# Factoring Methods. Example 1: 2x * x + 2 * 1 2(x + 1)

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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 it is a Special Case Common Factor or Greatest Common Factor (GCF) A Common Factor is something you can divide each term of a polynomial by evenly. The Greatest Common Factor (GCF) is the largest common factor you can evenly divide each term of a polynomial by. When you are factoring, you always want to remove the GCF. Remember that 1 is not a GCF. An easy way to see what is common to all the terms is to individually factor each term. After factoring each term in Example 1, it easy to see that 2 is the largest number that is common to all the terms in the polynomial and therefore the GCF. The next step is to put the GCF on the outside of the parenthesis and put what is left inside the parenthesis. Example 1: 2x * x + 2 * 1 2(x + 1) In Example 2, the same process is done, but in this case it is not a number that is common to both terms. It is the variable x that is common, so x can be moved to the outside of the parenthesis and 3x + 4 can be left inside the parenthesis. Example 2: 3x 2 + 4x 3 * x * x + 4 * x x(3x + 4) In Example 3, we again follow the same process. First, each term is factored individually. Then we determine what is common in all the terms. In this case, it is 5x 2, so 5x 2 will be put on the outside of the parenthesis and x 2 + 7x + 2 will be put inside the parenthesis. Example 3: 5x 4-35x x 2 5x 4 = 5 * x * x * x * x 35x 3 = 7 * 5 * x * x * x 10x 2 = 5 * 2 * x * x GCF = 5x 2 Factored form: 5x 2 (x 2-7x + 2) Factoring Methods Provided by Tutoring Services 1 Updated April 2013

2 Factoring by Grouping Before Factoring by Grouping, remember to always see if there is a common factor. If there is one, factor out the GCF before trying to factor by grouping. Example 4: 10abx + 15bx - 8ax - 12x x(10ab + 15b - 8a - 12) x[(10ab + 15b) + (-8a - 12)] x[5b(2a + 3) + -4(2a + 3)] x(2a + 3)(5b - 4) Factoring by grouping requires that you have four terms. What you want to do is group the terms into two groups of two terms each. When deciding which terms to put together, think about which terms share a common factor. In Example 5, it does not matter which terms are grouped together because all the terms share a common factor other than one. However, in Example 6A, you cannot group 6xy and -5 together and -15x and 2y together because these two groups do not share a common factor other than one. At least one group must share a common factor other than one. Also note that in all the examples, a plus sign was put between the parentheses. It is important that there be a plus sign between them. The next step is to factor out the GCF in each set of parentheses. See above for how to do this. After factoring out the GCF, the insides of the parentheses should be the same. If the groups are not the same as in Example 6A and 7, then either you need to change the grouping to make it work as in Example 6 or the problem is not factorable by grouping as in Example 7. If the insides of the parentheses match, then write what is in the parentheses one time and then write what is outside the parentheses in another set of parentheses. Example 5: 2ab + 12a + 3b + 18 (2ab + 12a) + (3b + 18) 2a(b + 6) + 3(b + 6) (b + 6)(2a + 3) Example 6A: 6xy x + 2y 6B: 6xy x + 2y (6xy - 5) + (-15x +2y) (6xy - 15x) + (-5 + 2y) 1(6xy - 5) + (-15x + 2y) 3x(2y - 5) + 1(-5 + 2y) The parentheses don t match. (2y - 5)(3x + 1) Example 7: x + 2xy + 3y + 5 (x + 2xy) + (3y + 5) x(1 + 2y) + (3y + 5) The parentheses don t match, and the problem can t be rearranged to work, so it is not factorable by grouping. Provided by Tutoring Services 2 Factoring Methods

