3. Power of a Product: Separate letters, distribute to the exponents and the bases

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "3. Power of a Product: Separate letters, distribute to the exponents and the bases"

Transcription

1 Chapter 5 : Polynomials and Polynomial Functions 5.1 Properties of Exponents Rules: 1. Product of Powers: Add the exponents, base stays the same 2. Power of Power: Multiply exponents, bases stay the same 3. Power of a Product: Separate letters, distribute to the exponents and the bases 4. Zero exponents: Any number to the zero power = 1, zero to any power is o 5. Negative exponents: If base has ( ), change to reciprocal, no ( ) in final answer 6. Quotient of Powers: Subtract exponents, simplify bases if possible 7. Power of a Quotient: Find the power of the numerator and denominator Hints: If a base doesn t have an exponent, it is to the 1st power Follow order of operations (PEMDAS) Reciprocals as last step

2 Multiply straight across Reduce everything to lowest terms The answer should have only one of each base remaining No negative exponents in answer To Simplify Expressions: Use the properties to get rid of ( ). Multiply fractions straight across, line up common bases. Reduce the numbers, move negative exponents, get to one of each base. Scientific Notation Negative powers, start small Ex * 10 ³ = Positive powers, start big Ex * 10³ = 1250 Exponential Growth: A = C( 1 + r)t Exponential Decay: A = C(1 r)t

3 5.2 Evaluate and Graph Polynomial Functions 5. 3 Add, subtract and multiply polynomials Monomial: a single term with exponent as integer Polynomial: Has many(all) terms with exponents as integers Coefficients: numbers that are in front of terms Leading Coefficient: coefficient of term with highest exponent (+) opens up, ( ) opens down Terms: Variables separated by addition term the amount of terms determines what type (monomial, binomial, trinomial, etc) Degree: Value of the highest exponent (Linear is single term) 3 = cubic, 2 = quadratic 1 = monomial Constant: the single number at the end (no letters attached to it) Adding : Combine like terms (2x + 3) + ( 3x + 4) = 5x + 7 Subtracting: Change subtraction to adding the opposite ( 2x + 3) ( 3x + 4) ( 2x +3) + ( 3x 4) = 1x 1 Multiply: Use foil, combine like terms First, outside, Inside, last or distribute each term singlely. With multiple terms, just distribute and combine like terms.

4 (2x+3)(3x+4) = 6x² + 8x + 9x + 12 = 6x² + 17x + 2 Watch the signs Line up terms, and combine like terms Multiplication patterns and short cuts: 2 (a + b) = a 2 + 2ab + b 2 Instead of using foil, replace values into formula. 3 (a + b) = a 3 + 3a 2 b + 3ab + b^3 the a value decreases left to right, b value increases left to right 5.4 Factor and Solve Polynomial Equations Rules to follow: 1. Look for common monomials 2. Use factoring Pattern *difference of squares a^2 b^2 = (a + b) ( a b) ex. 2x^4 32x^2 2x^2( x^2 16) 2x^2 (x + 4) ( x 4) *sum of cubes a^3 + b^3 = ( a + b) (a^2 ab + b^2) *diff of cubes a^3 b^3 = ( a b) ( a^2 + ab+ b^2) ex. 8x^3 125 (something cubed minus something cubed) a= (2x) ^3 b = (5)^3 ( 2x 5) ( 4x^2 + 10x + 25)

5 3. Factorable trinomial ( like what we ve done with X and super box) ex. x^2 8x 20 ( x 10 ) ( x + 2) ex. 2x^2 3x + 1 (2x 1) ( x 1) 4. Factor by grouping ( 4 terms break into two different groups, factor out and rewrite, will get two sets of () that have same terms) ex. x^3 7x^2 4x + 28 x^2( x 7) 4( x 7) (x^2 4 )(x 7) = still need to simplify x^2 term because its a difference of squares (x 2) (x +2)(x 7) Set all terms equal to zero to solve 5. Any combination of 1 4 Hints: Determine common factors Determine common letters Pull out the lowest in each term Must be in every term, to factor it out 20k 10k² + 70kz² = 10k( 2 5k + 7kz) Once the common values are pulled out, determine what is left. 2 terms, use the factoring patterns. 3 terms use trinomial, 4 terms factor by grouping. The degree tells how many solutions there will be Grouping example:

