8X 9 Build a rectangle and find the area (product). 7. (X )(X ) = 8. (X 4)(X 3) = 9.
More informationLESSON 6.1 EXPONENTS LESSON 6.1 EXPONENTS 253
LESSON 6.1 EXPONENTS LESSON 6.1 EXPONENTS 5 OVERVIEW Here's what you'll learn in this lesson: Properties of Exponents Definition of exponent, power, and base b. Multiplication Property c. Division Property
More informationName: Chapter 7: Exponents and Polynomials
Name: Chapter 7: Exponents and Polynomials 7-1: Integer Exponents Objectives: Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents. You
More informationTopics Covered in Math 115
Topics Covered in Math 115 Basic Concepts Integer Exponents Use bases and exponents. Evaluate exponential expressions. Apply the product, quotient, and power rules. Polynomial Expressions Perform addition
More informationNever leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!
1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a
More informationSection 5.2 Polynomials, Sums, and Differences
Department of Mathematics Grossmont College October 2, 2012 4.1 Systems of Linear Equations in Two Variables Learning Objectives: Give the degree of a polynomial Add and subract polynomials evaluate a
More informationA field trips costs $800 for the charter bus plus $10 per student for x students. The cost per student is represented by: 10x x
LEARNING STRATEGIES: Activate Prior Knowledge, Shared Reading, Think/Pair/Share, Note Taking, Group Presentation, Interactive Word Wall A field trips costs $800 for the charter bus plus $10 per student
More informationLesson 2: Introduction to Variables
Lesson 2: Introduction to Variables Topics and Objectives: Evaluating Algebraic Expressions Some Vocabulary o Variable o Term o Coefficient o Constant o Factor Like Terms o Identifying Like Terms o Combining
More informationSimplifying Radical Expressions
Simplifying Radical Expressions Product Property of Radicals For any real numbers a and b, and any integer n, n>1, 1. If n is even, then When a and b are both nonnegative. n ab n a n b 2. If n is odd,
More informationAlgebra 2 (3) Apex - Semester 1 Part A
Algebra 2 (3) Apex - Semester 1 Part A Name Algebra I Literacy Advantage Unit 1 : The Need to Read Lesson 1.1 : Reading and Vocabulary Activity 1.1.1 : Study - Active Reading Learn skills and strategies
More informationSomething that can have different values at different times. A variable is usually represented by a letter in algebraic expressions.
Lesson Objectives: Students will be able to define, recognize and use the following terms in the context of polynomials: o Constant o Variable o Monomial o Binomial o Trinomial o Polynomial o Numerical
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 informationDay 131 Practice. What Can You Do With Polynomials?
Polynomials Monomial - a Number, a Variable or a PRODUCT of a number and a variable. Monomials cannot have radicals with variables inside, quotients of variables or variables with negative exponents. Degree
More informationLESSON 9.1 ROOTS AND RADICALS
LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical
More information6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property
6.1 Using Properties of Exponents Objectives 1. Use properties of exponents to evaluate and simplify expressions involving powers. 2. Use exponents and scientific notation to solve real life problems.
More informationMATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline
MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline 1. Real Numbers (33 topics) 1.3 Fractions (pg. 27: 1-75 odd) A. Simplify fractions. B. Change mixed numbers
More informationReady To Go On? Skills Intervention 7-1 Integer Exponents
7A Evaluating Expressions with Zero and Negative Exponents Zero Exponent: Any nonzero number raised to the zero power is. 4 0 Ready To Go On? Skills Intervention 7-1 Integer Exponents Negative Exponent:
More informationIntermediate Algebra Student Workbook
North Seattle College Intermediate Algebra Student Workbook Development Team (Scottsdale C.C.) Donna Gaudet William Meacham Jenifer Bohart Amy Volpe Linda Knop Donna Guhse Fourth Edition 2014 Page 1 This
More informationIntermediate Algebra Student Workbook
North Seattle College Intermediate Algebra Student Workbook Development Team (Scottsdale C.C.) Donna Gaudet William Meacham Jenifer Bohart Amy Volpe Linda Knop Donna Guhse Fourth Edition 2014 This work
More informationPOLYNOMIAL EXPRESSIONS PART 1
POLYNOMIAL EXPRESSIONS PART 1 A polynomial is an expression that is a sum of one or more terms. Each term consists of one or more variables multiplied by a coefficient. Coefficients can be negative, so
More informationBeginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions
1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:
More informationMath ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying
Math 1050 2 ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying General Tips for Studying: 1. Review this guide, class notes, the
More informationProperties of Real Numbers
Pre-Algebra Properties of Real Numbers Identity Properties Addition: Multiplication: Commutative Properties Addition: Multiplication: Associative Properties Inverse Properties Distributive Properties Properties
More informationReteach Simplifying Algebraic Expressions
1-4 Simplifying Algebraic Expressions To evaluate an algebraic expression you substitute numbers for variables. Then follow the order of operations. Here is a sentence that can help you remember the order
More informationMath 3 Variable Manipulation Part 3 Polynomials A
Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does
More informationA-2. Polynomials and Factoring. Section A-2 1
A- Polynomials and Factoring Section A- 1 What you ll learn about Adding, Subtracting, and Multiplying Polynomials Special Products Factoring Polynomials Using Special Products Factoring Trinomials Factoring
More informationChapter 6. Polynomials
Chapter 6 Polynomials How to Play the Stock Market 6.1 Monomials: Multiplication and Division 6.2 Polynomials 6.3 Addition and Subtraction of Polynomials 6.4 Multiplication of Polynomials Chapter Review
More informationUnit 3 Vocabulary. An algebraic expression that can contains. variables, numbers and operators (like +, An equation is a math sentence stating
Hart Interactive Math Algebra 1 MODULE 2 An algebraic expression that can contains 1 Algebraic Expression variables, numbers and operators (like +,, x and ). 1 Equation An equation is a math sentence stating
More informationName. Use Two-Color Counters to model each addition problem. Make pairs of red and yellow counters. Find the sum.
Lesson 1 The Number System Name Use Two-Color Counters to model each addition problem. Make pairs of red and yellow counters. Find the sum. 1. 2. 9 + ( 10) 18 + 9 Using Two-Color Counters, model each addition
More informationSections 7.1, 7.2: Sums, differences, products of polynomials CHAPTER 7: POLYNOMIALS
Sections 7.1, 7.2: Sums, differences, products of polynomials CHAPTER 7: POLYNOMIALS Quiz results Average 73%: high h score 100% Problems: Keeping track of negative signs x = + = + Function notation f(x)
More informationChapter 7: Exponents
Chapter : Exponents Algebra Chapter Notes Name: Notes #: Sections.. Section.: Review Simplify; leave all answers in positive exponents:.) m -.) y -.) m 0.) -.) -.) - -.) (m ) 0.) 0 x y Evaluate if a =
More information5.3. Polynomials and Polynomial Functions
5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a
More informationControlling the Population
Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1
More informationSolving Equations Quick Reference
Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number
More informationAccuplacer Review Workshop. Elementary Algebra Part II. Week Three. Includes internet links to instructional videos for additional resources:
Accuplacer Review Workshop Elementary Algebra Part II Week Three Includes internet links to instructional videos for additional resources: http://www.mathispower4u.com (Arithmetic Video Library) http://www.purplemath.com
More informationI CAN classify polynomials by degree and by the number of terms.
13-1 Polynomials I CAN classify polynomials by degree and by the number of terms. 13-1 Polynomials Insert Lesson Title Here Vocabulary monomial polynomial binomial trinomial degree of a polynomial 13-1
More informationPolynomials: Adding, Subtracting, & Multiplying (5.1 & 5.2)
Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.) Determine if the following functions are polynomials. If so, identify the degree, leading coefficient, and type of polynomial 5 3 1. f ( x) =
More informationBeginning Algebra MAT0024C. Professor Sikora. Professor M. J. Sikora ~ Valencia Community College
Beginning Algebra Professor Sikora MAT002C POLYNOMIALS 6.1 Positive Integer Exponents x n = x x x x x [n of these x factors] base exponent Numerical: Ex: - = where as Ex: (-) = Ex: - = and Ex: (-) = Rule:
More informationIn this lesson, we will focus on quadratic equations and inequalities and algebraic methods to solve them.