3 Factoring by Grouping is also used to factor problems in the form ax 2 + bx + c. The letters a, b, and c represent numbers. If you do not see a number in front of one of the x terms, then the number is 1. Example 8: 4x 2 + x - 34 a = 4 b = 1 c = -34 The way to factor a trinomial, a polynomial with three terms, is to change it from three terms to four. The first step is to find two numbers that add together to get b and multiply together to get ac. An easy way to find the numbers is to list the factors of ac. If you cannot find two numbers that will do this, then the problem is not factorable by grouping. Once you find the two numbers, break the b into those numbers. After that, the problems are just like Examples 4-7. Example 9: x 2-5x = -5 and * = 6 * 1 = 6 x 2-3x -2x +6 Factors of 6 : (x 2-3x) + (-2x + 6) 1 6 x(x - 3) + -2(x - 3) -1-6 (x -3)(x - 2) = -5-2 * -3 = 6 Example 10: 2x x = 13 and * = 2 * 15 = 30 2x 2 + 3x + 10x + 15 Factors of 30: (2x 2 + 3x) + (10x + 15) x(2x + 3) + 5(2x + 3) (2x + 3)(x + 5) = 13 3 * 10 = Example 11: x 2 + 4x = 4 and * = 1 * 9 = 9 Factors of 9: No factor of 9 can add to 4, so the 1 9 trinomial is not factorable over the -1-9 integers Provided by Tutoring Services 3 Factoring Methods

4 Special Case Factoring There are some special cases for factoring that you just have to memorize. a 2 - b 2 = (a + b)(a - b) Difference of Perfect Squares a 3 + b 3 = (a + b)(a 2 - ab + b 2 ) Sum of Cubes a 3 - b 3 = (a - b)(a 2 + ab + b 2 ) Difference of Cubes You need to be able to recognize a special case and be able to follow the rule. Example 12: x 2-4 Case: a 2 - b 2 = (a + b)(a - b) Both x 2 and 4 are perfect squares. x 2 = x * x and 4 = 2 2 = 2 * 2 Answer: (x + 2)(x - 2) Example 13: x Case: a 3 + b 3 = (a + b)(a 2 - ab + b 2 ) Both x 3 and 27 are perfect cubes. x 3 = x * x * x and 27 = 3 3 = 3 * 3 * 3 Answer: (x + 3)(x 2 3x ) (x + 3)(x 2-3x + 9) Example 14: 8x Case: a 3 - b 3 = (a - b)(a 2 + ab + b 2 ) Both 8x 3 and 125 are perfect cubes. 8x 3 = 2 * 2 * 2 * x * x * x and 125 = 5 3 = 5 * 5 * 5 Answer: (2x 5)[(2x) x + (5) 2 ] (2x - 5)(4x x + 25) Provided by Tutoring Services 4 Factoring Methods

5 Substitution Substitution is used to factor trinomials that are in the form: ax 2m + bx m +c. Again, the letters a, b, and c represent numbers, and m also represents a number. What you do is substitute u for x 2 in the problem, then you factor normally. After you have your answer, you then substitute x 2 back into the problem. Example 15 shows how to use substitution in a problem. Example 15: x 4-4x 2-45 u = x 2 u 2-4u = -4 and * = -45 u 2-9u + 5u - 45 u(u - 9) + 5(u - 9) (u + 5)(u - 9) (x 2 + 5)(x 2-9) (x 2 + 5)(x + 3)(x - 3) = -4 and (5)(-9) = -45 Example 16: x 4-81 u = x 2 u 2-81 (u + 9)(u - 9) (x 2 + 9)(x 2-9) Difference of Perfect Squares on (x 4-81) (x 2 + 9)(x + 3)(x - 3) Difference of Perfects Squares on (x 2-9) Example 17: (x - 2) 2-9 u = x - 2 u 2-9 (u + 3)(u - 3) (x )(x - 2-3) (x + 1)(x - 5) Provided by Tutoring Services 5 Factoring Methods

6 Quick Steps to Factoring 1. Common Factor (Examples 1-3) a. If there is a common factor in the problem, factor it out. 2. Factor by Grouping a. There are three terms (Examples 9-11) i. Find two numbers that + = b and * = ac ii. Split the b term into the two numbers. iii. Follow the steps in 2b. b. There are four terms (Examples 5-7) i. Group the terms into two sets of parentheses. ii. Factor the GCF out of each parenthesis. iii. If the parentheses are the same after removing the GCF, then write the parentheses once and write what you factored out of the parentheses in a second set of parenthesis. 3. Special Factoring (Examples 12-14) a. Is it a special case of factoring? If so follow the appropriate rule. i. (a 2 + b 2 ) = (a + b) (a - b) ii. a 3 + b 3 = (a + b)(a 2 - ab + b 2 ) iii. a 3 - b 3 = (a - b)(a 2 + ab + b 2 ) Provided by Tutoring Services 6 Factoring Methods