6 5.5 Apply the remainder and factor Theorems Long Division:

7 Always look at first term to determine how many times it goes into term in question If there is a remainder, put it over the divisor as part of the answer Be sure to place the values you choose, above the appropriate terms If you are missing a term, account for it anyway by using 0x^value, to keep its place in answer Synthetic Division: Only works for linear polynomial divisors

8 Division in terms of (x r): Use the opposite value of r. Write coefficients of each term in order. Use zero if a term is not given. Add down, multiply divisor diagonally, put new answer under next term. Continue through last term. The last value is the remainder, put it over the divisor term. Going from right to left: remainder, constant, coefficents. Rewrite expression with using coefficients and terms. Each coefficient will be one degree less than you started with. 2 ways to evaluate a function: 1. Direct Substitution plug values in for x and solve ex. f(x) = 3x^2 2x + 7 find f( 4) plug 4 in for x and solve.

9 f( 4) = 3( 4)^2 2( 4) + 7 = Synthetic Substitution Follow steps of synthetic division, but do not use opposite of x f(x) = 3x^2 2x + 7 f( 4) = 63 The remainder is the answer= 63 How to make a general graph of polynomials: f(x) = 2x^5 + 3x^ Determine if it is odd or even (number of degrees) 2. Determine how many times it will cross the x intercept (same number as degrees) 3. Determine how many turns (one less than the degrees) 4. Determine y intercept (the constant number at the end) 5. Determine what the left and right behavior is (what quadrant it begins in) Even positive up / up Even negative down / down Odd positive down / up Odd negative up / down 6. Graph the general line, crossing through the x intercept with certain number of turns and passing through the y intercept. Applying synthetic division with story problems/figures: When given total volume of a shape: divide the volume by one of the sides. Factor out to find missing side (use x or super box)

10 OR Divide the first answer by the other side. Will get the three sides. When dividing, you will always get zero for a remainder as they are all factors.

Chapter 5 - Polynomials and Polynomial Functions

Chapter 5 - Polynomials and Polynomial Functions Math 233 - Spring 2009 Chapter 5 - Polynomials and Polynomial Functions 5.1 Addition and Subtraction of Polynomials Definition 1. A polynomial is a finite sum of terms in which all variables have whole

More information

Date: Section P.2: Exponents and Radicals. Properties of Exponents: Example #1: Simplify. a.) 3 4. b.) 2. c.) 3 4. d.) Example #2: Simplify. b.) a.

Date: Section P.2: Exponents and Radicals. Properties of Exponents: Example #1: Simplify. a.) 3 4. b.) 2. c.) 3 4. d.) Example #2: Simplify. b.) a. Properties of Exponents: Section P.2: Exponents and Radicals Date: Example #1: Simplify. a.) 3 4 b.) 2 c.) 34 d.) Example #2: Simplify. a.) b.) c.) d.) 1 Square Root: Principal n th Root: Example #3: Simplify.

More information

CD 1 Real Numbers, Variables, and Algebraic Expressions

CD 1 Real Numbers, Variables, and Algebraic Expressions CD 1 Real Numbers, Variables, and Algebraic Expressions The Algebra I Interactive Series is designed to incorporate all modalities of learning into one easy to use learning tool; thereby reinforcing learning

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

More information

2.1 Algebraic Expressions and Combining like Terms

2.1 Algebraic Expressions and Combining like Terms 2.1 Algebraic Expressions and Combining like Terms Evaluate the following algebraic expressions for the given values of the variables. 3 3 3 Simplify the following algebraic expressions by combining like

More information

MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

PRECALCULUS Semester I Exam Review Sheet

PRECALCULUS Semester I Exam Review Sheet PRECALCULUS Semester I Exam Review Sheet Chapter Topic P.1 Real Numbers {1, 2, 3, 4, } Natural (aka Counting) Numbers {0, 1, 2, 3, 4, } Whole Numbers {, -3, -2, -2, 0, 1, 2, 3, } Integers Can be expressed

