University of Phoenix Material
|
|
- Emil Boyd
- 6 years ago
- Views:
Transcription
1 1 University of Phoenix Material Factoring and Radical Expressions The goal of this week is to introduce the algebraic concept of factoring polynomials and simplifying radical expressions. Think of factoring as breaking a very complicated problem into its simple or basic parts. What Is Factoring? The basic idea of factoring is to rewrite an additionsubtraction problem as a multiplication problem. Factors are the numbers that we multiply. For example, 2 and 5 are the factors of 10 because 2 5 = 10. The addition of 6 and 4 can be written into the multiplication of 2 and 5: In other words, we factored the addition into the multiplication 2 5. The addition of 6 and 4 could have also been written into the multiplication of 1 x 10; however, the goal is always to find prime factors (go to CME > Building Math Confidence > Phoenix Math Topic 62). The factors 1 and 10 are not prime factors of 10 (the answer of adding 6 and 4). Factoring by GCF The factoring process involves applying the distributive property that states. where a, b, and c are terms in an expression. Here is how was factored into 2 5 using the distributive property: From the demonstration above, the key is to find the greatest common factor (GCF) of each term to apply the distributive property. This same idea can be used to factor polynomials. See the following examples: GCF = 4 GCF = 5 GCF = 3 GCF = GCF = To learn more about how to find the GCF of two or more terms, see the video "Monomial Greatest Common Factor" from Khan Academy (go to CME > Building Math Confidence > Math Videos > Khan Academy).
2 2 Factoring by Grouping When a polynomial has four terms (or more) and all the terms have no common factors, we can try using a factoring technique called factoring by grouping. This technique consists of grouping the terms of the polynomial and then factoring each group by its GCF. Here is an example: Group (adding parenthesis) the first two terms and last two terms. Factor by GCF each group. The GCF of the first group is and for the second group is 5. Now the GCF is ( ), so the final factoring is Factoring Trinomials by Grouping In Cognitive Tutor, you will learn how to factor trinomials of the forms and using the Guess and Check Method or Undoing FOIL Method. However, these methods can sometimes be time consuming. An alternate method is to factor trinomials by grouping. The basic idea is to convert a trinomial into a polynomial of four terms, so factoring by grouping can be applied. Here is an example: First, multiply the coefficient of the first term by the last term:. Next, find factors of that, when these are added, the result is. In our case, and. Now, rewrite the middle term of the trinomial using the factors of and. We converted the trinomial into a polynomial with four terms, so we can factor by grouping.
3 3 Factoring Trinomials by ac-method The ac-method is a standard technique for factoring trinomials of the form ax² + bx + c. This method can be used for all non-prime trinomials of that form that can be written as the product of two binomials. Factor 27x² + 6x 5 using the ac-method. First, we label a, b, and c. a = 27 b = 6 c = -5 We ask what two numbers have a product of ac and a sum of b. ac = (27)(-5) = -135 The factors of -1 are as follows (-3)(45) (3)(-45) (-5)(27) (5)(-27) (-15)(9) (15)(-9) Now, let s see which of these has a sum of the value of b. (-3)+(45)= 42 (3)+(-45)= -42 (-5)+(27)= 22 (5)+(-27)= -22 (-15)+(9)= -6 (15)+(-9)= 6 This is the correct one! So, we rewrite our polynomial as follows 27x² 9x + 15x 5 = 9x(3x1) + 5(3x1) = (9x+5)(3x1)
4 4 Factoring by Special Methods There are strategies that can help you streamline the factoring process of special trinomials and binomials. The key is knowing how to identify these special polynomials. Trinomials Perfect Squares These are trinomials that have the form or and these can be factored as a binomial square with the form or, respectively. In other words, the first and last terms of the trinomial are perfect squares, and the middle term is twice the product of the square roots of the first and last terms. Example: To learn more about factoring trinomial perfect squares, see the video Factoring Perfect Square Trinomials from Khan Academy (go to CME > Building Math Confidence > Math Videos > Khan Academy). Difference of Squares A difference of squares is a binomial in the form. In other words, it is the subtraction of two square terms. This special binomial can be factored as. Example: To learn more about factoring trinomial perfect squares, see the video Factoring Difference of Squares from Khan Academy (go to CME > Building Math Confidence > Math Videos > Khan Academy). Sum and Difference of Cubes Here is the form of sum and difference of cubes and its factoring: Example: To learn more about factoring trinomial perfect squares, see the video Factoring Sum and Difference of Cubes from Khan Academy (go to CME > Building Math Confidence > Math Videos > Khan Academy). Radical Expressions