7 Sample Problems 1. 2x y a a 2-30a a a x x ax - bx + ay - by 8. 2ax x + 6a 9. m 3 + n r 3 - s x x y y a 2-2ax - 3a + 2x 16. a 2-2a + ab - 2b y x x 2-4x n 2 + 5n n 2-10n m 2 + 3ms - 4s y 2 + 4y y 2 - y t 2-14t x - x x - x s s x x a 4-10a 3-72a x 3-3x 2-2x (x - 1) (x + 2) 2 - (y - 3) (2x - 1) a 2-4ab b a 3-16a a 19. x m 3 n 6 + d a 3 - a 2 b - a + b 22. x 2 + 6x + 5 Provided by Tutoring Services 7 Factoring Methods

8 Answers: 1. 2(x - 3)(x + 3) 2. 3(y - 4)(y + 4) 3. (a - 2)(a + 2)(a 2 + 4) 4. 5(a - 3) (a + 2) (x - 25) 2 7. (x + y)(a - b) 8. (x + 3)(2a + 1) 9. (m + n)(m 2 - mn + n 2 ) 10. (r - s)(r 2 + rs + s 2 ) 11. (4x - 1)(16x 2 + 4x + 1) 12. (2x + 1)(4x 2-2x + 1) 13. (3y - 1)(9y 2 + 3y + 1) 14. (5y - 1)(25y 2 + 5y + 1) 15. (a - 1)(3a - 2x) 16. (a - 2)(a + b) 17. (5 - y)(25 + 5y + y 2 ) 18. (x 2-3)(x 4 + 3x 2 + 9) 19. (x 2 + 5)(x 4-5x ) 23. (x - 3)(x - 1) 24. (n + 2)(n + 3) 25. (n - 5) (m - s)(m + 4s) 27. (y + 6)(y - 2) 28. (y - 6)(y + 5) 29. (t - 18)(t + 4) 30. (3 + x)(2 - x) 31. (9 - x)(4 + x) 32. (6s + 1) (x + 15)(x - 10) 34. 2a 2 (a - 9)(a + 4) 35. (x - 1)(x + 1)(2x - 3) 36. (x + 1)(x - 3) 37. (x - y + 5)(x + y - 1) 38. (5-2x)(3 + 2x) 39. 4a 2-4ab + b 2-36 (2a - b) 2-36 (2a - b + 6)(2a - b - 6) 40. 2a(a - 4) (mn 2 + d)(m 2 n 4 - mn 2 d + d 2 ) 21. (a - 1)(a + 1)(a - b) 22. (x + 5)(x + 1) Provided by Tutoring Services 8 Factoring Methods

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### EAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an

### How To Solve Factoring Problems 05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

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

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

### Topic: Special Products and Factors Subtopic: Rules on finding factors of polynomials Quarter I: Special Products and Factors and Quadratic Equations Topic: Special Products and Factors Subtopic: Rules on finding factors of polynomials Time Frame: 20 days Time Frame: 3 days Content Standard:

### Simplifying Algebraic Fractions 5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

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### 1.1 Practice Worksheet Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)

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

### FACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project 9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : Module

### CHAPTER 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

### Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions: Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?

### Polynomials and Factoring; More on Probability Polynomials and Factoring; More on Probability Melissa Kramer, (MelissaK) Anne Gloag, (AnneG) Andrew Gloag, (AndrewG) Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)

### Veterans Upward Bound Algebra I Concepts - Honors Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER

### BEGINNING ALGEBRA ACKNOWLEDMENTS BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science

### 1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods

### 6706_PM10SB_C4_CO_pp192-193.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

### 4.4 Factoring ax 2 + bx + c 4.4 Factoring ax 2 + bx + c From the last section, we now know a trinomial should factor as two binomials. With this in mind, we need to look at how to factor a trinomial when the leading coefficient is

### Sect 6.1 - Greatest Common Factor and Factoring by Grouping Sect 6.1 - Greatest Common Factor and Factoring by Grouping Our goal in this chapter is to solve non-linear equations by breaking them down into a series of linear equations that we can solve. To do this,

### Finding Solutions of Polynomial Equations DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE 2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution 1. Sit anywhere in the concentric circles. Do not move the desks. 2. Take out chapter 6, HW/notes #1-#5, a pencil, a red pen, and your calculator. 3. Work on opener #6 with the person sitting across from