More information

Unit 3 Polynomials Study Guide

Unit 3 Polynomials Study Guide Unit Polynomials Study Guide 7-5 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial

More information

Answers to Basic Algebra Review

Answers to Basic Algebra Review Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

More information

ALGEBRA I A PLUS COURSE OUTLINE

ALGEBRA I A PLUS COURSE OUTLINE ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best

More information

Table of Contents Sequence List

Table of Contents Sequence List Table of Contents Sequence List 368-102215 Level 1 Level 5 1 A1 Numbers 0-10 63 H1 Algebraic Expressions 2 A2 Comparing Numbers 0-10 64 H2 Operations and Properties 3 A3 Addition 0-10 65 H3 Evaluating

More information

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , ) Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.1-1.4, 1.6-1.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

Mth 95 Module 2 Spring 2014

Mth 95 Module 2 Spring 2014 Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

More information

Monomials with the same variables to the same powers are called like terms, If monomials are like terms only their coefficients can differ.

Monomials with the same variables to the same powers are called like terms, If monomials are like terms only their coefficients can differ. Chapter 7.1 Introduction to Polynomials A monomial is an expression that is a number, a variable or the product of a number and one or more variables with nonnegative exponents. Monomials that are real

More information

ModuMath Algebra Lessons

ModuMath Algebra Lessons ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations

More information

Exponents, Polynomials and Functions. Copyright Cengage Learning. All rights reserved.

Exponents, Polynomials and Functions. Copyright Cengage Learning. All rights reserved. Exponents, Polynomials and Functions 3 Copyright Cengage Learning. All rights reserved. 3.1 Rules for Exponents Copyright Cengage Learning. All rights reserved. Rules for Exponents The basic concept of

More information

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2 Pure Math 0 Notes Unit : Polynomials Unit : Polynomials -: Reviewing Polynomials Epressions: - mathematical sentences with no equal sign. Eample: Equations: - mathematical sentences that are equated with

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

Course Title: Math A Elementary Algebra

Course Title: Math A Elementary Algebra Course Title: Math A Elementary Algebra Unit 0: Course Introduction In this unit you will get familiar with important course documents, such as syllabus and communication policy. You will register on MyMathLab

More information

Chapter R - Basic Algebra Operations (69 topics, due on 05/01/12)

Chapter R - Basic Algebra Operations (69 topics, due on 05/01/12) Course Name: College Algebra 001 Course Code: R3RK6-CTKHJ ALEKS Course: College Algebra with Trigonometry Instructor: Prof. Bozyk Course Dates: Begin: 01/17/2012 End: 05/04/2012 Course Content: 288 topics

More information

Chapter 6: Polynomials and Polynomial Functions. Chapter 6.1: Using Properties of Exponents Goals for this section. Properties of Exponents:

Chapter 6: Polynomials and Polynomial Functions. Chapter 6.1: Using Properties of Exponents Goals for this section. Properties of Exponents: Chapter 6: Polynomials and Polynomial Functions Chapter 6.1: Using Properties of Exponents Properties of Exponents: Products and Quotients of Powers activity on P323 Write answers in notes! Using Properties

More information

Solving Logarithmic Equations

Solving Logarithmic Equations Solving Logarithmic Equations Deciding How to Solve Logarithmic Equation When asked to solve a logarithmic equation such as log (x + 7) = or log (7x + ) = log (x + 9), the first thing we need to decide

More information

FACTORING POLYNOMIALS

FACTORING POLYNOMIALS 296 (5-40) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated

More information

1.3 Polynomials and Factoring

1.3 Polynomials and Factoring 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

More information

HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

More information

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if 1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

More information

MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

More information

Algebra II Pacing Guide First Nine Weeks

Algebra II Pacing Guide First Nine Weeks First Nine Weeks SOL Topic Blocks.4 Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole and natural. 7. Recognize that the

