2.4. Factoring Quadratic Expressions. Goal. Explore 2.4. Launch 2.4


 Barnaby Cobb
 4 years ago
 Views:
Transcription
1 2.4 Factoring Quadratic Epressions Goal Use the area model and Distributive Property to rewrite an epression that is in epanded form into an equivalent epression in factored form The area of a rectangle is used to write the factored form for the quadratic epression. Students use the epanded form of quadratic epressions to subdivide a rectangle into parts and to label each part with an appropriate epression for its area. These epressions provide clues on how to split the middle term, b, of the epression 2 + b+c to find the dimensions of the rectangle or the two linear factors for the factored form using the Distributive Property. We do not epect students to be eperts in factoring. More factoring will be done in Say It With Symbols. We do want students to develop understanding of the Distributive Property and its connection to the area of a rectangle. Launch 2.4 You can use the eample in the student book to talk about combining like terms if you did not do it in the summary of the last problem. In the last problem, Carminda claimed she did not need the rectangular model to multiply (+3) by (+2). You may want to review these steps with the students. Suggested Question Could you reverse Carminda s method to write the epression in factored form? (This is more difficult, but some students may notice that you have to split the 8 to get the first step in reversing Carminda s method. Some may notice that you have to split the 8 to get the area of two of the smaller rectangles in a rectangle whose area is represented by Also, the product of the coefficients of must be 2. Therefore, 8=2+6 and = ) Let the class work in pairs. Eplore 2.4 Look for interesting strategies for Questions A, B, and C. The purpose of these questions is to help students see that given the first and last terms of the epanded form 2 + b+c, such as 2 + b+9, there are many possible factorings of 2 + b+c which all depend on which factors of c you choose. The factors of c one chooses then determine the value of b. Starting with the factors of c might be a good strategy for splitting b into two parts. Students can put their work on large poster paper to use during the summary. If students find one pair of factors that work, ask them if other pairs might work. For Question A, wholenumber factors that will work are: and 9 and 3 and 3. Students can also use rational number factor pairs such as and 8. There are many factor pairs. Here it suffices to focus on wholenumber pairs. Some students may need more guidance on how to factor the epressions. The area model can be used to help students recall the steps needed to multiply two binomials like (+)(+4) or students can use the Distributive Property. This is true with factoring. Students can use the area model to factor; this method is illustrated with the epression However, the distributive property is a more useful tool when factoring epression like a 2 + b+c especially when negative coefficients for a, b, and/or c are involved. To factor an epression like , you can use the metaphor of subparts of the rectangle to get the four epressions needed: This then becomes the first step in factoring a quadratic epression in epanded form. We are reversing the process of multiplying two binomials. Do an eample, , to help students see how to use the area model to do the factoring. Sketch a rectangle whose area is represented by the epression I N V E S T I G AT I O N 2 Investigation 2 Quadratic Epressions 69
2 Subdivide it into four parts. Suggested Questions Which part has an area represented by 2? By 5? How can you split 8 into two pieces so that the pieces represent the remaining two subparts? What clues can you use to split 8? (The sum of the coefficients of each part must equal 8 and their product must equal 5.) So what two numbers have a sum of 8 and a product of 5? (3 and 5. So the areas of the two sub parts are 3 and 5. Students may try several different pairs. ) What are the dimensions of the rectangle? (+3 and +5) How can you use the Distributive Property to show that (+3)(+5) is equal to ? Have some groups put their work on large sheets of paper. Make sure you get all of the different strategies that might occur in A C. In Question D, ask: Is Alyse correct? How do you know? What did she do in step? Step 2? Step 3? Summarize students work in Question D before you have students do Question E. 5 Summarize 2.4 Call on different students to discuss their strategies for factoring epressions. Refer to the large posters of the student work. Many will have drawn diagrams. Have the class ask questions about the work. Be sure to relate the sub areas of the rectangle to the parts of the epanded epression and then relate the factored form to the dimensions and area of the whole rectangle. Collect several different rectangles for Question A. Ask the students to show: A sketch of their rectangle with the area of its subparts appropriately labeled. The dimensions of the rectangle. The two equivalent epressions for the area of the rectangle. You can tell the students that writing an epanded form in factored form is called factoring. In Questions D and E, ask students to eplain their method and why it is correct. For eample, = = (+3)+4(+3)=(+3)(+4). Check for Understanding Describe the steps you used to factor the epanded form of a quadratic. Use the following epressions as eamples: (Students should start by looking for factors of the constant term. Then they choose the pair whose sum is equal to the coefficient of the middle term or to b in b. Some may start with finding various sums for b and then choosing the two whose product is equal to the constant term, c. Note: If the coefficient of 2 is not, then finding the coefficient of is more complicated.) Suggested Question How can you check that your answers are equivalent to the original epressions? (Multiply the two factors to see if you get the original epanded form or use a rectangular model to check your work or check the graphs of each equation y=factored form and y=epanded form for each epression. The graphs should be identical.) 70 Frogs, Fleas, and Painted Cubes
