( ) is called the dependent variable because its
|
|
- John Jones
- 5 years ago
- Views:
Transcription
1 page 1 of 16 CLASS NOTES: 3 8 thru 4 3 and 11 7 Functions, Exponents and Polynomials 3 8: Function Notation A function is a correspondence between two sets, the domain (x) and the range (y). An example of a function is f ( x ) = 2 x + 3 f ( x ) is the same as y The domain refers to x. x is called the independent variable because it can be ANY number well mostly! The range refers to y. y or f x ( ) is called the dependent variable because its value depends on the value of x that we plug in. These mean the same thing!! Evaluate the given expression if x = 4. Ex 3 x 1 If f ( x ) = 3 x 1, find the indicated value. ( ) Ex f 4 If x = 4, then 3 x 1 = 3 ( 4 ) 1 = 12 1 = 11 If f ( x ) = 3 x 1, then ( ) = 3 ( 4 ) 1 f 4 = 12 1 = 11 Therefore, f ( 4 ) = 11
2 page 2 of 16 EX 1 Find the indicated values for the function f ( x ) = x (a) f ( 3 ) = Hint: Sometimes you may see functions written as: f : x x (b) f ( 5) = (c) f ( 1 ) = (d) f ( 0 ) = (e) f ( a ) = (f) f ( 3b ) = Finding the RANGE of a function EX 2 Given f ( x ) = x 3 with the domain D = { 0, 1, 2, 3, 4}, find the range of f ( x ). Plug in each of the five values for x. Those five answers are the members of the range, R = {,,, }. Even if a range value occurs twice, you only include it once in the range set.
3 page 3 of 16 Finding the DOMAIN of a function This means finding the values that x can be. This may seem strange since x is the value that we plug into the function. You may be thinking, Don t they usually tell us what to plug in? And yes, they do! However, finding the domain means asking ourselves, Are there any numbers that I cannot plug into the function? And yes, there might be! There are TWO situations that may cause us to exclude values from the domain. [ 1] If the function is a fraction, any value of x that makes the denominator = zero must be excluded from the domain. Ex If f ( x ) = 2 x, then x 3 because the denominator would be 0. x 3 So, the domain of this function is R except x 3 [ 2 ] If the function includes a square root, then the domain is the set of values that make the value of what is under the square root sign 0. Ex If f ( x ) = 2 x + 1, then 2 x x 1 x 1 2 So, the domain of this function is x 1 2 If neither of these two situations apply, then the domain is simply R (all real numbers). EX 3 Give the domain of each function. (a) f ( x ) = x 2 x + 1 (b) f ( x ) = 3 4 x + 5 6
4 page 4 of 16 EX 4 Give the domain of each function. (a) f ( x ) = x 7 3 (b) f ( x ) = 2 x x (c) f ( x ) = x ( x + 2 ) x 3 ( ) (d) f ( x ) = 3 x 4 (e) f ( x ) = x + 5 (f) f ( x ) = x x
5 page 5 of 16 COMPOSITION OF FUNCTIONS This means TWO or more functions combined by +, --, or a function inside of a function EX 5 Given f ( x ) = 3 x 2 and g x (a) f ( 1 ) + g ( 2 ) = ( ) = x 2, find the indicated values. (b) f ( g ( 5 )) = Hint: Find the value of the innermost function first!! (c) g ( f ( 1 )) = (d) f ( f ( 3 )) = (e) g ( 2 f ( 1) ) = (f) f ( g ( a )) =
6 page 6 of : Linear Functions LINEAR FUNCTIONS Functions that are in the form f ( x ) = mx + b. m is the slope of the line. ( 0, b ) are the coordinates for the y-intercept. This is also denoted as f ( 0 ) = b f ( x ) = y Function Notation: OLD Equation Notation y = 2 x 1 f x NEW Function Notation ( ) = 2 x 1 Complete the ordered pair ( 4, ) to form a solution for the equation. Plug in 4 for x and solve for y. Find the value for f ( 4 ). Plug in 4 for x and simplify. f ( 4 ) = 2 ( 4 ) 1 = 8 1 = 7 The solution is ( 4, 7) f ( 4 ) = 7 EX 1 Find an equation of each linear function f using the given information. (a) m = 4, b = 3