More information

x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent.

x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent. Rules of Exponents: If n > 0, m > 0 are positive integers and x, y are any real numbers, then: x m x n = x m+n x m x n = xm n, if m n (x m ) n = x mn (xy) n = x n y n ( x y ) n = xn y n 1 Can we make sense

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

More information

Rockhurst High School Algebra 1 Topics

Rockhurst High School Algebra 1 Topics Rockhurst High School Algebra 1 Topics Chapter 1- Pre-Algebra Skills Simplify a numerical expression using PEMDAS. Substitute whole numbers into an algebraic expression and evaluate that expression. Substitute

More information

CLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column.

CLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column. SYNTHETIC DIVISION CLASS NOTES When factoring or evaluating polynomials we often find that it is convenient to divide a polynomial by a linear (first degree) binomial of the form x k where k is a real

More information

Placement Test Review Materials for

Placement Test Review Materials for Placement Test Review Materials for 1 To The Student This workbook will provide a review of some of the skills tested on the COMPASS placement test. Skills covered in this workbook will be used on the

More information

1.3 Algebraic Expressions

1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

ALGEBRA 2 SCOPE AND SEQUENCE. Writing Quadratic Equations Quadratic Regression

ALGEBRA 2 SCOPE AND SEQUENCE. Writing Quadratic Equations Quadratic Regression ALGEBRA 2 SCOPE AND SEQUENCE UNIT 1 2 3 4 5 6 DATES & NO. OF DAYS 8/22-9/16 19 days 9/19-9/30 9 days 10/3-10/14 9 days 10/20-10/25 4 days 10/27-12/2 17 days 12/5-12/16 10 days UNIT NAME Foundations for

More information

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

More information

6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

More information

Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4)

Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) Chapter 2: Functions and Linear Functions 1. Know the definition of a relation. Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) 2. Know the definition of a function. 3. What

More information

I. Terms A. Polynomial B. Terms C. Degree of the Polynomial D. Examples E. Find the degree of the polynomials F. Polynomial or Not?

I. Terms A. Polynomial B. Terms C. Degree of the Polynomial D. Examples E. Find the degree of the polynomials F. Polynomial or Not? Section 6.1 Operations with Polynomials Objectives: Multiply, divide and simplify monomials and expressions involving powers. Add, subtract, and multiply polynomials. I. Terms A. Polynomial B. Terms C.

More information

COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

More information

Determinants can be used to solve a linear system of equations using Cramer s Rule.

Determinants can be used to solve a linear system of equations using Cramer s Rule. 2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

More information

Math 002 Unit 5 - Student Notes

Math 002 Unit 5 - Student Notes Sections 7.1 Radicals and Radical Functions Math 002 Unit 5 - Student Notes Objectives: Find square roots, cube roots, nth roots. Find where a is a real number. Look at the graphs of square root and cube

More information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

Grade 8 Math. Content Skills Learning Targets Assessment Resources & Technology

Grade 8 Math. Content Skills Learning Targets Assessment Resources & Technology St. Michael-Albertville Middle School East Teacher: Dawn Tveitbakk Grade 8 Math September 2014 UEQ: (new) CEQ: WHAT IS THE LANGUAGE OF ALGEBRA? HOW ARE FUNCTIONS USED? HOW CAN ALGEBRA BE USED TO SOLVE

More information

1. Use Properties of Exponents

1. Use Properties of Exponents A. Polynomials Polynomials are one of the most fundamental types of functions used in mathematics. They are very simple to use, primarily because they are formed entirely by multiplication (exponents are

More information

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

More information

CAHSEE on Target UC Davis, School and University Partnerships

CAHSEE on Target UC Davis, School and University Partnerships UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

More information

Chapter 7 - Roots, Radicals, and Complex Numbers

Chapter 7 - Roots, Radicals, and Complex Numbers Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

Algebra Unit 6 Syllabus revised 2/27/13 Exponents and Polynomials

Algebra Unit 6 Syllabus revised 2/27/13 Exponents and Polynomials Algebra Unit 6 Syllabus revised /7/13 1 Objective: Multiply monomials. Simplify expressions involving powers of monomials. Pre-assessment: Exponents, Fractions, and Polynomial Expressions Lesson: Pages

More information

Pre-Calculus II Factoring and Operations on Polynomials

Pre-Calculus II Factoring and Operations on Polynomials Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...