3 2.4 Factoring Quadratic Epressions At a Glance PACING 2 days Mathematical Goal Use the area model and Distributive Property to rewrite an epression that is in epanded form into an equivalent epression in factored form Launch In the last problem, Carminda claimed she did not need the rectangular model to multiply (+3) by (+2). You may want to review these steps with the students. Could you reverse Carminda s method to write the epression in factored form? Let the class work in pairs. Eplore Look for interesting strategies for Questions A, B, and C. Students could put their work on large poster paper to use during the summary. If students find one pair of factors that works, ask them if other factor pairs might work. Some students may need more guidance on how to factor the epressions. We are reversing the process of multiplying two binomials. Do an eample, , to help students see how to use the area model to do the factoring. Sketch a rectangle whose area is represented by the epression Subdivide it into four parts. Which part has an area represented by 2? By 5? How can you split 8 into two pieces so that the pieces represent the remaining two subparts? What clues can you use to split 8? What two numbers have a sum of 8 and a product of 5? What are the dimensions of the rectangle? How can you use the Distributive Property to show that (+3)(+5) is equal to ? Have some groups put their work on large sheets of paper. Make sure you get all of the different strategies that might occur in Questions A, B, and C. In Question D, ask: Is Alyse correct? How do you know? What did she do in step? Step 2? Step 3? Summarize Question D before you have students do Question E. Materials Poster Paper (optional) Investigation 2 Quadratic Epressions 7
4 Summarize Call on different students to discuss their strategies for factoring epressions. Refer to the large posters of the student work. Relate the sub areas of the rectangle to the parts of the epanded epression and then relate the factored form to the dimensions and area of the whole rectangle. In Questions D and E, ask students to eplain their method and why it is correct. For eample, = = (+3)+4(+3)=(+3)(+4). Describe the steps you used to factor the epanded form of a quadratic. How can you check that your answers are equivalent to the original epressions? Materials Student notebooks ACE Assignment Guide for Problem 2.4 Core Other Connections 56, Etensions 62 64; unassigned choices from previous problems Adapted For suggestions about adapting ACE eercises, see the CMP Special Needs Handbook. Connecting to Prior Units 56: Moving Straight Ahead Answers to Problem 2.4 A.. Possible drawing: Since the lengths of the two segments marked with question marks seem to be the same length, students may argue that 3 and 3 are the only factors of 9 that will work, while others will claim that and 9 are possibilities as well. Either case is acceptable. 2. Possible answers: Factored form: (+3)(+3); Epanded: or ; Factored form: (+)(+9); Epanded: or Note: There are infinitely more factor pairs that will work for 9 such as and 8 2 or and 36. For this problem we can focus 4 on the wholenumber factors. B.. If b=9, then in factored form is (+)(+8). If b=6, then in factored form is (+2)(+4). If b=9, then in factored form is ()(8) Frogs, Fleas, and Painted Cubes
5 If b=6, then in factored form is (2)(4). Note: Not every b value that students pick will result in a quadratic epression that is factorable at this time by students. For eample a b value of 0 gives an epression of This epression cannot be factored by students unless they know the quadratic formula which they may not learn until high school. In this problem, we want students to realize that they are looking at the factor pairs of 8 and finding a pair whose sum is the value for b that will make 2 + b+8 an epression that can be factored. Restricting the factor pairs of 8 to integers, we have ( and 8), ( and 8), (2 and 4), and (2 and 4). So the values for b are 9,9, 6 and 6, respectively, as illustrated above. For the case where b=6, a possible student eplanation may include the use of an area model such as the one shown below: The factor pair consisting of 2 and 4 yields a b value of 6 and the factored form of (+2)(+4) for Students will probably find that not all of their classmates found the same epressions. However the epressions that students will have will be either (+2)(+4), (+)(+8), ()(8) or (2)(4). C.. r= r=3, s=8 (or r=8, s=3) 3. r= and s=24 or r=24 and s= 4. A common strategy is to compare the c term of the 2 + b+c form to the constant terms in the factors; r and s must be factors of c. Meanwhile, the sum of r and s must make b. D. Yes; Students can make an area model to check that =(+2)(+8). They may also reason that Alyse applied the Distributive Property. In step, she separated 0 into 2+8 and in step 2 she used the Distributive Property to rewrite as (+2) and 8+6 as 8(+2). In step 3, she used the Distributive Property to see that (+2) is multiplied by both and 8 so she can factor the (+2) term out and multiply it by the remaining (+8). E.. Using the Distributive Property: = (+5)+2(+5)= (+5)(+2). Note: Students may also write this as (+2)(+5) which is equivalent because of the Commutative Property of Multiplication. In the answers for Question E, parts () and (4) the common factor has been pulled out in front since it seems the most natural. For part (), the factor of +5 was pulled out in front to obtain (+5)(+2). 2. Using the Distributive Property: = = (+0)+(+0)=(+0)(+). 3. Using the Distributive Property: = = (+5)2(+5)=(+5)(2). 4. Using the Distributive Property: = = (+5)+(+5)=(+5)(+). 5. Using the Distributive Property: 28+5= = (3)5(3)=(5)(3). 6. Using the Distributive Property: = = (6)+6(6)=(6)(6). Note: Students may need help seeing that 5 is a factor of 5 and 5 in part (5) or that 2 is a factor of 2 and 0 in part (3). Factoring out a negative quantity is represented somewhat differently in the part (6) solution. I N V E S T I G AT I O N 2 Investigation 2 Quadratic Epressions 73