7 page 7 of 16 EX 2 Find an equation of each linear function f using the given information. (a) m = 1 2, f ( 0 ) = 5 (b) slope is 1, f ( 2 ) = 3 Plug in what we know Because f ( x ) = y, ( ) = 3 is the same as the point 2, 3 f 2 x y ( ) f ( x ) = mx + b ( ) = 1 ( 2 ) + b f 2 3 = 2 + b 5 = b We need m and b [aka f 0 f ( x ) = mx + b Answer: f ( x ) = 1 x + 5 ( ) ] to write a linear function in the form (c) m = 4, f ( 2 ) = 7 (d) f ( 3 ) = 12, f ( 5 ) = 2 Hint: m is y 2 y 1 x 2 x 1 = f ( 5 ) f ( 3 ) = 5 3 = 14 = 7 2
8 page 8 of 16 EX 3 Find an equation of each linear function f using the given information. (a) f ( 3 ) = 2, f ( 12 ) = 4
9 page 9 of : Polynomials Examples: 3 x 4 + x 3 y + 4 x 2 y 2 2 xy 3 + 9y x 3 6 x 2 + x 4 Constant - a number (without a variable) Monomial - a constant, a variable or a product of constants and variables. Term another word for monomial Binomial the sum of two monomials. Polynomial - the sum of two or more monomials. Coefficient - the number (factor) in front of the variables of a monomial. Leading Coefficient the coefficient of the term with the highest power. Degree of a Variable the power (exponent) of a variable in a monomial. Degree of a Monomial the sum of the powers of the variables in a monomial. Degree of a Polynomial the largest of the degrees of the monomials in a polynomial. Descending Order If the polynomial is a single variable (one variable) polynomial, the terms are arranged so that the exponents of the variable decrease from left to right. Similar Monomials (also called like terms ) monomials that have the exact variables and powers of those variables. Combine Similar Terms by adding their coefficients Simplified Polynomial to simplify a polynomial (1) combine similar terms and (2) arrange terms in descending order.
10 page 10 of 16 Given the example: 7 x x + 4 x 3 1. Draw a circle around each monomial. 2. Rewrite the polynomial in descending order. 3. What is the value of the constant? 4. What is the coefficient of x? 5. What is the leading coefficient? 6. What is the degree of the term 7 x 2? 7. What is the degree of the polynomial? 8. What is the degree of the term 5 x 2 y 2? 9. What is the degree of the term 9 xy 2? 10. Which term is similar to 7 x 3? 11. Simplify the polynomial x 3 x x x ADDING and SUBTRACTING polynomials is as simple as combining similar terms. EX: Add 2 x 2 3 x + 5 and x 3 5 x x 5 EX: Subtract 2 x 2 3 x + 5 from x 3 5 x x 5
11 page 11 of : Using Laws of Exponents Simplify. EX: x 5 x 3 x a x b = x a +b EX: EX: ( xy ) 3 ( 5 x ) 2 ( xy ) a = x a y a EX: $ x 2 % ' 4 $ x a % ' b = x a b EX: $ x 3 y 5 % ' 4 EX: $ 2 x 2 y 3 % ' 5 $ x a y b % ' n = x an y bn EX: % x 3 $ ' ( 2 EX: $ xy 2 % ' 3
12 page 12 of 16 Simplify completely. EX: x $ x 2 % ' 3 x 5 EX: $ 3 xy 2 z 3 % ' 3 EX: % 3 x 2 y 3 $ ' ( % 4 xy 2 $ ( EX: 3 x ' ( ) $ 4 x 5 % ' EX: % x 2 $ ' ( % x 4 $ ( EX: x 2 x k ' EX: $ x 2 % ' k EX: $ x 2 % ' k $ x k % ' 3
13 page 13 of : Multiplying Polynomials (Using FOIL) FIRST OUTER INNER LAST Then COMBINE like terms! Multiply ( x + 2 )( x + 5) F + O + I + L x x x 5 2 x 2 5 x 2 5 x 2 x 10 x x + 10 EX 1 Multiply. ( x + 2 ) ( y ) EX 2 Multiply. ( x + 2 )( 3 x + 1) EX 3 Multiply. ( 2a 5) ( 3a + 4) EX 4 Multiply. 3 x + 1 ( ) $ x x + 6 % '
14 page 14 of 16 ( a + b ) 2 does NOT equal a 2 + b 2!!!!!!! Use F-O-I-L to multiply the following. ( a + b ) 2 ( a b ) 2 ( a + b )( a b ) EX 5 Multiply. ( x + 1) 2 EX 6 Multiply. ( x 3) 2 EX 7 Multiply. ( x + 5) ( x 5)
15 page 15 of : Powers of Binomials (Binomial Expansion) ( a + b) 0 = 1 ( a + b) 1 = 1a 1 + 1b 1 ( a + b) 2 = 1a 2 + 2ab + 1b 2 ( a + b) 3 = 1a 3 + 3a 2 b + 3ab 2 + 1b 3 ( a + b) 4 = 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + 1b 4 ( a + b) 5 = 1a 5 + 5a 4 b + 10a 3 b a 2 b 3 + 5ab 4 + 1b 5 Binomial Expansion - The sum of terms that you get when you multiply out a power of a binomial. The pattern of coefficients comes from a triangular array of numbers called Pascal s Triangle. EX 1 Expand ( a + b) 6 by using row 6 of Pascal s Triangle. EX 2 Find the first four terms in the expansion of ( x 2y) 7.