5 5 In Cognitive Tutor, you will learn that the root of a number is written as and it means What number raised to power results in? or put another way,. Examples: A. means Example A is read as the square root of and the answer is because 2 = 36. B. means Example B is read as the cube root of 7 and the answer because 3 = 27. C. means Finally, Example C is read as the fifth root of and the answer is because 5 = 32. These are examples of perfect roots because the roots are integers (see Phoenix Math: Topic 21). There will be many instances when the root of a number is not an integer. For example, has no integer root because there is no integer that you can square to get a result of 12. However, these types of roots can be simplified in radical form by using the product rule of radicals that is stated below. The basic idea consists on rewriting the radicand (expression inside of the radical) as a product of two integers, but one of the integers is a perfect root. Here is an example: Understanding how to find nonperfect roots in radical form is necessary to simplify the process of performing basic operations with radical expressions, which you will learn in Cognitive Tutor. Fractional Exponents The root of a number can also be written using fractional exponents instead of using the radical symbols. Here is how: and Examples: Writing radical expressions with fractional exponents provides an alternative method for performing operations with radicals using the exponent rules you learned in the Algebra 1a course, particularly
6 6 when operations with radicals seems impossible. For example, multiplying two radical expressions with different indexes:
Chapter 8: Factoring Polynomials. Algebra 1 Mr. Barr
p. 1 Chapter 8: Factoring Polynomials Algebra 1 Mr. Barr Name: p. 2 Date Schedule Lesson/Activity 8.1 Monomials & Factoring 8.2 Using the Distributive Property 8.3 Quadratics in the form x 2 +bx+c Quiz
More informationWe begin, however, with the concept of prime factorization. Example: Determine the prime factorization of 12.
Chapter 3: Factors and Products 3.1 Factors and Multiples of Whole Numbers In this chapter we will look at the topic of factors and products. In previous years, we examined these with only numbers, whereas
More informationIn the previous section, we added and subtracted polynomials by combining like terms. In this section, we extend that idea to radicals.
4.2: Operations on Radicals and Rational Exponents In this section, we will move from operations on polynomials to operations on radical expressions, including adding, subtracting, multiplying and dividing
More informationSection 7.1 Common Factors in Polynomials
Chapter 7 Factoring How Does GPS Work? 7.1 Common Factors in Polynomials 7.2 Difference of Two Squares 7.3 Perfect Trinomial Squares 7.4 Factoring Trinomials: (x 2 + bx + c) 7.5 Factoring Trinomials: (ax
More informationFactoring completely is factoring a product down to a product of prime factors. 24 (2)(12) (2)(2)(6) (2)(2)(2)(3)
Factoring Contents Introduction... 2 Factoring Polynomials... 4 Greatest Common Factor... 4 Factoring by Grouping... 5 Factoring a Trinomial with a Table... 5 Factoring a Trinomial with a Leading Coefficient
More informationStep one is identifying the GCF, and step two is dividing it out.
Throughout this course we will be looking at how to undo different operations in algebra. When covering exponents we showed how ( 3) 3 = 27, then when covering radicals we saw how to get back to the original
More informationFactor Trinomials When the Coefficient of the Second-Degree Term is 1 (Objective #1)
Factoring Trinomials (5.2) Factor Trinomials When the Coefficient of the Second-Degree Term is 1 EXAMPLE #1: Factor the trinomials. = = Factor Trinomials When the Coefficient of the Second-Degree Term
More informationPolynomial and Rational Expressions. College Algebra
Polynomial and Rational Expressions College Algebra Polynomials A polynomial is an expression that can be written in the form a " x " + + a & x & + a ' x + a ( Each real number a i is called a coefficient.
