Quadratics - Build Quadratics From Roots

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Quadratics - Build Quadratics From Roots"

Transcription

1 9.5 Quadratics - Build Quadratics From Roots Objective: Find a quadratic equation that has given roots using reverse factoring and reverse completing the square. Up to this point we have found the solutions to quadratics by a method such as factoring or completing the square. Here we will take our solutions and work backwards to find what quadratic goes with the solutions. We will start with rational solutions. If we have rational solutions we can use factoring in reverse, we will set each solution equal to x and then make the equation equal to zero by adding or subtracting. Once we have done this our expressions will become the factors of the quadratic. Example 1. The solutions are and Set each solution equal to x x = or x = Make each equation equal zero + + Subtractfrom first, add to second x =0 or x + =0 These expressions are the factors (x )(x + )=0 FOIL x +x x 8 Combine like terms x x 8=0 If one or both of the solutions are fractions we will clear the fractions by multiplying by the denominators. Example. The solution are and Set each solution equal tox x = or x = Clear fractions by multiplying by denominators x = or x = Make each equation equal zero Subtract from the first, subtractfrom the second x =0 or x =0 These expressions are the factors (x )(x ) =0 FOIL 1x 9x 8x + 6=0 Combine like terms 1x 17x + 6=0 1

2 If the solutions have radicals (or complex numbers) then we cannot use reverse factoring. In these cases we will use reverse completing the square. When there are radicals the solutions will always come in pairs, one with a plus, one with a minus, that can be combined into one solution using ±. We will then set this solution equal to x and square both sides. This will clear the radical from our problem. Example. The solutions are and x = ± x = x =0 Write as one expression equal tox Subtractfrom both sides We may have to isolate the term with the square root (with plus or minus) by adding or subtracting. With these problems, remember to square a binomial we use the formula (a +b) = a + ab +b Example. The solutions are 5 and +5 x =±5 x =±5 x x+=5 x x +=50 Write as one expression equal tox Isolate the square root term Subtract from both sides Subtract 50 x x 6 = 0 World View Note: Before the quadratic formula, before completing the square, before factoring, quadratics were solved geometrically by the Greeks as early as 00 BC! In 1079 Omar Khayyam, a Persian mathematician solved cubic equations geometrically! If the solution is a fraction we will clear it just as before by multiplying by the denominator. Example 5. The solutions are + and x = ± x = ± Write as one expresion equal tox Clear fraction by multiplying by Isolate the square root term

3 x =± 16x 16x + = 16x 16x + 1=0 Subtractfrom both sides Subtract The process used for complex solutions is identical to the process used for radicals. Example 6. The solutions are 5i and +5i Write as one expression equal tox x =±5i Isolate the i term Subtract from both sides x =±5i x 8x + 16 = 5i i = 1 x 8x + 16 = Add 5 to both sides x 8x + 1 =0 Example 7. The solutions are 5i and +5i x = ± 5i x =±5i x =±5i x 1x + 9=5i i = 1 x 1x +9= 5 Write as one expression equal to x Clear fraction by multiplying by denominator Isolate the i term Subtract from both sides Add 5 to both sides x 1x + =0 Attribution.0 Unported License. (

4 9.5 Practice - Build Quadratics from Roots From each problem, find a quadratic equation with those numbers as its solutions. 1), 5 ) 0, 5), 7) 0, 0 9), 11 11), 1 1) 1, 1 15) 7, 17) 1, ) 6, 1 9 1) ± 5 ) ± 1 5 5) ± 11 7) ± 9) ±i 1 1) ± 6 ) 1 ± i 5) 6 ±i 7) 1 ± 6 9) 6 ± i 8 ), 6 ) 1, 1 6) 0, 9 8), 5 10), 1 1) 5 8, 5 7 1) 1, 16), 9 18) 5, 1 0) 5,0 ) ± 1 ) ± 7 6) ± 8) ± 11i 0) ± 5i ) ± ) ± i 6) 9 ±i 5 8) ± 5i 0) ± i 15 Attribution.0 Unported License. (