16 page 16 of 16 EX 3 Find the first three terms of the expansion of ( 2x y) 6. EX 4 Expand and simplify the expression ( x 3) 4. EX 5 The first three terms of the expansion of a + b ( ) 20 are a a 19 b + 190a 18 b 2. Write the last three terms.
Algebra 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 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 informationGeometry 21 Summer Work Packet Review and Study Guide
Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the
More informationAdding and Subtracting Polynomials Add and Subtract Polynomials by doing the following: Combine like terms
POLYNOMIALS AND POLYNOMIAL OPERATIONS STUDY GUIDE Polynomials Polynomials are classified by two different categories: by the number of terms, and the degree of the leading exponent. Number Classification
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 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 informationSCIE 4101 Spring Math Review Packet #2 Notes Algebra I
SCIE 4101 Spring 011 Math Review Packet # Notes Algebra I I consider Algebra and algebraic thought to be the heart of mathematics everything else before that is arithmetic. The first characteristic of
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 informationSCIE 4101 Fall Math Review Packet #2 Notes Patterns and Algebra I Topics
SCIE 4101 Fall 014 Math Review Packet # Notes Patterns and Algebra I Topics I consider Algebra and algebraic thought to be the heart of mathematics everything else before that is arithmetic. The first
More informationGeometry Summer Assignment 2018
Geometry Summer Assignment 2018 The following packet contains topics and definitions that you will be required to know in order to succeed in Geometry this year. You are advised to be familiar with each
More informationSecondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics
Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together
More informationCombining Like Terms in Polynomials
Section 1 6: Combining Like Terms in Polynomials Polynomials A polynomial is an expression that has two or more terms each separated by a + or sign. If the expression has only one term it is called a monomial.
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 informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II. 2 nd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part II 2 nd Nine Weeks, 2016-2017 1 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
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 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 informationSection 1.4 Circles. Objective #1: Writing the Equation of a Circle in Standard Form.