More informationFactoring Trinomials of the Form
Section 7 3: Factoring Trinomials of the Form 1x 2 + Bx + C The FOIL process changes a product of 2 binomials into a polynomial. The reverse process starts with a polynomial and finds the 2 binomials whose
More information-5y 4 10y 3 7y 2 y 5: where y = -3-5(-3) 4 10(-3) 3 7(-3) 2 (-3) 5: Simplify -5(81) 10(-27) 7(9) (-3) 5: Evaluate = -200
Polynomials: Objective Evaluate, add, subtract, multiply, and divide polynomials Definition: A Term is numbers or a product of numbers and/or variables. For example, 5x, 2y 2, -8, ab 4 c 2, etc. are all
More informationAccuplacer Review Workshop. Intermediate Algebra. Week Four. Includes internet links to instructional videos for additional resources:
Accuplacer Review Workshop Intermediate Algebra Week Four Includes internet links to instructional videos for additional resources: http://www.mathispower4u.com (Arithmetic Video Library) http://www.purplemath.com
More information2 TERMS 3 TERMS 4 TERMS (Must be in one of the following forms (Diamond, Slide & Divide, (Grouping)
3.3 Notes Factoring Factoring Always look for a Greatest Common Factor FIRST!!! 2 TERMS 3 TERMS 4 TERMS (Must be in one of the following forms (Diamond, Slide & Divide, (Grouping) to factor with two terms)
More informationChapter 5 Polynomials
Department of Mathematics Grossmont College October 7, 2012 Multiplying Polynomials Multiplying Binomials using the Distributive Property We can multiply two binomials using the Distributive Property,
More informationDevelopmental Math An Open Program Unit 12 Factoring First Edition
Developmental Math An Open Program Unit 12 Factoring First Edition Lesson 1 Introduction to Factoring TOPICS 12.1.1 Greatest Common Factor 1 Find the greatest common factor (GCF) of monomials. 2 Factor
More information7.1 Review for Mastery
7.1 Review for Mastery Factors and Greatest Common Factors A prime number has exactly two factors, itself and 1. The number 1 is not a prime number. To write the prime factorization of a number, factor
More informationACCUPLACER Elementary Algebra Assessment Preparation Guide
ACCUPLACER Elementary Algebra Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre
More informationTool 1. Greatest Common Factor (GCF)
Chapter 7: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
More informationIs the following a perfect cube? (use prime factorization to show if it is or isn't) 3456
Is the following a perfect cube? (use prime factorization to show if it is or isn't) 3456 Oct 2 1:50 PM 1 Have you used algebra tiles before? X 2 X 2 X X X Oct 3 10:47 AM 2 Factor x 2 + 3x + 2 X 2 X X
More informationMath 101, Basic Algebra Author: Debra Griffin
Math 101, Basic Algebra Author: Debra Griffin Name Chapter 5 Factoring 5.1 Greatest Common Factor 2 GCF, factoring GCF, factoring common binomial factor 5.2 Factor by Grouping 5 5.3 Factoring Trinomials
More informationTERMINOLOGY 4.1. READING ASSIGNMENT 4.2 Sections 5.4, 6.1 through 6.5. Binomial. Factor (verb) GCF. Monomial. Polynomial.
Section 4. Factoring Polynomials TERMINOLOGY 4.1 Prerequisite Terms: Binomial Factor (verb) GCF Monomial Polynomial Trinomial READING ASSIGNMENT 4. Sections 5.4, 6.1 through 6.5 160 READING AND SELF-DISCOVERY
More informationHow can we factor polynomials?