5 9.5 Answers - Build Quadratics from Roots NOTE: There are multiple answers for each problem. Try checking your answers because your answer may also be correct. 1) x 7x + 10=0 ) x 9x + 18 =0 ) x x + 0 =0 ) x 1x + 1 = 0 5) x 8x + 16 =0 6) x 9x = 0 7) x = 0 8) x + 7x + 10 = 0 9) x 7x =0 10) x x =0 11) 16x 16x +=0 1) 56x 75x + 5 =0 1) 6x 5x + 1=0 1) 6x 7x + =0 15) 7x 1x + 1 = 0 16) 9x 0x +=0 17) 18x 9x 5=0 18) 6x 7x 5=0 19) 9x + 5x 6=0 0) 5x +x = 0 1) x 5 = 0 ) x 1=0 ) 5x 1=0 ) x 7=0 5) x 11 = 0 6) x 1 = 0 7) 16x =0 8) x + 11 = 0 9) x + 1 = 0 0) x + 50 = 0 1) x x =0 ) x + 6x +7=0 ) x x + 10 = 0 ) x + x + 0 = 0 5) x 1x + 9 = 0 6) x + 18x + 86 = 0 7) x + x 5=0 8) 9x 1x + 9 =0 9) 6x 96x + 8 =0 0) x + 8x + 19 = 0 Attribution.0 Unported License. ( 5

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

Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

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

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

This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.

This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the

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

A Quick Algebra Review

A Quick Algebra Review 1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

More 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

MTH124: Honors Algebra I

MTH124: Honors Algebra I MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,

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

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

Polynomials and Factoring

Polynomials and Factoring 7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of

More information

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

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

Factoring Quadratic Trinomials

Factoring Quadratic Trinomials Factoring Quadratic Trinomials Student Probe Factor Answer: Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials Part 1 of the lesson consists of circle puzzles

More information

Mohawk Valley Community College MVCC MA115 Mr. Bauer

Mohawk Valley Community College MVCC MA115 Mr. Bauer Mohawk Valley Community College MVCC MA115 Course description: This is a dual credit course. Successful completion of the course will give students 1 VVS Credit and 3 MVCC Credit. College credits do have

More information

Factoring ax 2 + bx + c - Teacher Notes

Factoring ax 2 + bx + c - Teacher Notes Southern Nevada Regional Professi onal D evel opment Program VOLUME 1, ISSUE 8 MAY 009 A N ewsletter from the Sec ondary Mathematic s Team Factoring ax + bx + c - Teacher Notes Here we look at sample teacher

More information

Tic Tac Toe Review or Partner Activity

Tic Tac Toe Review or Partner Activity Tic Tac Toe Math Factoring Practice & Solving Quadratics by Factoring Mied Problems HSA.REI.B.4b - Solve HSA.SSE.B.3a - Factor HSA.IF.C.8 - Factor Tic Tac Toe Review or Partner Activity Grade Level: Algebra

More information

A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it

A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply

More information

FOIL FACTORING. Factoring is merely undoing the FOIL method. Let s look at an example: Take the polynomial x²+4x+4.

FOIL FACTORING. Factoring is merely undoing the FOIL method. Let s look at an example: Take the polynomial x²+4x+4. FOIL FACTORING Factoring is merely undoing the FOIL method. Let s look at an example: Take the polynomial x²+4x+4. First we take the 3 rd term (in this case 4) and find the factors of it. 4=1x4 4=2x2 Now

More information

Factoring - Grouping

Factoring - Grouping 6.2 Factoring - Grouping Objective: Factor polynomials with four terms using grouping. The first thing we will always do when factoring is try to factor out a GCF. This GCF is often a monomial like in

More information

IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.

IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa. IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of $2,000. She also earns $500 for

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information

Factoring - Greatest Common Factor

Factoring - Greatest Common Factor 6.1 Factoring - Greatest Common Factor Objective: Find the greatest common factor of a polynomial and factor it out of the expression. The opposite of multiplying polynomials together is factoring polynomials.

More information

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

More information

0.4 FACTORING POLYNOMIALS

0.4 FACTORING POLYNOMIALS 36_.qxd /3/5 :9 AM Page -9 SECTION. Factoring Polynomials -9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use

More information

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

More information

ACCUPLACER. Testing & Study Guide. Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011

ACCUPLACER. Testing & Study Guide. Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011 ACCUPLACER Testing & Study Guide Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011 Thank you to Johnston Community College staff for giving permission to revise

More information

Mathematics Placement

Mathematics Placement Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

More information

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

More information

COLLEGE ALGEBRA. Paul Dawkins

COLLEGE ALGEBRA. Paul Dawkins COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5

More information

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

More information

Florida Math for College Readiness

Florida Math for College Readiness Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

More information

Algebra 2: Q1 & Q2 Review

Algebra 2: Q1 & Q2 Review Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short

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

Algebra I Credit Recovery

Algebra I Credit Recovery Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

More information

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

More information

Prerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.

Prerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303. Course Syllabus Math 1314 College Algebra Revision Date: 8-21-15 Catalog Description: In-depth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems

More information

Warm-Up Oct. 22. Daily Agenda:

Warm-Up Oct. 22. Daily Agenda: Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment

More information

MATH 90 CHAPTER 6 Name:.

MATH 90 CHAPTER 6 Name:. MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a

More information

Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Copyrighted Material. Chapter 1 DEGREE OF A CURVE Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

More information

Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai

Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for

More information

FSCJ PERT. Florida State College at Jacksonville. assessment. and Certification Centers

FSCJ PERT. Florida State College at Jacksonville. assessment. and Certification Centers FSCJ Florida State College at Jacksonville Assessment and Certification Centers PERT Postsecondary Education Readiness Test Study Guide for Mathematics Note: Pages through are a basic review. Pages forward

More information

SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014))

SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014)) SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014)) There are so far 8 most common methods to solve quadratic equations in standard form ax² + bx + c = 0.

More information

9.2 Summation Notation

9.2 Summation Notation 9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

More information

MATH 102 College Algebra

MATH 102 College Algebra FACTORING Factoring polnomials ls is simpl the reverse process of the special product formulas. Thus, the reverse process of special product formulas will be used to factor polnomials. To factor polnomials

More information

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year. This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

More information

ACCUPLACER Arithmetic & Elementary Algebra Study Guide

ACCUPLACER Arithmetic & Elementary Algebra Study Guide ACCUPLACER Arithmetic & Elementary Algebra Study Guide Acknowledgments We would like to thank Aims Community College for allowing us to use their ACCUPLACER Study Guides as well as Aims Community College

More information

Factoring Special Polynomials

Factoring Special Polynomials 6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These

More information

Factoring A Quadratic Polynomial

Factoring A Quadratic Polynomial Factoring A Quadratic Polynomial If we multiply two binomials together, the result is a quadratic polynomial: This multiplication is pretty straightforward, using the distributive property of multiplication

More information

SECTION P.5 Factoring Polynomials

SECTION P.5 Factoring Polynomials BLITMCPB.QXP.0599_48-74 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The

More information

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

More information

Lecture 5 Rational functions and partial fraction expansion

Lecture 5 Rational functions and partial fraction expansion S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion (review of) polynomials rational functions pole-zero plots partial fraction expansion repeated poles nonproper rational functions

More information

Polynomial Degree and Finite Differences

Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

More information

Algebra and Geometry Review (61 topics, no due date)

Algebra and Geometry Review (61 topics, no due date) Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

More information

Sample Problems. Practice Problems

Sample Problems. Practice Problems Lecture Notes Factoring by the AC-method page 1 Sample Problems 1. Completely factor each of the following. a) 4a 2 mn 15abm 2 6abmn + 10a 2 m 2 c) 162a + 162b 2ax 4 2bx 4 e) 3a 2 5a 2 b) a 2 x 3 b 2 x

More information

Section A-3 Polynomials: Factoring APPLICATIONS. A-22 Appendix A A BASIC ALGEBRA REVIEW

Section A-3 Polynomials: Factoring APPLICATIONS. A-22 Appendix A A BASIC ALGEBRA REVIEW A- Appendi A A BASIC ALGEBRA REVIEW C In Problems 53 56, perform the indicated operations and simplify. 53. ( ) 3 ( ) 3( ) 4 54. ( ) 3 ( ) 3( ) 7 55. 3{[ ( )] ( )( 3)} 56. {( 3)( ) [3 ( )]} 57. Show by

More information

Polynomial Operations and Factoring

Polynomial Operations and Factoring Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

More information

The majority of college students hold credit cards. According to the Nellie May

The majority of college students hold credit cards. According to the Nellie May CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials

More information

Factoring. Factoring Polynomial Equations. Special Factoring Patterns. Factoring. Special Factoring Patterns. Special Factoring Patterns

Factoring. Factoring Polynomial Equations. Special Factoring Patterns. Factoring. Special Factoring Patterns. Special Factoring Patterns Factoring Factoring Polynomial Equations Ms. Laster Earlier, you learned to factor several types of quadratic expressions: General trinomial - 2x 2-5x-12 = (2x + 3)(x - 4) Perfect Square Trinomial - x