6 74 Frogs, Fleas, and Painted Cubes
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 informationPredicting the Ones Digit
. Predicting the Ones Digit Goals Eamine patterns in the eponential and standard forms of powers of whole numbers Use patterns in powers to estimate the ones digits for unknown powers In this problem,
More informationMathematics More Visual Using Algebra Tiles
www.cpm.org Chris Mikles CPM Educational Program A California Nonprofit Corporation 33 Noonan Drive Sacramento, CA 958 (888) 80876 fa: (08) 7778605 email: mikles@cpm.org An Eemplary Mathematics Program
More informationCPM Educational Program
CPM Educational Program A California, NonProfit Corporation Chris Mikles, National Director (888) 8084276 email: mikles @cpm.org CPM Courses and Their Core Threads Each course is built around a few
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationUnit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials
Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial
More informationThe 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 information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationA.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 informationFACTORING ax 2 bx c WITH a 1
296 (6 20) Chapter 6 Factoring 6.4 FACTORING a 2 b c WITH a 1 In this section The ac Method Trial and Error Factoring Completely In Section 6.3 we factored trinomials with a leading coefficient of 1. In
More informationPolynomials 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 informationLINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,
LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are
More information6706_PM10SB_C4_CO_pp192193.qxd 5/8/09 9:53 AM Page 192 192 NEL
92 NEL Chapter 4 Factoring Algebraic Epressions GOALS You will be able to Determine the greatest common factor in an algebraic epression and use it to write the epression as a product Recognize different
More informationUsing the Area Model to Teach Multiplying, Factoring and Division of Polynomials
visit us at www.cpm.org Using the Area Model to Teach Multiplying, Factoring and Division of Polynomials For more information about the materials presented, contact Chris Mikles mikles@cpm.org From CCA
More informationEquations Involving Fractions
. Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation
More informationIntegrating algebraic fractions
Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate
More information1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient
Section 3.1 Systems of Linear Equations in Two Variables 163 SECTION 3.1 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES Objectives 1 Determine whether an ordered pair is a solution of a system of linear
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationMath 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 informationPolynomial 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 informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationSystems of Equations Involving Circles and Lines
Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system
More informationSect. 1.3: Factoring
Sect. 1.3: Factoring MAT 109, Fall 2015 Tuesday, 1 September 2015 Algebraic epression review Epanding algebraic epressions Distributive property a(b + c) = a b + a c (b + c) a = b a + c a Special epansion
More informationA 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 informationUsing Algebra Tiles from Polynomials to Factoring
Using Algebra Tiles from Polynomials to Factoring For more information about the materials you find in this packet, contact: Chris Mikles (888) 8084276 mikles@cpm.org CPM Educational Program 203, all
More informationChris Yuen. Algebra 1 Factoring. Early High School 810 Time Span: 5 instructional days
1 Chris Yuen Algebra 1 Factoring Early High School 810 Time Span: 5 instructional days Materials: Algebra Tiles and TI83 Plus Calculator. AMSCO Math A Chapter 18 Factoring. All mathematics material and
More informationApplications of the Pythagorean Theorem
9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem
More informationMath 25 Activity 6: Factoring Advanced
Instructor! Math 25 Activity 6: Factoring Advanced Last week we looked at greatest common factors and the basics of factoring out the GCF. In this second activity, we will discuss factoring more difficult
More informationNegative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2
4 (4) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a year
More informationMEP Y8 Practice Book A. In this section we consider how to expand (multiply out) brackets to give two or more terms, as shown below: ( ) = +
8 Algebra: Brackets 8.1 Epansion of Single Brackets In this section we consider how to epand (multiply out) brackets to give two or more terms, as shown below: = + 3 + 6 3 18 First we revise negative numbers
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 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 informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationNSM100 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 informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More information3. Solve the equation containing only one variable for that variable.