1 Section 1. Circles Objective #1: Writing the Equation of a Circle in Standard Form. We begin by giving a definition of a circle: Definition: A Circle is the set of all points that are equidistant from
More informationMA094 Part 2 - Beginning Algebra Summary
MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page
More informationPart 2 - Beginning Algebra Summary
Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian
More informationALGEBRA 2 Summer Review Assignments Graphing
ALGEBRA 2 Summer Review Assignments Graphing To be prepared for algebra two, and all subsequent math courses, you need to be able to accurately and efficiently find the slope of any line, be able to write
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 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 informationReview Notes - Solving Quadratic Equations
Review Notes - Solving Quadratic Equations What does solve mean? Methods for Solving Quadratic Equations: Solving by using Square Roots Solving by Factoring using the Zero Product Property Solving by Quadratic
More informationLESSON 7.2 FACTORING POLYNOMIALS II
LESSON 7.2 FACTORING POLYNOMIALS II LESSON 7.2 FACTORING POLYNOMIALS II 305 OVERVIEW Here s what you ll learn in this lesson: Trinomials I a. Factoring trinomials of the form x 2 + bx + c; x 2 + bxy +
More informationAlgebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials
Algebra 1: Hutschenreuter Chapter 10 Notes Name 10.1 Adding and Subtracting Polynomials Polynomial- an expression where terms are being either added and/or subtracted together Ex: 6x 4 + 3x 3 + 5x 2 +
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 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 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 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 information7-7 Multiplying Polynomials
Example 1: Multiplying Monomials A. (6y 3 )(3y 5 ) (6y 3 )(3y 5 ) (6 3)(y 3 y 5 ) 18y 8 Group factors with like bases together. B. (3mn 2 ) (9m 2 n) Example 1C: Multiplying Monomials Group factors with
More informationUNIT 2 FACTORING. M2 Ch 11 all
UNIT 2 FACTORING M2 Ch 11 all 2.1 Polynomials Objective I will be able to put polynomials in standard form and identify their degree and type. I will be able to add and subtract polynomials. Vocabulary
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 informationMath 1 Variable Manipulation Part 6 Polynomials
Name: Math 1 Variable Manipulation Part 6 Polynomials Date: 1 VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does not have
More informationSect Addition and Subtraction of Polynomials
Sect 5.5 - Addition and Subtraction of Polynomials Concept #1 Introduction to Polynomials Before we begin discussing polynomials, let s review some items from chapter 1 with the following example: Complete
More informationRadicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize).
Summer Review Packet for Students Entering Prealculus Radicals: To simplify means that 1) no radicand has a perfect square factor and ) there is no radical in the denominator (rationalize). Recall the
More informationCHAPTER 1 POLYNOMIALS
1 CHAPTER 1 POLYNOMIALS 1.1 Removing Nested Symbols of Grouping Simplify. 1. 4x + 3( x ) + 4( x + 1). ( ) 3x + 4 5 x 3 + x 3. 3 5( y 4) + 6 y ( y + 3) 4. 3 n ( n + 5) 4 ( n + 8) 5. ( x + 5) x + 3( x 6)
More informationSolving Multi-Step Equations
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 terms to both sides of the equation to get the
More informationSections 7.2, 7.3, 4.1
Sections 7., 7.3, 4.1 Section 7. Multiplying, Dividing and Simplifying Radicals This section will discuss the rules for multiplying, dividing and simplifying radicals. Product Rule for multiplying radicals
More informationMCR3U Unit 7 Lesson Notes
7.1 Arithmetic Sequences Sequence: An ordered list of numbers identified by a pattern or rule that may stop at some number or continue indefinitely. Ex. 1, 2, 4, 8,... Ex. 3, 7, 11, 15 Term (of a sequence):
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 informationGrade 9 Mathematics Unit #2 Patterns & Relations Sub-Unit #1 Polynomials
Grade 9 Mathematics Unit #2 Patterns & Relations Sub-Unit #1 Polynomials Lesson Topic I Can 1 Definitions Define Polynomials Identify Polynomials Identify different parts of a polynomial Identify monomials,
More informationTo factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x
Factoring trinomials In general, we are factoring ax + bx + c where a, b, and c are real numbers. To factor an expression means to write it as a product of factors instead of a sum of terms. The expression
More informationTopic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3
Topic 7: Polynomials Table of Contents 1. Introduction to Polynomials. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Special Products of Binomials 5. Factoring Polynomials 6. Factoring
More informationCoach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers
Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}
More informationWhat is a constant? A Constant is a number representing a quantity or value that does not change.
Worksheet -: Algebraic Expressions What is a constant? A Constant is a number representing a quantity or value that does not change. What is a variable? A variable is a letter or symbol representing a
More informationMAT30S Grade 10 Review Mr. Morris
GRADE 11 PRECALCULUS REVIEW OF GRADE 10 The following Grade 10 concepts should be reviewed for Grade 11 Precal: 1. Slopes of the Graphs of Linear Functions 2. Powers and Roots 3. Simplifying Radicals 4.
More informationAlgebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals
Algebra 1 Math Review Packet Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals 2017 Math in the Middle 1. Clear parentheses using the distributive
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 informationSummer Review. For Students Entering. Algebra 2 & Analysis
Lawrence High School Math Department Summer Review For Students Entering Algebra 2 & Analysis Fraction Rules: Operation Explanation Example Multiply Fractions Multiply both numerators and denominators
More informationNote: A file Algebra Unit 09 Practice X Patterns can be useful to prepare students to quickly find sum and product.