More information

YOU CAN COUNT ON NUMBER LINES

YOU CAN COUNT ON NUMBER LINES Key Idea 2 Number and Numeration: Students use number sense and numeration to develop an understanding of multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and

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

FACTORING QUADRATIC EQUATIONS

FACTORING QUADRATIC EQUATIONS FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods

More information

Cover Page. MTT 1, MTT 2, MTT 3, MTT 4 Developmental Mathematics I, II, III, IV MTE 1, MTE 2, MTE 3, MTE 4, MTE 5, MTE 6, MTE 7, MTE 8, MTE 9

Cover Page. MTT 1, MTT 2, MTT 3, MTT 4 Developmental Mathematics I, II, III, IV MTE 1, MTE 2, MTE 3, MTE 4, MTE 5, MTE 6, MTE 7, MTE 8, MTE 9 Cover Page MTT 1, MTT 2, MTT 3, MTT 4 Developmental Mathematics I, II, III, IV MTE 1, MTE 2, MTE 3, MTE 4, MTE 5, MTE 6, MTE 7, MTE 8, Faculty Name: Jodi Clingenpeel Program Head: Amy Williamson Dean s

More information

Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:

Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as: Limits Limits: Graphical Solutions Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as: The limits are defined as the value that the function approaches as it goes

More information

Algebra II New Summit School High School Diploma Program

Algebra II New Summit School High School Diploma Program Syllabus Course Description: Algebra II is a two semester course. Students completing this course will earn 1.0 unit upon completion. Required Materials: 1. Student Text Glencoe Algebra 2: Integration,

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

COLLEGE ALGEBRA LEARNING COMMUNITY

COLLEGE ALGEBRA LEARNING COMMUNITY COLLEGE ALGEBRA LEARNING COMMUNITY Tulsa Community College, West Campus Presenter Lori Mayberry, B.S., M.S. Associate Professor of Mathematics and Physics lmayberr@tulsacc.edu NACEP National Conference

More information

COURSE SYLLABUS. Intermediate Algebra DMTH 0200. 2-2 - 3 Lecture - Lab - Credit. Prerequisite

COURSE SYLLABUS. Intermediate Algebra DMTH 0200. 2-2 - 3 Lecture - Lab - Credit. Prerequisite COURSE SYLLABUS Intermediate Algebra 2-2 - 3 Lecture - Lab - Credit Prerequisite Beginning Algebra or a MATH THEA/ACCUPLACER score at the Remediation Standard This syllabus has been reviewed and is current

More information

Factoring Algebra- Chapter 8B Assignment Sheet

Factoring Algebra- Chapter 8B Assignment Sheet Name: Factoring Algebra- Chapter 8B Assignment Sheet Date Section Learning Targets Assignment Tues 2/17 Find the prime factorization of an integer Find the greatest common factor (GCF) for a set of monomials.

More information

Completing the square

Completing the square Completing the square mc-ty-completingsquare-009-1 In this unit we consider how quadratic expressions can be written in an equivalent form using the technique known as completing the square. This technique

More information

Algebra 1 Advanced Mrs. Crocker. Final Exam Review Spring 2014

Algebra 1 Advanced Mrs. Crocker. Final Exam Review Spring 2014 Name: Mod: Algebra 1 Advanced Mrs. Crocker Final Exam Review Spring 2014 The exam will cover Chapters 6 10 You must bring a pencil, calculator, eraser, and exam review flip book to your exam. You may bring

More information

(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its

(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its (1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:

More information

Factor Diamond Practice Problems

Factor Diamond Practice Problems Factor Diamond Practice Problems 1. x 2 + 5x + 6 2. x 2 +7x + 12 3. x 2 + 9x + 8 4. x 2 + 9x +14 5. 2x 2 7x 4 6. 3x 2 x 4 7. 5x 2 + x -18 8. 2y 2 x 1 9. 6-13x + 6x 2 10. 15 + x -2x 2 Factor Diamond Practice

More information

A Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles

A Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...

More information

6.5 Factoring Special Forms

6.5 Factoring Special Forms 440 CHAPTER 6. FACTORING 6.5 Factoring Special Forms In this section we revisit two special product forms that we learned in Chapter 5, the first of which was squaring a binomial. Squaring a binomial.

More information

MEP Pupil Text 12. A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued.

MEP Pupil Text 12. A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued. MEP Pupil Text Number Patterns. Simple Number Patterns A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued. Worked Example Write down the

More information

MATH 90 CHAPTER 1 Name:.