Lesson 7 Quadratic Equations, Inequalities, and Factoring Lesson 7 Quadratic Equations, Inequalities and Factoring In this lesson, we will focus on quadratic equations and inequalities and algebraic methods
More informationIntermediate Algebra with Applications
Lakeshore Technical College 10-804-118 Intermediate Algebra with Applications Course Outcome Summary Course Information Alternate Title Description Total Credits 4 Total Hours 72 Pre/Corequisites Prerequisite
More information1.3 Algebraic Expressions. Copyright Cengage Learning. All rights reserved.
1.3 Algebraic Expressions Copyright Cengage Learning. All rights reserved. Objectives Adding and Subtracting Polynomials Multiplying Algebraic Expressions Special Product Formulas Factoring Common Factors
More informationMA 22000, Lesson 2 Functions & Addition/Subtraction Polynomials Algebra section of text: Sections 3.5 and 5.2, Calculus section of text: Section R.
MA 000, Lesson Functions & Addition/Subtraction Polynomials Algebra section of tet: Sections.5 and 5., Calculus section of tet: Section R.1 Definition: A relation is any set of ordered pairs. The set of
More informationSECTION 1.4 PolyNomiAls feet. Figure 1. A = s 2 = (2x) 2 = 4x 2 A = 2 (2x) 3 _ 2 = 1 _ = 3 _. A = lw = x 1. = x
SECTION 1.4 PolyNomiAls 4 1 learning ObjeCTIveS In this section, you will: Identify the degree and leading coefficient of polynomials. Add and subtract polynomials. Multiply polynomials. Use FOIL to multiply
More informationSection 5.1 Polynomial Functions and Models
Term: A term is an expression that involves only multiplication and/or division with constants and/or variables. A term is separated by + or Polynomial: A polynomial is a single term or the sum of two
More informationFunctions: Polynomial, Rational, Exponential
Functions: Polynomial, Rational, Exponential MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Spring 2014 Objectives In this lesson we will learn to: identify polynomial expressions,
More informationCURRICULUM CATALOG. Algebra I (3130) VA
2018-19 CURRICULUM CATALOG Table of Contents COURSE OVERVIEW... 1 UNIT 1: FOUNDATIONS OF ALGEBRA... 1 UNIT 2: LINEAR EQUATIONS... 2 UNIT 3: FUNCTIONS... 2 UNIT 4: INEQUALITIES AND LINEAR SYSTEMS... 2 UNIT
More informationUNIT 4: RATIONAL AND RADICAL EXPRESSIONS. 4.1 Product Rule. Objective. Vocabulary. o Scientific Notation. o Base
UNIT 4: RATIONAL AND RADICAL EXPRESSIONS M1 5.8, M2 10.1-4, M3 5.4-5, 6.5,8 4.1 Product Rule Objective I will be able to multiply powers when they have the same base, including simplifying algebraic expressions
More informationPre-Algebra 2. Unit 9. Polynomials Name Period
Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:
More informationAlg 1B Chapter 7 Final Exam Review
Name: Class: Date: ID: A Alg B Chapter 7 Final Exam Review Please answer all questions and show your work. Simplify ( 2) 4. 2. Simplify ( 4) 4. 3. Simplify 5 2. 4. Simplify 9x0 y 3 z 8. 5. Simplify 7w0
More informationAlgebra One Dictionary
Algebra One Dictionary Page 1 of 17 A Absolute Value - the distance between the number and 0 on a number line Algebraic Expression - An expression that contains numbers, operations and at least one variable.
More informationUnit Essential Questions. How can you represent quantities, patterns, and relationships? How are properties of real numbers related to algebra?