Note: This unit can be used as needed (review or introductory) to practice operations on polynomials. Math Background Previously, you Identified monomials and their characteristics Applied the laws of
More informationChapter 3-1 Polynomials
Chapter 3 notes: Chapter 3-1 Polynomials Obj: SWBAT identify, evaluate, add, and subtract polynomials A monomial is a number, a variable, or a product of numbers and variables with whole number exponents
More informationLESSON 6.3 POLYNOMIAL OPERATIONS II
LESSON 6.3 POLYNOMIAL OPERATIONS II LESSON 6.3 POLYNOMIALS OPERATIONS II 277 OVERVIEW Here's what you'll learn in this lesson: Multiplying Binomials a. Multiplying binomials by the FOIL method b. Perfect
More informationSUMMER REVIEW PACKET. Name:
Wylie East HIGH SCHOOL SUMMER REVIEW PACKET For students entering Regular PRECALCULUS Name: Welcome to Pre-Calculus. The following packet needs to be finished and ready to be turned the first week of the
More informationMultiplication of Polynomials
Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is
More informationMath 2 Variable Manipulation Part 3 Polynomials A
Math 2 Variable Manipulation Part 3 Polynomials A 1 MATH 1 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 not
More informationFactoring Review Types of Factoring: 1. GCF: a. b.
Factoring Review Types of Factoring: 1. GCF: a. b. Ex. A. 4 + 2 8 B. 100 + 25 2. DOS: a. b. c. Ex. A. 9 B. 2 32 3. Plain x Trinomials: Start Signs Factors 1. 2. 3. 4. Ex. A. + 7 + 12 B. 2 3 4. Non-Plain
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 informationUnit 2, Ongoing Activity, Little Black Book of Algebra II Properties
Unit 2, Ongoing Activity, Little Black Book of Algebra II Properties Little Black Book of Algebra II Properties Unit 2 - Polynomial Equations & Inequalities 2.1 Laws of Exponents - record the rules for
More informationLT1: Adding and Subtracting Polynomials. *When subtracting polynomials, distribute the negative to the second parentheses. Then combine like terms.
LT1: Adding and Subtracting Polynomials *When adding polynomials, simply combine like terms. *When subtracting polynomials, distribute the negative to the second parentheses. Then combine like terms. 1.
More informationMath 10-C Polynomials Concept Sheets
Math 10-C Polynomials Concept Sheets Concept 1: Polynomial Intro & Review A polynomial is a mathematical expression with one or more terms in which the exponents are whole numbers and the coefficients
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 informationA monomial is measured by its degree To find its degree, we add up the exponents of all the variables of the monomial.
UNIT 6 POLYNOMIALS Polynomial (Definition) A monomial or a sum of monomials. A monomial is measured by its degree To find its degree, we add up the exponents of all the variables of the monomial. Ex. 2
More informationHow to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials
PRC Ch P_3.notebook How to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials How to remove common factors from polynomials
More informationAlgebra. Practice Pack
Algebra Practice Pack WALCH PUBLISHING Table of Contents Unit 1: Algebra Basics Practice 1 What Are Negative and Positive Numbers?... 1 Practice 2 Larger and Smaller Numbers................ 2 Practice
More informationP.1: Algebraic Expressions, Mathematical Models, and Real Numbers
Chapter P Prerequisites: Fundamental Concepts of Algebra Pre-calculus notes Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Algebraic expression: a combination of variables and
More informationMath 3 Variable Manipulation Part 4 Polynomials B COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number:
Math 3 Variable Manipulation Part 4 Polynomials B COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number: 1 Examples: 1 + i 39 + 3i 0.8.i + πi + i/ A Complex Number
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 informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More informationThere are two main properties that we use when solving linear equations. Property #1: Additive Property of Equality
Chapter 1.1: Solving Linear and Literal Equations Linear Equations Linear equations are equations of the form ax + b = c, where a, b and c are constants, and a zero. A hint that an equation is linear is
More information2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY
2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you
More informationREAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y?
REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} REAL WORLD SCENARIOS 1. If 4x + y = 110 where 10 < x < 0, what is the least possible value of y? WORK AND ANSWER SECTION. Evaluate
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 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 informationMaintaining Mathematical Proficiency
Chapter 7 Maintaining Mathematical Proficiency Simplify the expression. 1. 5x 6 + 3x. 3t + 7 3t 4 3. 8s 4 + 4s 6 5s 4. 9m + 3 + m 3 + 5m 5. 4 3p 7 3p 4 1 z 1 + 4 6. ( ) 7. 6( x + ) 4 8. 3( h + 4) 3( h
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 informationMATH 60 Course Notebook Chapter #1
MATH 60 Course Notebook Chapter #1 Integers and Real Numbers Before we start the journey into Algebra, we need to understand more about the numbers and number concepts, which form the foundation of Algebra.
More informationNorthwood High School Algebra 2/Honors Algebra 2 Summer Review Packet
Northwood High School Algebra 2/Honors Algebra 2 Summer Review Packet This assignment should serve as a review of the Algebra 1 skills necessary for success. Our hope is that this review will keep your
More informationUnit 7: Factoring Quadratic Polynomials
Unit 7: Factoring Quadratic Polynomials A polynomial is represented by: where the coefficients are real numbers and the exponents are nonnegative integers. Side Note: Examples of real numbers: Examples
More information5-1 Study Guide and Intervention
5-1 Study Guide and Intervention Multiply and Divide Monomials Negative exponents are a way of expressing the multiplicative inverse of a number. Negative Exponents a -n = 1 a n and 1 a -n = a n for any
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 informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in
More informationUnderstand the vocabulary used to describe polynomials Add polynomials Subtract polynomials Graph equations defined by polynomials of degree 2
Section 5.1: ADDING AND SUBTRACTING POLYNOMIALS When you are done with your homework you should be able to Understand the vocabulary used to describe polynomials Add polynomials Subtract polynomials Graph
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 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 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 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 information5.1 Monomials. Algebra 2
. Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific
More informationModule 2 Study Guide. The second module covers the following sections of the textbook: , 4.1, 4.2, 4.5, and
Module 2 Study Guide The second module covers the following sections of the textbook: 3.3-3.7, 4.1, 4.2, 4.5, and 5.1-5.3 Sections 3.3-3.6 This is a continuation of the study of linear functions that we
More informationAlgebra II Notes Polynomial Functions Unit Introduction to Polynomials. Math Background
Introduction to Polynomials Math Background Previously, you Identified the components in an algebraic epression Factored quadratic epressions using special patterns, grouping method and the ac method Worked
More informationMath Precalculus I University of Hawai i at Mānoa Spring
Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2013 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents
More informationDear Future Pre-Calculus Students,
Dear Future Pre-Calculus Students, Congratulations on your academic achievements thus far. You have proven your academic worth in Algebra II (CC), but the challenges are not over yet! Not to worry; this
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 information= 9 = x + 8 = = -5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives:
Math 65 / Notes & Practice #1 / 20 points / Due. / Name: Home Work Practice: Simplify the following expressions by reducing the fractions: 16 = 4 = 8xy =? = 9 40 32 38x 64 16 Solve the following equations
More informationGraphing Linear Equations: Warm Up: Brainstorm what you know about Graphing Lines: (Try to fill the whole page) Graphing
Graphing Linear Equations: Warm Up: Brainstorm what you know about Graphing Lines: (Try to fill the whole page) Graphing Notes: The three types of ways to graph a line and when to use each: Slope intercept
More informationChetek-Weyerhaeuser High School
Chetek-Weyerhaeuser High School Unit 1 Variables and Expressions Math RtI Units and s Math RtI A s 1. I can use mathematical properties to evaluate expressions. I can use mathematical properties to evaluate
More informationWhich one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x ) A) x = 5 B) x = -6 C) x = -5 D) x = 6
Review for Final Exam Math 124A (Flatley) Name Which one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x - 14 1) A) x = 5 B) x = -6 C) x = -5 D) x = 6 Solve the linear equation.
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 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 informationPacing Guide Algebra 1
Pacing Guide Algebra Chapter : Equations and Inequalities (one variable) Section Section Title Learning Target(s) I can. Evaluate and Simplify Algebraic Expressions. Evaluate and simplify numeric and algebraic
More informationMath Precalculus I University of Hawai i at Mānoa Spring
Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2014 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents
More information