MATH 90 CHAPTER 1 Name:. MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.

More information

SPECIAL PRODUCTS AND FACTORS

SPECIAL PRODUCTS AND FACTORS CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

More information

MATH 0310 Course Document - Revised Spring 2014

MATH 0310 Course Document - Revised Spring 2014 Page 1 of 6 1. Course Title: Intermediate Algebra MATH 0310 Course Document - Revised Spring 2014 2. Prerequisite: Introductory Algebra (Math 0308) or placement by testing 3. Credit: Three (3) Semester

More information

2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2

2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2 00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the

More information

Algebra 1 Course Information

Algebra 1 Course Information Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

More information

5. Factoring by the QF method

5. Factoring by the QF method 5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the

More information

Transition To College Mathematics

Transition To College Mathematics Transition To College Mathematics In Support of Kentucky s College and Career Readiness Program Northern Kentucky University Kentucky Online Testing (KYOTE) Group Steve Newman Mike Waters Janis Broering

More information

parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL

parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does

More information

Approximating functions by Taylor Polynomials.

Approximating functions by Taylor Polynomials. Chapter 4 Approximating functions by Taylor Polynomials. 4.1 Linear Approximations We have already seen how to approximate a function using its tangent line. This was the key idea in Euler s method. If

More information

M0312 Assignments on Grade Tally Sheet Spring 2011 Section # Name

M0312 Assignments on Grade Tally Sheet Spring 2011 Section # Name M0312 Assignments on Grade Tally Sheet Spring 2011 Section # Name Date Packet Work Points MyMathLab Sections 1/18/11 Line 1: Pop Test 1.1, page 1-7 1.1 Pop Test 1.2, page 1-8 1.2 Scientific Notation Worksheet,

More information

A. Factoring out the Greatest Common Factor.

A. Factoring out the Greatest Common Factor. DETAILED SOLUTIONS AND CONCEPTS - FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

Factoring. Factoring Monomials Monomials can often be factored in more than one way.

Factoring. Factoring Monomials Monomials can often be factored in more than one way. Factoring Factoring is the reverse of multiplying. When we multiplied monomials or polynomials together, we got a new monomial or a string of monomials that were added (or subtracted) together. For example,

More information

9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.

9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11. 9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

More information

Course Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)

Course Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,

More information

Teaching & Learning Plans. Complex Number Operations. Leaving Certificate Syllabus

Teaching & Learning Plans. Complex Number Operations. Leaving Certificate Syllabus Teaching & Learning Plans Complex Number Operations Leaving Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve.

More information

The generation and supply of electricity within the U.K is achieved through the use of a 3-phase system.

The generation and supply of electricity within the U.K is achieved through the use of a 3-phase system. Three Phase Electricity Supplies and Systems The generation and supply of electricity within the U.K is achieved through the use of a 3-phase system. This consists of 3 separate phase conductors along

More information

Cost-Volume-Profit Analysis

Cost-Volume-Profit Analysis Chapter 6 Notes Page 1 Cost-Volume-Profit Analysis Understanding the relationship between a firm s costs, profits and its volume levels is very important for strategic planning. When you are considering

More information

College Readiness Math MOOC

College Readiness Math MOOC College Readiness Math MOOC Instructor Information: Dr. Jennifer Kosiak, jkosiak@uwlax.edu General email: mathmooc@uwlax.edu Mathematics Department, University of Wisconsin- La Crosse Description: The

More information

Johnson County Community College

Johnson County Community College Johnson County Community College Course Syllabus Mathematics Division Math 115: Elementary Algebra Section 450 (Self-Paced) Fall 2015 (CRN 80682) Instructor Information Primary Instructor: Mary Deas Campus

More information

REVIEW EXERCISES DAVID J LOWRY

REVIEW EXERCISES DAVID J LOWRY REVIEW EXERCISES DAVID J LOWRY Contents 1. Introduction 1 2. Elementary Functions 1 2.1. Factoring and Solving Quadratics 1 2.2. Polynomial Inequalities 3 2.3. Rational Functions 4 2.4. Exponentials and

More information

Find all of the real numbers x that satisfy the algebraic equation:

Find all of the real numbers x that satisfy the algebraic equation: Appendix C: Factoring Algebraic Expressions Factoring algebraic equations is the reverse of expanding algebraic expressions discussed in Appendix B. Factoring algebraic equations can be a great help when

More information