More information

Developmental Math Course Outcomes and Objectives

Developmental Math Course Outcomes and Objectives Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/Pre-Algebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and

More information

Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called.

Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called. Chapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rate

More information

2 is the BASE 5 is the EXPONENT. Power Repeated Standard Multiplication. To evaluate a power means to find the answer in standard form.

2 is the BASE 5 is the EXPONENT. Power Repeated Standard Multiplication. To evaluate a power means to find the answer in standard form. Grade 9 Mathematics Unit : Powers and Exponent Rules Sec.1 What is a Power 5 is the BASE 5 is the EXPONENT The entire 5 is called a POWER. 5 = written as repeated multiplication. 5 = 3 written in standard

More information

Algebra 1 Course Title

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

Text: A Graphical Approach to College Algebra (Hornsby, Lial, Rockswold)

Text: A Graphical Approach to College Algebra (Hornsby, Lial, Rockswold) Students will take Self Tests covering the topics found in Chapter R (Reference: Basic Algebraic Concepts) and Chapter 1 (Linear Functions, Equations, and Inequalities). If any deficiencies are revealed,

More information

Advanced Algebra 2. I. Equations and Inequalities

Advanced Algebra 2. I. Equations and Inequalities Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

More information

Chapter 3. Algebra. 3.1 Rational expressions BAa1: Reduce to lowest terms

Chapter 3. Algebra. 3.1 Rational expressions BAa1: Reduce to lowest terms Contents 3 Algebra 3 3.1 Rational expressions................................ 3 3.1.1 BAa1: Reduce to lowest terms...................... 3 3.1. BAa: Add, subtract, multiply, and divide............... 5

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

MAT 0950 Course Objectives

MAT 0950 Course Objectives MAT 0950 Course Objectives 5/15/20134/27/2009 A student should be able to R1. Do long division. R2. Divide by multiples of 10. R3. Use multiplication to check quotients. 1. Identify whole numbers. 2. Identify

More information

Academic Algebra II Curriculum Crawford Central School District

Academic Algebra II Curriculum Crawford Central School District Concept Competency Vocabulary Strategy Simplify numerical evaluate expressions formulas. Apply the order of operations absolute value. sum, difference, product, quotient, power, base, exponent, grouping

More information

Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

More information

Algebra Tiles Activity 1: Adding Integers

Algebra Tiles Activity 1: Adding Integers Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting

More information

Dividing Polynomials VOCABULARY

Dividing Polynomials VOCABULARY - Dividing Polynomials TEKS FOCUS TEKS ()(C) Determine the quotient of a polynomial of degree three and degree four when divided by a polynomial of degree one and of degree two. TEKS ()(A) Apply mathematics

More information

FACTORING OUT COMMON FACTORS

FACTORING OUT COMMON FACTORS 278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

More information

Sect 3.2 Synthetic Division

Sect 3.2 Synthetic Division 94 Objective 1: Sect 3.2 Synthetic Division Division Algorithm Recall that when dividing two numbers, we can check our answer by the whole number (quotient) times the divisor plus the remainder. This should

More information

Chapter 4 Fractions and Mixed Numbers

Chapter 4 Fractions and Mixed Numbers Chapter 4 Fractions and Mixed Numbers 4.1 Introduction to Fractions and Mixed Numbers Parts of a Fraction Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used.

More information

Radicals - Multiply and Divide Radicals

Radicals - Multiply and Divide Radicals 8. Radicals - Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals

More information

Prep for College Algebra

Prep for College Algebra Prep for College Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet

More information

Algebra 1-2. A. Identify and translate variables and expressions.

Algebra 1-2. A. Identify and translate variables and expressions. St. Mary's College High School Algebra 1-2 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used

More information

Algebra Cheat Sheets

Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95

REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95 REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts which are taught in the specified math course. The sheets

More information

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts.

