2 TERMS 3 TERMS 4 TERMS (Must be in one of the following forms (Diamond, Slide & Divide, (Grouping)
|
|
- Quentin Brown
- 5 years ago
- Views:
Transcription
1 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) or Guess-and-check) Difference of Two Perfect Squares a 2 b 2 = (a + b)( a b) O R Sum Of Cubes - SOAP a 3 + b 3 = (a+b)(a 2 - ab + b 2 ) Difference of Cubes - SOAP a 3 - b 3 = (a-b)(a 2 + ab + b 2 ) O Guess and check (or use some other method) Guess & Check 1. List the factors for the first coefficient and the constant. 2x 2 + 3x 5 2 = 2, 1 5 = 5, 1 2. Place the 1 set of factors in the parenthesis. FOIL to check. (2x + 1)(x 5) or (2x 5)(x + 1) 2x 2 9x 5 2x 2 3x 5 3. If your numbers are correct except for the signs, simply change the signs. (2x + 5)(x 1) 2x 2 + 3x 5 Group first two & last two terms and see if each pair has a GCF. (note: you may need to change order of the terms) 2x 3 8x 2 + 3x 12 THEN If the GCF of each pair results in a common binomial, factor out the binomial. 2x 2 (x 4) + 3(x 4) THEN Write the binomial times the binomial created by the terms left when the GCF binomial was pulled out. (x 4 )(2x 2 + 3) 1. If nothing can be done to the original expression, then it is PRIME. 2. Check to see if any of your final answers will factor further. 3. Check your answer by multiplying.
2 Factoring Binomials Binomials have two terms! If the binomial is the difference of 2 squares, use x 2 y 2 = (x + y)(x - y). NOTE: The sum of 2 squares cannot be factored!!! x 2 + y 2 is prime! Factor: a) x 2 4 b) x c) 4x 2 49 d) 2x 2 72 e) 36a 2 b 2 81x 4 y 6 f) x 4 16 If the binomial is the sum or difference of 2 cubes, use x 3 + y 3 = (x + y)(x 2 xy + y 2 ) OR x 3 - y 3 = (x - y)(x 2 + xy + y 2 ). Hint: Think SOAP a) x b) x c) x 3 27 d) 8x 3 1 e) 125x y 3 z 3
3 Factoring GCFs & by Grouping GCF means greatest common factor Working backwards to figure out which polynomials multiplied together will produce a certain product is called factoring. When factoring, always check first to see if the terms of the polynomial you are factoring have a greatest common factor (GCF). The greatest common factor, GCF, is the product of the greatest common factor of the numerical coefficients and the highest power of every variable that is a factor of each monomial. For example, consider the monomials 10x 2 y 3 and 15xy 2. The GCF is 5xy 2 because 5 is the highest integer that will divide evenly into both 10 and 15, the highest power of x that is contained in both terms is x, and the highest power of y that is contained in both terms is y 2. When factoring polynomials, first look for the largest monomial which is a factor of each term of the polynomial. Factor out (divide each term by) this largest monomial. Examples Factor by finding the GCF. a. 16m 2 n + 12mn 2 b. 5k 3 p 3kp 2 + k 3 p 5 c. 6x 2 y + 3xy 5 9x 3 y 3 d. 22m 3 n 5 11m 2 p e. -3x 2 9 f. 10x 3 y 2 z 2 + 5x 2 y 2 z 5xyz
4 In some cases, there is not a GCF for ALL of the terms in a polynomials. If you have 4 terms with no GCF, then try factoring by grouping. Factoring by Grouping 1. Group the first 2 terms together and the last 2 terms together. 2. Factor out the GCF from each binomial. 3. Factor out the common binomial. a) x 3 + 5x 2-2x 10 b) x 3 + 3x 2-9x 27 c) 3x 3-2x 2-6x + 4 d) x 3 - x 2-5x + 5 e) 5x 3-3x 2-10x + 6 f) 2x 3 3x 2 8x + 12
5 Factoring Trinomials ax 2 + bx + c when a = 1 If the coefficient of x 2 is 1, then x 2 + bx + c = (x + m)(x + n) where m and n multiply to give c AND m and n add to give b. g. x 2 + 3x + 2 h. x 2 + 6x + 5 i. x 2 + 9x + 18 j. x 2 + 6x + 9 k. x x + 45 If the coefficient of x 2 is 1, then x 2 - bx + c = (x - m)(x - n) where m and n multiply to give c AND m and n add to give b. a. x 2-6x + 8 b. x 2-11x + 30 c. x 2-9x + 8 d. x 2-8x + 16 e. x 2-12x + 27