How can we factor polynomials? Factoring refers to writing something as a product. Factoring completely means that all of the factors are relatively prime (they have a GCF of 1). Methods of factoring:
More informationP.1 Algebraic Expressions, Mathematical models, and Real numbers. Exponential notation: Definitions of Sets: A B. Sets and subsets of real numbers:
P.1 Algebraic Expressions, Mathematical models, and Real numbers If n is a counting number (1, 2, 3, 4,..) then Exponential notation: b n = b b b... b, where n is the Exponent or Power, and b is the base
More informationPOD. Combine these like terms: 1) 3x 2 4x + 5x x 7x ) 7y 2 + 2y y + 5y 2. 3) 5x 4 + 2x x 7x 4 + 3x x
POD Combine these like terms: 1) 3x 2 4x + 5x 2 6 + 9x 7x 2 + 2 2) 7y 2 + 2y 3 + 2 4y + 5y 2 3) 5x 4 + 2x 5 5 10x 7x 4 + 3x 5 12 + 2x 1 Definitions! Monomial: a single term ex: 4x Binomial: two terms separated
More informationSection 13-1: The Distributive Property and Common Factors
Section 13-1: The Distributive Property and Common Factors Factor: 4y 18z 4y 18z 6(4y 3z) Identify the largest factor that is common to both terms. 6 Write the epression as a product by dividing each term
More informationUnit 8 Notes: Solving Quadratics by Factoring Alg 1
Unit 8 Notes: Solving Quadratics by Factoring Alg 1 Name Period Day Date Assignment (Due the next class meeting) Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday
More informationCCAC ELEMENTARY ALGEBRA
CCAC ELEMENTARY ALGEBRA Sample Questions TOPICS TO STUDY: Evaluate expressions Add, subtract, multiply, and divide polynomials Add, subtract, multiply, and divide rational expressions Factor two and three
More informationThe two meanings of Factor 1. Factor (verb) : To rewrite an algebraic expression as an equivalent product
At the end of Packet #1we worked on multiplying monomials, binomials, and trinomials. What we have to learn now is how to go backwards and do what is called factoring. The two meanings of Factor 1. Factor
More informationAlg2A Factoring and Equations Review Packet
1 Factoring using GCF: Take the greatest common factor (GCF) for the numerical coefficient. When choosing the GCF for the variables, if all the terms have a common variable, take the one with the lowest
More informationSection 7.4 Additional Factoring Techniques
Section 7.4 Additional Factoring Techniques Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Factor trinomials when a = 1. Multiplying binomials
More informationSection 5.6 Factoring Strategies
Section 5.6 Factoring Strategies INTRODUCTION Let s review what you should know about factoring. (1) Factors imply multiplication Whenever we refer to factors, we are either directly or indirectly referring
More informationUnit 8: Quadratic Expressions (Polynomials)
Name: Period: Algebra 1 Unit 8: Quadratic Expressions (Polynomials) Note Packet Date Topic/Assignment HW Page Due Date 8-A Naming Polynomials and Combining Like Terms 8-B Adding and Subtracting Polynomials
More informationLesson 7.1: Factoring a GCF
Name Lesson 7.1: Factoring a GCF Date Algebra I Factoring expressions is one of the gateway skills that is necessary for much of what we do in algebra for the rest of the course. The word factor has two
More information(8m 2 5m + 2) - (-10m 2 +7m 6) (8m 2 5m + 2) + (+10m 2-7m + 6)
Adding Polynomials Adding & Subtracting Polynomials (Combining Like Terms) Subtracting Polynomials (if your nd polynomial is inside a set of parentheses). (x 8x + ) + (-x -x 7) FIRST, Identify the like
More informationSlide 1 / 128. Polynomials
Slide 1 / 128 Polynomials Slide 2 / 128 Table of Contents Factors and GCF Factoring out GCF's Factoring Trinomials x 2 + bx + c Factoring Using Special Patterns Factoring Trinomials ax 2 + bx + c Factoring
More informationLesson 3 Factoring Polynomials Skills
Lesson 3 Factoring Polynomials Skills I can common factor polynomials. I can factor trinomials like where a is 1. ie. I can factor trinomials where a is not 1. ie. I can factor special products. Common
More informationName Class Date. Adding and Subtracting Polynomials
8-1 Reteaching Adding and Subtracting Polynomials You can add and subtract polynomials by lining up like terms and then adding or subtracting each part separately. What is the simplified form of (3x 4x
More informationMultiply the binomials. Add the middle terms. 2x 2 7x 6. Rewrite the middle term as 2x 2 a sum or difference of terms. 12x 321x 22
Section 5.5 Factoring Trinomials 349 Factoring Trinomials 1. Factoring Trinomials: AC-Method In Section 5.4, we learned how to factor out the greatest common factor from a polynomial and how to factor
More informationSelected Worked Homework Problems. Step 1: The GCF must be taken out first (if there is one) before factoring the hard trinomial.
Section 7 4: Factoring Trinomials of the form Ax 2 + Bx + C with A >1 Selected Worked Homework Problems 1. 2x 2 + 5x + 3 Step 1: The GCF must be taken out first (if there is one) before factoring the hard
More informationFactoring is the process of changing a polynomial expression that is essentially a sum into an expression that is essentially a product.