Unit Essential Questions How can you represent quantities, patterns, and relationships? How are properties of real numbers related to algebra? Williams Math Lessons TARGET RATING 3 VARIABLES AND EXPRESSIONS
More informationVariables and Expressions
Variables and Expressions A variable is a letter that represents a value that can change. A constant is a value that does not change. A numerical expression contains only constants and operations. An algebraic
More informationAlgebra 31 Summer Work Packet Review and Study Guide
Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the
More informationUnit 2: Polynomials Guided Notes
Unit 2: Polynomials Guided Notes Name Period **If found, please return to Mrs. Brandley s room, M 8.** Self Assessment The following are the concepts you should know by the end of Unit 1. Periodically
More informationPolynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms
Polynomials Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms Polynomials A polynomial looks like this: Term A number, a variable, or the
More informationStudy Guide for Math 095
Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.
More informationA polynomial is an algebraic expression that has many terms connected by only the operations of +, -, and of variables.
A polynomial is an algebraic expression that has many terms connected by only the operations of +, -, and of variables. 2x + 5 5 x 7x +19 5x 2-7x + 19 x 2 1 x + 2 2x 3 y 4 z x + 2 2x The terms are the
More informationSolving Quadratic Equations
Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic
More informationEquations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero
Solving Other Types of Equations Rational Equations Examine each denominator to find values that would cause the denominator to equal zero Multiply each term by the LCD or If two terms cross-multiply Solve,
More informationAlgebra I Polynomials
Slide 1 / 217 Slide 2 / 217 Algebra I Polynomials 2014-04-24 www.njctl.org Slide 3 / 217 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying
More informationDue for this week. Slide 2. Copyright 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
MTH 09 Week 1 Due for this week Homework 1 (on MyMathLab via the Materials Link) The fifth night after class at 11:59pm. Read Chapter 6.1-6.4, Do the MyMathLab Self-Check for week 1. Learning team coordination/connections.
More informationAFM Review Test Review
Name: Class: Date: AFM Review Test Review Multiple Choice Identify the choice that best completes the statement or answers the question. What are the solutions of the inequality?. q + (q ) > 0 q < 3 q
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 informationASSIGNMENT. Please complete only the assignment for the class you will begin in September 2018.
ASSIGNMENT Attached is an assignment containing items necessary for you to have mastered to do well in Algebra II. Please complete only the assignment for the class you will begin in September 2018. Practicing
More informationUnit 1 Notes. Polynomials
Unit 1 Notes 1 Day Number Date Topic Problem Set 1 Wed. Sept. Operations with Signed Numbers and Order of Operations Pre-Unit Review PS (1 ) Thurs. Sept. Working with Exponents Start Exponent Laws P.S.
More informationDegree of a polynomial
Variable Algebra Term Polynomial Monomial Binomial Trinomial Degree of a term Degree of a polynomial Linear A generalization of arithmetic. Letters called variables are used to denote numbers, which are
More informationCURRICULUM UNIT MAP 1 ST QUARTER. COURSE TITLE: Applied Algebra 1 GRADE: 9
1 ST QUARTER Unit 1: Exploring Rational Numbers WEEK 1-3 Objectives: Write equations and formulas to solve application problems Compare order and plot rational and irrational numbers, including square
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 informationCurriculum Map: Mathematics
Curriculum Map: Mathematics Course: Honors Algebra II Grade(s): 9/10 Unit 1: Expressions, Equations, and Inequalities In this unit, students review basics concepts and skills of algebra studied in previous
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 informationAlgebra Performance Level Descriptors
Limited A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Algebra. A student at this level has an emerging ability to A student whose performance
More informationGrade AM108R 7 + Mastering the Standards ALGEBRA. By Murney R. Bell
Hayes AM108R Mastering the Standards ALGEBRA By Murney R. Bell Grade 7 + Mastering the Standards Algebra By Murney R. Bell Illustrated by Reneé Yates Copyright 2008, Hayes School Publishing Co., Inc.,
More informationMathematics Textbook Correlation to the 2016 Algebra I Standards of Learning and Curriculum Framework