Question : How do you solve a system of linear equations? There are two basic strategies for solving a system of two linear equations and two variables. In each strategy, one of the variables is eliminated
More informationFactoring 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 informationMultiplying Binomials and Factoring Trinomials Using Algebra Tiles and Generic Rectangles
Multiplying Binomials Standard: Algebra 10.0 Time: 55 mins. Multiplying Binomials and Factoring Trinomials Using Algebra Tiles and s Materials: Class set of Algebra Tiles or access to a computer for each
More informationSection A3 Polynomials: Factoring APPLICATIONS. A22 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 informationPreCalculus 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 informationFactoring 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 informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationFactoring Quadratic Trinomials
Factoring Quadratic Trinomials Student Probe Factor x x 3 10. Answer: x 5 x Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials. Part 1 of the lesson consists
More informationIOWA EndofCourse Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.
IOWA EndofCourse 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 informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationSolving Equations by the Multiplication Property
2.2 Solving Equations by the Multiplication Property 2.2 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the multiplication property to solve equations. Find the mean
More informationRoots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian)
Roots, Linear Factors, and Sign Charts review of background material for Math 16A (Barsamian) Contents 1. Introduction 1. Roots 1. Linear Factors 4. Sign Charts 5 5. Eercises 8 1. Introduction The sign
More informationCHAPTER 7: FACTORING POLYNOMIALS
CHAPTER 7: FACTORING POLYNOMIALS FACTOR (noun) An of two or more quantities which form a product when multiplied together. 1 can be rewritten as 3*, where 3 and are FACTORS of 1. FACTOR (verb)  To factor
More informationFactoring 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 informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationFACTORING QUADRATICS 8.1.1 through 8.1.4
Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationIntroduction to Quadratic Functions
Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
More informationGrade 6 Mathematics Common Core State Standards
Grade 6 Mathematics Common Core State Standards Standards for Mathematical Practice HOW make sense of problems, persevere in solving them, and check the reasonableness of answers. reason with and flexibly
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationHIBBING 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 informationFactorising quadratics
Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationUnit 7 Quadratic Relations of the Form y = ax 2 + bx + c
Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationA 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 informationAPPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS
APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /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 informationWentzville School District Algebra 1: Unit 8 Stage 1 Desired Results
Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Unit Title: Quadratic Expressions & Equations Course: Algebra I Unit 8  Quadratic Expressions & Equations Brief Summary of Unit: At
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationWe start with the basic operations on polynomials, that is adding, subtracting, and multiplying.
R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract
More informationSystems of Linear Equations: Solving by Substitution
8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing
More informationHFCC Math Lab Beginning Algebra 13 TRANSLATING ENGLISH INTO ALGEBRA: WORDS, PHRASE, SENTENCES
HFCC Math Lab Beginning Algebra 1 TRANSLATING ENGLISH INTO ALGEBRA: WORDS, PHRASE, SENTENCES Before being able to solve word problems in algebra, you must be able to change words, phrases, and sentences
More informationFactors and Products
CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square
More informationHigher 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 informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity 8G18G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More informationDeveloping Conceptual Understanding of Number. Set J: Perimeter and Area
Developing Conceptual Understanding of Number Set J: Perimeter and Area Carole Bilyk cbilyk@gov.mb.ca Wayne Watt wwatt@mts.net Perimeter and Area Vocabulary perimeter area centimetres right angle Notes
More informationSECTION A3 Polynomials: Factoring
A3 Polynomials: Factoring A23 thick, write an algebraic epression in terms of that represents the volume of the plastic used to construct the container. Simplify the epression. [Recall: The volume 4
More informationIn this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).
CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,
More informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
More informationConvert between units of area and determine the scale factor of two similar figures.
CHAPTER 5 Units of Area c GOAL Convert between units of area and determine the scale factor of two. You will need a ruler centimetre grid paper a protractor a calculator Learn about the Math The area of
More informationSECTION 1.6 Other Types of Equations
BLITMC1B.111599_11174 12//2 1:58 AM Page 11 Section 1.6 Other Types of Equations 11 12. A person throws a rock upward from the edge of an 8foot cliff. The height, h, in feet, of the rock above the water
More informationSECTION 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 information7 Literal Equations and
CHAPTER 7 Literal Equations and Inequalities Chapter Outline 7.1 LITERAL EQUATIONS 7.2 INEQUALITIES 7.3 INEQUALITIES USING MULTIPLICATION AND DIVISION 7.4 MULTISTEP INEQUALITIES 113 7.1. Literal Equations
More informationSolve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers
More informationx x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =
Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More informationMathematical goals. Starting points. Materials required. Time needed
Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between
More informationAnswers 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 informationUnit. Area Model Factoring. Researchbased National Science Foundationfunded. Learning transforms lives.
Unit 10 Area Model Factoring Researchbased National Science Foundationfunded Learning transforms lives. Dear Student, When we multiply two factors, we get their product. If we start with the product,
More informationCopyrighted 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