Ch. 8 Polynomial Factoring Sec. 1 Factoring is the process of changing a polynomial expression that is essentially a sum into an expression that is essentially a product. Factoring polynomials is not much
More informationPrerequisites. Introduction CHAPTER OUTLINE
Prerequisites 1 Figure 1 Credit: Andreas Kambanls CHAPTER OUTLINE 1.1 Real Numbers: Algebra Essentials 1.2 Exponents and Scientific Notation 1.3 Radicals and Rational Expressions 1.4 Polynomials 1.5 Factoring
More informationSimplifying and Combining Like Terms Exponent
Simplifying and Combining Like Terms Exponent Coefficient 4x 2 Variable (or Base) * Write the coefficients, variables, and exponents of: a) 8c 2 b) 9x c) y 8 d) 12a 2 b 3 Like Terms: Terms that have identical
More informationAlgebra 7-4 Study Guide: Factoring (pp & 487) Page 1! of 11!
Page 1! of 11! Attendance Problems. Find each product. 1.(x 2)(2x + 7) 2. (3y + 4)(2y + 9) 3. (3n 5)(n 7) Factor each trinomial. 4. x 2 +4x 32 5. z 2 + 15z + 36 6. h 2 17h + 72 I can factor quadratic trinomials
More informationFactoring Methods. Example 1: 2x * x + 2 * 1 2(x + 1)
Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if
More informationReview Journal 6 Assigned Work: See Website
MFM2P Polynomial Checklist 1 Goals for this unit: I can apply the distributive law to the product of binomials. I can complete the following types of factoring; common, difference of squares and simple
More informationIn this section we revisit two special product forms that we learned in Chapter 5, the first of which was squaring a binomial.
5B. SPECIAL PRODUCTS 11 5b Special Products 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 informationSection 5.5 Factoring Trinomials, a = 1
Section 5.5 Factoring Trinomials, a = 1 REVIEW Each of the following trinomials have a lead coefficient of 1. Let s see how they factor in a similar manner to those trinomials in Section 5.4. Example 1:
More information2.01 Products of Polynomials
2.01 Products of Polynomials Recall from previous lessons that when algebraic expressions are added (or subtracted) they are called terms, while expressions that are multiplied are called factors. An algebraic
More informationSection 1.5: Factoring Special Products
Objective: Identify and factor special products including a difference of two perfect squares, perfect square trinomials, and sum and difference of two perfect cubes. When factoring there are a few special
More informationUnit: Polynomials and Factoring
Unit: Polynomials: Multiplying and Factoring Name Dates Taught Specific Outcome 10I.A.1 Demonstrate an understanding of factors of whole numbers by determining: Prime factors Greatest common factor Least
More informationAlgebra Module A33. Factoring - 2. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Algebra Module A33 Factoring - 2 Copyright This publication The Northern Alberta Institute of Technology 2002. All Rights Reserved. LAST REVISED November, 2008 Factoring - 2 Statement of Prerequisite
More informationFACTORING HANDOUT. A General Factoring Strategy
This Factoring Packet was made possible by a GRCC Faculty Excellence grant by Neesha Patel and Adrienne Palmer. FACTORING HANDOUT A General Factoring Strategy It is important to be able to recognize the
More informationFactoring Quadratic Expressions VOCABULARY
5-5 Factoring Quadratic Expressions TEKS FOCUS Foundational to TEKS (4)(F) Solve quadratic and square root equations. TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil,
More informationMTH 110-College Algebra
MTH 110-College Algebra Chapter R-Basic Concepts of Algebra R.1 I. Real Number System Please indicate if each of these numbers is a W (Whole number), R (Real number), Z (Integer), I (Irrational number),
More informationAlgebra. Chapter 8: Factoring Polynomials. Name: Teacher: Pd:
Algebra Chapter 8: Factoring Polynomials Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Factor polynomials by using the GCF. Pgs: 1-6 HW: Pages 7-8 o Day 2: SWBAT: Factor quadratic trinomials of
More informationFactoring is the process of changing a polynomial expression that is essentially a sum into an expression that is essentially a product.