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts. Chapter 10-1 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax

More information

Greatest Common Factor (GCF) Factoring

Greatest Common Factor (GCF) Factoring Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication

More information

Portable Assisted Study Sequence ALGEBRA IIA

Portable Assisted Study Sequence ALGEBRA IIA SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of

More information

Identify examples of field properties: commutative, associative, identity, inverse, and distributive.

Identify examples of field properties: commutative, associative, identity, inverse, and distributive. Topic: Expressions and Operations ALGEBRA II - STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers

More information

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

More information

Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

More information

Operations on Decimals

Operations on Decimals 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. Then write the decimal

More information

Factoring and Applications

Factoring and Applications Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

More information

1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation.

1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation. 1.1 Basic Concepts Write sets using set notation. Objectives A set is a collection of objects called the elements or members of the set. 1 2 3 4 5 6 7 Write sets using set notation. Use number lines. Know

More information

Rules for Exponents and the Reasons for Them

Rules for Exponents and the Reasons for Them Print this page Chapter 6 Rules for Exponents and the Reasons for Them 6.1 INTEGER POWERS AND THE EXPONENT RULES Repeated addition can be expressed as a product. For example, Similarly, repeated multiplication

More information

5.1 The Remainder and Factor Theorems; Synthetic Division

5.1 The Remainder and Factor Theorems; Synthetic Division 5.1 The Remainder and Factor Theorems; Synthetic Division In this section you will learn to: understand the definition of a zero of a polynomial function use long and synthetic division to divide polynomials

More information

Algebra 1 Course Objectives

Algebra 1 Course Objectives Course Objectives The Duke TIP course corresponds to a high school course and is designed for gifted students in grades seven through nine who want to build their algebra skills before taking algebra in

More information

Review of Key Concepts: 1.2 Characteristics of Polynomial Functions

Review of Key Concepts: 1.2 Characteristics of Polynomial Functions Review of Key Concepts: 1.2 Characteristics of Polynomial Functions Polynomial functions of the same degree have similar characteristics The degree and leading coefficient of the equation of the polynomial

More information

MATH REFRESHER I. Notation, Solving Basic Equations, and Fractions. Rockefeller College MPA Welcome Week 2016

MATH REFRESHER I. Notation, Solving Basic Equations, and Fractions. Rockefeller College MPA Welcome Week 2016 MATH REFRESHER I Notation, Solving Basic Equations, and Fractions Rockefeller College MPA Welcome Week 2016 Lucy C. Sorensen Assistant Professor of Public Administration and Policy 1 2 Agenda Why Are You

More information

Order of Operations - PEMDAS. Rules for Multiplying or Dividing Positive/Negative Numbers

Order of Operations - PEMDAS. Rules for Multiplying or Dividing Positive/Negative Numbers Order of Operations - PEMDAS *When evaluating an expression, follow this order to complete the simplification: Parenthesis ( ) EX. (5-2)+3=6 (5 minus 2 must be done before adding 3 because it is in parenthesis.)

More information

Polynomials Classwork

Polynomials Classwork Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th -degree polynomial function Def.: A polynomial function of degree n in the vaiable

More information

DMA 060 Polynomials and Quadratic Applications

DMA 060 Polynomials and Quadratic Applications DMA 060 Polynomials and Quadratic Applications Brief Description This course provides a conceptual study of problems involving graphic and algebraic representations of quadratics. Topics include basic

More information

Accuplacer Elementary Algebra Study Guide for Screen Readers

Accuplacer Elementary Algebra Study Guide for Screen Readers Accuplacer Elementary Algebra Study Guide for Screen Readers The following sample questions are similar to the format and content of questions on the Accuplacer Elementary Algebra test. Reviewing these

More information

Operations with Algebraic Expressions: Multiplication of Polynomials

Operations with Algebraic Expressions: Multiplication of Polynomials Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the

More information

2.5 Zeros of a Polynomial Functions

2.5 Zeros of a Polynomial Functions .5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and

More information