and Curriculum Framework Publisher: McGraw-Hill School Education Text: Algebra 1 Copyright date 2018 A.1 The student will a) represent verbal quantitative situations algebraically; and TE: 5-9, 23-29,
More informationPolynomials: Add and Subtract
GSE Advanced Algebra Operations with Polynomials Polynomials: Add and Subtract Let's do a quick review on what polynomials are and the types of polynomials. A monomial is an algebraic expression that is
More informationChapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring
Chapter Six Polynomials Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Properties of Exponents The properties below form the basis
More informationUnit 2: Polynomials Guided Notes
Unit 2: Polynomials Guided Notes Name Period **If found, please return to Mrs. Brandley s room, M 8.** Self Assessment The following are the concepts you should know by the end of Unit 1. Periodically
More informationWe say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:
R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)
More informationPolynomial Operations Polly s Pasta
Polynomial Operations ACTIVITY 4.2 SUGGESTED LEARNING STRATEGIES: Close Reading, Discussion Group, Create Representations, Think/Pair/Share, Self/Peer Revision and Pizza Supply sells wholesale goods to
More informationMathB65 Ch 4 IV, V, VI.notebook. October 31, 2017
Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest
More informationx y x y ax bx c x Algebra I Course Standards Gap 1 Gap 2 Comments a. Set up and solve problems following the correct order of operations (including proportions, percent, and absolute value) with rational
More informationFastTrack - MA109. Exponents and Review of Polynomials
FastTrack - MA109 Exponents and Review of Polynomials Katherine Paullin, Ph.D. Lecturer, Department of Mathematics University of Kentucky katherine.paullin@uky.edu Monday, August 15, 2016 1 / 25 REEF Question
More informationAssignment 2.1. Exponent Properties: The Product Rule
Assignment.1 NAME: Exponent Properties: The Product Rule What is the difference between x and x? Explain in complete sentences and with examples. Product Repeated Multiplication Power of the form a b 5
More informationEx.1 identify the terms and coefficients of the expression.
Modeling with expressions An expression is a mathematical phrase that contains numbers or variables. Terms are the parts being added. Coefficient is the number in front of the variable. A constant is a
More informationUnit 2: Polynomials Guided Notes
Unit 2: Polynomials Guided Notes Name Period **If found, please return to Mrs. Brandley s room, M-8.** 1 Self-Assessment The following are the concepts you should know by the end of Unit 1. Periodically
More informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic
More informationExample #3: 14 (5 + 2) 6 = = then add = 1 x (-3) then. = 1.5 = add
Grade 9 Curricular content Operations with rational numbers (addition, subtraction, multiplication, division and order of operations) -incudes brackets and exponents (exponent laws) -exponents includes
More informationUnit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power.
MODULE 1 1 Polynomial A function that contains 1 or more or terms. The variables may be to any non-negative power. 1 Modeling Mathematical modeling is the process of using, and to represent real world
More informationHarbor Creek School District. Algebra II Advanced. Concepts Timeframe Skills Assessment Standards Linear Equations Inequalities
Algebra II Advanced and Graphing and Solving Linear Linear Absolute Value Relation vs. Standard Forms of Linear Slope Parallel & Perpendicular Lines Scatterplot & Linear Regression Graphing linear Absolute
More informationAlgebra 2 Summer Work Packet Review and Study Guide
Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the nine specific concepts covered in the
More informationA monomial or sum of monomials
Polynomial: A monomial or sum of monomials Polynomial in x is an expression of the form a n x n + a n 1 x n 1 + a n 2 x n 2 +. a 1 x 1 + a 0 where n is a positive integer and a n 0 Example: 6x 3 + 2x 8x
More informationWest Windsor-Plainsboro Regional School District Math A&E Grade 7
West Windsor-Plainsboro Regional School District Math A&E Grade 7 Page 1 of 24 Unit 1: Introduction to Algebra Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale
More informationRising 8th Grade Math. Algebra 1 Summer Review Packet
Rising 8th Grade Math Algebra 1 Summer Review Packet 1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving Multi-Step Equations 3. Add/subtract
More informationLESSON #1: VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II
1 LESSON #1: VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II Mathematics has developed a language all to itself in order to clarify concepts and remove ambiguity from the analysis of problems.
More information