Ch. 8 Polynomial Factoring Sec. 1 Factoring is the process of changing a polynomial expression that is essentially a sum into an expression that is essentially a product. Factoring polynomials is not much
More informationFactoring. Difference of Two Perfect Squares (DOTS) Greatest Common Factor (GCF) Factoring Completely Trinomials. Factor Trinomials by Grouping
Unit 6 Name Factoring Day 1 Difference of Two Perfect Squares (DOTS) Day Greatest Common Factor (GCF) Day 3 Factoring Completely Binomials Day 4 QUIZ Day 5 Factor by Grouping Day 6 Factor Trinomials by
More informationa*(variable) 2 + b*(variable) + c
CH. 8. Factoring polynomials of the form: a*(variable) + b*(variable) + c Factor: 6x + 11x + 4 STEP 1: Is there a GCF of all terms? NO STEP : How many terms are there? Is it of degree? YES * Is it in the
More informationMultiplication of Polynomials
Multiplication of Polynomials In multiplying polynomials, we need to consider the following cases: Case 1: Monomial times Polynomial In this case, you can use the distributive property and laws of exponents
More informationSkills Practice Skills Practice for Lesson 10.1
Skills Practice Skills Practice for Lesson 10.1 Name Date Water Balloons Polynomials and Polynomial Functions Vocabulary Match each key term to its corresponding definition. 1. A polynomial written with
More informationSection 5.3 Factor By Grouping
Section 5.3 Factor By Grouping INTRODUCTION In the previous section you were introduced to factoring out a common monomial factor from a polynomial. For example, in the binomial 6x 2 + 15x, we can recognize
More informationSection 13.1 The Greatest Common Factor and Factoring by Grouping. to continue. Also, circle your answer to each numbered exercise.
Algebra Foundations First Edition, Elayn Martin-Gay Sec. 13.1 Section 13.1 The Greatest Common Factor and Factoring by Grouping Complete the outline as you view Video Lecture 13.1. Pause the video as needed
More informationName. 5. Simplify. a) (6x)(2x 2 ) b) (5pq 2 )( 4p 2 q 2 ) c) (3ab)( 2ab 2 )(2a 3 ) d) ( 6x 2 yz)( 5y 3 z)
3.1 Polynomials MATHPOWER TM 10, Ontario Edition, pp. 128 133 To add polynomials, collect like terms. To subtract a polynomial, add its opposite. To multiply monomials, multiply the numerical coefficients.
More informationChapter 6.1: Introduction to parabolas and solving equations by factoring
Chapter 6 Solving Quadratic Equations and Factoring Chapter 6.1: Introduction to parabolas and solving equations by factoring If you push a pen off a table, how does it fall? Does it fall like this? Or
More informationMATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 3 (New Material From: , , and 10.1)
NOTE: In addition to the problems below, please study the handout Exercise Set 10.1 posted at http://www.austincc.edu/jbickham/handouts. 1. Simplify: 5 7 5. Simplify: ( ab 5 c )( a c 5 ). Simplify: 4x
More informationMATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 3 (New Material From: , , and 10.1)
NOTE: In addition to the problems below, please study the handout Exercise Set 10.1 posted at http://www.austin.cc.tx.us/jbickham/handouts. 1. Simplify: 5 7 5. Simplify: ( 6ab 5 c )( a c 5 ). Simplify:
More informationSection R.4 Review of Factoring. Factoring Out the Greatest Common Factor
1 Section R.4 Review of Factoring Objective #1: Factoring Out the Greatest Common Factor The Greatest Common Factor (GCF) is the largest factor that can divide into the terms of an expression evenly with
More informationChapter 6: Quadratic Functions & Their Algebra
Chapter 6: Quadratic Functions & Their Algebra Topics: 1. Quadratic Function Review. Factoring: With Greatest Common Factor & Difference of Two Squares 3. Factoring: Trinomials 4. Complete Factoring 5.
More informationThe two meanings of Factor
Name Lesson #3 Date: Factoring Polynomials Using Common Factors Common Core Algebra 1 Factoring expressions is one of the gateway skills necessary for much of what we do in algebra for the rest of the
More information5.1 Exponents and Scientific Notation
5.1 Exponents and Scientific Notation Definition of an exponent a r = Example: Expand and simplify a) 3 4 b) ( 1 / 4 ) 2 c) (0.05) 3 d) (-3) 2 Difference between (-a) r (-a) r = and a r a r = Note: The
More informationUnit 8: Polynomials Chapter Test. Part 1: Identify each of the following as: Monomial, binomial, or trinomial. Then give the degree of each.