6 Continue to think m and n multiply to give c (it s negative) AND m and n (one is positive, the other is negative) add to give b for the forms x 2 + bx - c OR x 2 - bx - c = (x + m)(x n). a) x 2 + x 12 b) x 2 + 2x 15 c) x 2-2x - 15 d) x 2-4x 5 e) x 2 + 5x 36 f) x 2-2x 15 g) x 2 - x 72 h) x 2 + 5x 14 i) x 2 + 8x - 9 SPECIAL CASE: If you recognize that the trinomial is a perfect square trinomial, then you can factor it as a binomial squared. In other words, x 2 + 2xy + y 2 = (x + y)(x + y) = (x + y) 2 OR x 2-2xy + y 2 = (x - y)(x - y) = (x - y) 2. a) x 2 + 6x + 9 b) x x + 36 c) x 2-8x + 16 Challenge: Can you factor these when a 1? 9x x x 2-90x x x + 9
7 Factoring Trinomials ax 2 + bx + c when a 1 If the coefficient of x 2 is not 1, then use a method below to factor the trinomial. Method 1: Guess and Check Method 2: Grouping Factor 2x x + 15 Factor 2x x + 15 Method 3: Slide and Divide Method 4: Diamond Factor 2x x + 15 Factor 2x x + 15
8 Another Example... Method 1: Guess and Check Method 2: Grouping Factor 4x 2-4x - 15 Factor 4x 2-4x - 15 Method 3: Slide and Divide Method 4: Diamond Factor 4x 2-4x - 15 Factor 4x 2-4x - 15 a) 3x 2-16x + 5 b) 4x 2 + 7x + 3 c) 2x 2 + 7x 9 d) 4x 2 + 5x 6
9 If you are given a polynomial expression to factor completely, ask yourself: 1. Is there a GCF? 2. Is the expression a binomial? 3. Is the expression a trinomial? 4. Does the expression have 4 terms? Examples Factor completely: a) 3x 2-24x 27 b) 9x x + 64 c) 9x d) -x 2 + 5x 4 e) x 2-7x + 1 f) x 2-18x + 81 g) 2x 3 3x 2 8x + 12 h) 2x 4 32 i) 16x
Factoring 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 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 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 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 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 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 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 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 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 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 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 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 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 informationUniversity of Phoenix Material
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
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationChapter 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 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 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 informationWeek 20 Algebra 1 Assignment:
Week 0 Algebra 1 Assignment: Day 1: pp. 38-383 #-0 even, 3-7 Day : pp. 385-386 #-18 even, 1-5 Day 3: pp. 388-389 #-4 even, 7-9 Day 4: pp. 39-393 #1-37 odd Day 5: Chapter 9 test Notes on Assignment: Pages
More informationDownloaded from
9. Algebraic Expressions and Identities Q 1 Using identity (x - a) (x + a) = x 2 a 2 find 6 2 5 2. Q 2 Find the product of (7x 4y) and (3x - 7y). Q 3 Using suitable identity find (a + 3)(a + 2). Q 4 Using
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 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 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 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 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 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 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 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 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 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 informationSection 5.3 Practice Exercises Vocabulary and Key Concepts
Section 5.3 Practice Exercises Vocabulary and Key Concepts 1. a. To multiply 2(4x 5), apply the property. b. The conjugate of 4x + 7 is. c. When two conjugates are multiplied the resulting binomial is
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 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 informationFactor out the common numerical and variable factors from each term.