Unit 8: Polynomials Chapter Test Part 1: Identify each of the following as: Monomial, binomial, or trinomial. Then give the degree of each. 1. 9x 2 2 2. 3 3. 2x 2 + 3x + 1 4. 9y -1 Part 2: Simplify each
More informationSection R.5 Review of Factoring. Factoring Out the Greatest Common Factor
1 Section R.5 Review of Factoring Objective #1: Factoring Out the Greatest Common Factor The Greatest Common Factor (GCF) is the largest factor that can divide into the terms of an expression evenly with
More informationName Date
NEW DORP HIGH SCHOOL Deirdre A. DeAngelis, Principal MATHEMATICS DEPARTMENT Li Pan, Assistant Principal Name Date Summer Math Assignment for a Student whose Official Class starts with 7, 8, and 9 Directions:
More information3.1 Factors and Multiples of Whole Numbers
3.1 Factors and Multiples of Whole Numbers LESSON FOCUS: Determine prime factors, greatest common factors, and least common multiples of whole numbers. The prime factorization of a natural number is the
More informationAlgebra/Geometry Blend Unit #5: Factoring and Quadratic Functions Lesson 2: Factoring Trinomials. What does factoring really mean?
Algebra/Geometry Blend Unit #5: Factoring and Quadratic Functions Lesson 2: Factoring Trinomials Name Period Date [page 1] Before you embark on your next factoring adventure, it is important to ask yourself
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction
Prerequisite Skills This lesson requires the use of the following skills: multiplying polynomials working with complex numbers Introduction 2 b 2 A trinomial of the form x + bx + that can be written as
More informationAlg2A Factoring and Equations Review Packet
1 Multiplying binomials: We have a special way of remembering how to multiply binomials called FOIL: F: first x x = x 2 (x + 7)(x + 5) O: outer x 5 = 5x I: inner 7 x = 7x x 2 + 5x +7x + 35 (then simplify)
More informationHFCC Math Lab Beginning Algebra -19. In this handout we will discuss one method of factoring a general trinomial, that is an
HFCC Math Lab Beginning Algebra -19 FACTORING TRINOMIALS a + b+ c ( a In this handout we will discuss one method of factoring a general trinomial, that is an epression of the form a + b+ c where a, b,
More informationMini-Lecture 6.1 The Greatest Common Factor and Factoring by Grouping
Copyright 01 Pearson Education, Inc. Mini-Lecture 6.1 The Greatest Common Factor and Factoring by Grouping 1. Find the greatest common factor of a list of integers.. Find the greatest common factor of
More information8-4 Factoring ax 2 + bx + c. (3x + 2)(2x + 5) = 6x x + 10
When you multiply (3x + 2)(2x + 5), the coefficient of the x 2 -term is the product of the coefficients of the x-terms. Also, the constant term in the trinomial is the product of the constants in the binomials.
More information2-4 Completing the Square
2-4 Completing the Square Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Write each expression as a trinomial. 1. (x 5) 2 x 2 10x + 25 2. (3x + 5) 2 9x 2 + 30x + 25 Factor each expression. 3.
More informationMATH 181-Quadratic Equations (7 )
MATH 181-Quadratic Equations (7 ) 7.1 Solving a Quadratic Equation by Factoring I. Factoring Terms with Common Factors (Find the greatest common factor) a. 16 1x 4x = 4( 4 3x x ) 3 b. 14x y 35x y = 3 c.
More informationQuadratic Algebra Lesson #2
Quadratic Algebra Lesson # Factorisation Of Quadratic Expressions Many of the previous expansions have resulted in expressions of the form ax + bx + c. Examples: x + 5x+6 4x 9 9x + 6x + 1 These are known
More information5.06 Rationalizing Denominators
.0 Rationalizing Denominators There is a tradition in mathematics of eliminating the radicals from the denominators (or numerators) of fractions. The process is called rationalizing the denominator (or
More information1. Which pair of factors of 8 has a sum of 9? 1 and 8 2. Which pair of factors of 30 has a sum of. r 2 4r 45
Warm Up 1. Which pair of factors of 8 has a sum of 9? 1 and 8 2. Which pair of factors of 30 has a sum of 17? 2 and 15 Multiply. 3. (x +2)(x +3) x 2 + 5x + 6 4. (r + 5)(r 9) r 2 4r 45 Objective Factor
More informationMath Final Examination STUDY GUIDE Fall Name Score TOTAL Final Grade
Math 10006 Final Examination STUDY GUIDE Fall 010 Name Score TOTAL Final Grade The Use of a calculator is permitted on this exam. Duration of the test is 13 minutes and will have less number of questions
More informationName: Algebra Unit 7 Polynomials
Name: Algebra Unit 7 Polynomials Monomial Binomial Trinomial Polynomial Degree Term Standard Form 1 ((2p 3 + 6p 2 + 10p) + (9p 3 + 11p 2 + 3p) TO REMEMBER Adding and Subtracting Polynomials TO REMEMBER
More information6.1 Greatest Common Factor and Factor by Grouping *
OpenStax-CNX module: m64248 1 6.1 Greatest Common Factor and Factor by Grouping * Ramon Emilio Fernandez Based on Greatest Common Factor and Factor by Grouping by OpenStax This work is produced by OpenStax-CNX
More informationPolynomial is a general description on any algebraic expression with 1 term or more. To add or subtract polynomials, we combine like terms.
Polynomials Lesson 5.0 Re-Introduction to Polynomials Let s start with some definition. Monomial - an algebraic expression with ONE term. ---------------------------------------------------------------------------------------------
More information6.3 Factor Special Products *
OpenStax-CNX module: m6450 1 6.3 Factor Special Products * Ramon Emilio Fernandez Based on Factor Special Products by OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons
More informationFactor Trinomials of the Form ax^2+bx+c
OpenStax-CNX module: m6018 1 Factor Trinomials of the Form ax^+bx+c Openstax Elementary Algebra This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 By
More informationName Class Date. Multiplying Two Binomials Using Algebra Tiles. 2x(x + 3) = x 2 + x. 1(x + 3) = x +
Name Class Date Multiplying Polynomials Going Deeper Essential question: How do you multiply polynomials? A monomial is a number, a variable, or the product of a number and one or more variables raised
More informationPolynomials. Factors and Greatest Common Factors. Slide 1 / 128. Slide 2 / 128. Slide 3 / 128. Table of Contents
Slide 1 / 128 Polynomials Table of ontents Slide 2 / 128 Factors and GF Factoring out GF's Factoring Trinomials x 2 + bx + c Factoring Using Special Patterns Factoring Trinomials ax 2 + bx + c Factoring
More informationChapter 4 Factoring and Quadratic Equations
Chapter 4 Factoring and Quadratic Equations Lesson 1: Factoring by GCF, DOTS, and Case I Lesson : Factoring by Grouping & Case II Lesson 3: Factoring by Sum and Difference of Perfect Cubes Lesson 4: Solving
More informationElementary Algebra Review for Exam 3
Elementary Algebra Review for Exam ) After receiving a discount of 5% on its bulk order of typewriter ribbons, John's Office Supply pays $5882. What was the price of the order before the discount? Round
More informationUNIT 1 RELATIONSHIPS BETWEEN QUANTITIES AND EXPRESSIONS Lesson 1: Working with Radicals and Properties of Real Numbers
Guided Practice Example 1 Reduce the radical expression result rational or irrational? 80. If the result has a root in the denominator, rationalize it. Is the 1. Rewrite each number in the expression as
More informationFactors of 10 = = 2 5 Possible pairs of factors:
Factoring Trinomials Worksheet #1 1. b 2 + 8b + 7 Signs inside the two binomials are identical and positive. Factors of b 2 = b b Factors of 7 = 1 7 b 2 + 8b + 7 = (b + 1)(b + 7) 2. n 2 11n + 10 Signs
More informationFinal Exam Review - MAT 0028
Final Exam Review - MAT 0028 All questions on the final exam are multiple choice. You will be graded on your letter choices only - no partial credit will be awarded. To maximize the benefit of this review,
More informationSect General Factoring Summary
111 Concept #1 Sect 6.6 - General Factoring Summary Factoring Strategy The flow chart on the previous page gives us a visual picture of how to attack a factoring problem. We first start at the top and
More informationGetting Ready for Algebra 2 - Test 3 Review
Getting Ready for Algebra 2 - Test 3 Review Short Answer 1. Simplify the expression. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Simplify the product using FOIL. 15. 16. Find the square. 17. Find the product.
More informationTopic 12 Factorisation
Topic 12 Factorisation 1. How to find the greatest common factors of an algebraic expression. Definition: A factor of a number is an integer that divides the number exactly. So for example, the factors
More information