CLEP Precalculus - Problem Drill 05: Polynomials No. 1 of 10 1. What is the greatest common factor among the terms of the polynomial? 21m 2 n 2 x 3 y 4 + 63mnx 2 y 2 49mx 2 y 4 + 28mn 2 xy 3 (A) 7mnxy
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: 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationUnit 9 Notes: Polynomials and Factoring. Unit 9 Calendar: Polynomials and Factoring. Day Date Assignment (Due the next class meeting) Monday Wednesday
Name Period Unit 9 Calendar: Polynomials and Factoring Day Date Assignment (Due the next class meeting) Monday Wednesday 2/26/18 (A) 2/28/18 (B) 9.1 Worksheet Adding, Subtracting Polynomials, Multiplying
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 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 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 informationChapter 5 Self-Assessment
Chapter 5 Self-Assessment. BLM 5 1 Concept BEFORE DURING (What I can do) AFTER (Proof that I can do this) 5.1 I can multiply binomials. I can multiply trinomials. I can explain how multiplication of binomials
More informationStudy P.5 CVC 1 7, # 1, 5, 9,...37, 39 55, 59, 65, 69, 73,
GOALS: Factor Polynomials using: 1. Distributive Property (common factors) 2. Trial and Error (trinomials) 3. Factor by Grouping (trinomials) Study P.5 CVC 1 7, # 1, 5, 9,...37, 39 55, 59, 65, 69, 73,...
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 informationFactor Quadratic Expressions of the Form ax 2 + bx + c. How can you use a model to factor quadratic expressions of the form ax 2 + bx + c?
5.5 Factor Quadratic Expressions of the Form ax 2 + bx + c The Ontario Summer Games are held every two years in even-numbered years to provide sports competition for youth between the ages of 11 and 22.
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 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 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 informationALGEBRAIC EXPRESSIONS AND IDENTITIES
9 ALGEBRAIC EXPRESSIONS AND IDENTITIES Exercise 9.1 Q.1. Identify the terms, their coefficients for each of the following expressions. (i) 5xyz 3zy (ii) 1 + x + x (iii) 4x y 4x y z + z (iv) 3 pq + qr rp
More informationFactoring Simple Trinomials February 24, What's Going On? What's the Pattern? Working Backwards. Finding Factors
What's Going On? What's the Pattern? Working Backwards Finding Factors Learning Goal I will be able to factor standard form equations when a = 1. What's the Pattern? (x + 2)(x + 3) = x 2 + 5x + 6 (x +
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 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 informationIdentifying & Factoring: x 2 + bx + c
Identifying & Factoring: x 2 + bx + c Apr 13 11:04 AM 1 May 16 8:52 AM 2 A polynomial that can be simplified to the form ax + bx + c, where a 0, is called a quadratic polynomial. Linear term. Quadratic
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 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 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 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 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 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 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 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 informationPolynomials. Unit 10 Polynomials 2 of 2 SMART Board Notes.notebook. May 15, 2013
Oct 19 9:41 M errick played basketball for 5 out of the 10 days for four hours each. How many hours did errick spend playing basketball? Oct 19 9:41 M Polynomials Polynomials 1 Table of ontents Factors
More informationMathematics 10C. UNIT THREE Polynomials. 3x 3-6x 2. 3x 2 (x - 2) 4x 2-3x - 1. Unit. Student Workbook. FOIL (2x - 3)(x + 1) A C = -4.
Mathematics 10C FOIL (2x - 3)(x + 1) Student Workbook Lesson 1: Expanding Approximate Completion Time: 4 Days Unit 3 3x 3-6x 2 Factor Expand 3x 2 (x - 2) Lesson 2: Greatest Common Factor Approximate Completion
More informationMultiplying Polynomials
14 Multiplying Polynomials This chapter will present problems for you to solve in the multiplication of polynomials. Specifically, you will practice solving problems multiplying a monomial (one term) and
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 informationChapter 2 Algebra Part 1
Chapter 2 Algebra Part 1 Section 2.1 Expansion (Revision) In Mathematics EXPANSION really means MULTIPLY. For example 3(2x + 4) can be expanded by multiplying them out. Remember: There is an invisible
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 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 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 information