MTH-4106-1 C1-C4 Factorization 1/31/12 11:38 AM Page 1 F MTH-4106-1 actoring and Algebraic Fractions
FACTORING AND ALGEBRAIC FUNCTIONS
Project Coordinator: Jean-Paul Groleau Authors: Nicole Perreault Suzie Asselin Content Revision: Jean-Paul Groleau Alain Malouin Updated Version: Line Régis Translator: Claudia de Fulviis Linguistic Revision: Johanne St-Martin Desktop Publishing: P.P.I. inc. Cover Page: Daniel Rémy First Printing: 2005 Printing: 2005 Reprint: 2006 Société de formation à distance des commissions scolaires du Québec All rights for translation, adaptation, in whole or in part, reserved for all countries. Any reproduction by mechanical or electronic means, including microreproduction, is forbidden without the written permission of a duly authorized representative of the Société de formation à distance des commissions scolaires du Québec (SOFAD). Legal Deposit 2005 Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada ISBN 978-2-89493-284-1
TABLE OF CONTENTS Introduction to the Program Flowchart... 0.4 The Program Flowchart... 0.5 How to Use This Guide... 0.6 General Introduction... 0.9 Intermediate and Terminal Objectives of the Module... 0.11 Diagnostic Test on the Prerequisites... 0.15 Answer Key for the Diagnostic Test on the Prerequisites... 0.19 Analysis of the Diagnostic Test Results... 0.21 Information for Distance Education Students... 0.23 UNITS 1. Factoring by Removing the Common Factor... 1.1 2. Factoring by Grouping... 2.1 3. Factoring Trinomials of the Form x 2 + bx + c or x 2 + bxy + cy 2... 3.1 4. Factoring Trinomials of the Form ax 2 + bx + c or ax 2 + bxy + cy 2... 4.1 5. Factoring Differences of Two Squares... 5.1 6. Factoring Polynomials... 6.1 7. Simplifying Algebraic Fractions... 7.1 8. Multiplying and Dividing Two Algebraic Fractions... 8.1 9. Adding and Subtracting Two Algebraic Fractions and Comparing Algebraic Expressions... 9.1 Final Review... 10.1 Answer Key for the Final Review... 10.6 Terminal Objectives... 10.8 Self-Evaluation Test... 10.11 Answer Key for the Self-Evaluation Test... 10.17 Analysis of the Self-Evaluation Test Results... 10.21 Final Evaluation... 10.22 Answer Key for the Exercises... 10.23 Glossary... 10.77 List of Symbols... 10.82 Bibliography... 10.83 Review Activities... 11.1 0.3
INTRODUCTION TO THE PROGRAM FLOWCHART Welcome to the World of Mathematics! This mathematics program has been developed for the adult students of the Adult Education Services of school boards and distance education. The learning activities have been designed for individualized learning. If you encounter difficulties, do not hesitate to consult your teacher or to telephone the resource person assigned to you. The following flowchart shows where this module fits into the overall program. It allows you to see how far you have progressed and how much you still have to do to achieve your vocational goal. There are several possible paths you can take, depending on your chosen goal. The first path consists of modules MTH-3003-2 (MTH-314) and MTH-4104-2 (MTH-416), and leads to a Diploma of Vocational Studies (DVS). The second path consists of modules MTH-4109-1 (MTH-426), MTH-4111-2 (MTH-436) and MTH-5104-1 (MTH-514), and leads to a Secondary School Diploma (SSD), which allows you to enroll in certain Cegep-level programs that do not call for a knowledge of advanced mathematics. The third path consists of modules MTH-5109-1 (MTH-526) and MTH-5111-2 (MTH-536), and leads to Cegep programs that call for a solid knowledge of mathematics in addition to other abiliies. If this is your first contact with this mathematics program, consult the flowchart on the next page and then read the section How to Use This Guide. Otherwise, go directly to the section entitled General Introduction. Enjoy your work! 0.4
CEGEP PROGRAM FLOWCHART MTH-5112-1 Logic MTH-536 MTH-5111-2 MTH-5110-1 Complement and Synthesis II Introduction to Vectors MTH-5109-1 Geometry IV MTH-514 MTH-5104-1 MTH-5103-1 MTH-526 Optimization II Probability II MTH-5108-1 MTH-5107-1 MTH-5106-1 MTH-5105-1 Trigonometric Functions and Equations Exponential and Logarithmic Functions and Equations Real Functions and Equations Conics MTH-5102-1 Statistics III MTH-5101-1 Optimization I Trades DVS MTH-436 MTH-4111-2 MTH-4110-1 Complement and Synthesis I The Four Operations on Algebraic Fractions MTH-4109-1 Sets, Relations and Functions MTH-426 MTH-4108-1 MTH-4107-1 Quadratic Functions Straight Lines II MTH-4106-1 Factoring and Algebraic Functions You ar e here MTH-4105-1 Exponents and Radicals MTH-416 MTH-4104-2 MTH-4103-1 MTH-4102-1 MTH-4101-2 Statistics II Trigonometry I Geometry III Equations and Inequalities II MTH-3003-2 Straight Lines I MTH-314 MTH-3002-2 MTH-3001-2 Geometry II The Four Operations on Polynomials MTH-216 MTH-2008-2 MTH-2007-2 Statistics and Probabilities I Geometry I MTH-2006-2 Equations and Inequalities I MTH-1007-2 Decimals and Percent MTH-116 MTH-1006-2 MTH-1005-2 The Four Operations on Fractions The Four Operations on Integers 25 hours = 1 credit 50 hours = 2 credits 0.5
HOW TO USE THIS GUIDE Hi! My name is Monica and I have been asked to tell you about this math module. What s your name? I m Andy. You ll see that with this method, math is a real breeze! Whether you are registered at an adult education center or pursuing distance education,...... you have probably taken a placement test which tells you exactly which module you should start with. My results on the test indicate that I should begin with this module. Now, the module you have in your hand is divided into three sections. The first section is...... the entry activity, which contains the test on the prerequisites. By carefully correcting this test using the corresponding answer key, and recording your results on the analysis sheet... 0.6
... you can tell if you re well enough prepared to do all the activities in the module. And if I m not, if I need a little review before moving on, what happens then? In that case, before you start the activities in the module, the results analysis chart refers you to a review activity near the end of the module. Good! In this way, I can be sure I have all the prerequisites for starting. Exactly! The second section contains the learning activities. It s the main part of the module. START The starting line shows where the learning activities begin.? The little white question mark indicates the questions for which answers are given in the text. The target precedes the objective to be met. The memo pad signals a brief reminder of concepts which you have already studied. Look closely at the box to the right. It explains the symbols used to identify the various activities.? The boldface question mark indicates practice exercises which allow you to try out what you have just learned. The calculator symbol reminds you that you will need to use your calculator.? The sheaf of wheat indicates a review designed to reinforce what you have just learned. A row of sheaves near the end of the module indicates the final review, which helps you to interrelate all the learning activities in the module. FINISH Lastly, the finish line indicates that it is time to go on to the self-evaluation test to verify how well you have understood the learning activities. 0.7
There are also many fun things in this module. For example, when you see the drawing of a sage, it introduces a Did you know that... A Did you know that...? Yes, for example, short tidbits on the history of mathematics and fun puzzles. They are interesting and relieve tension at the same time. Must I memorize what the sage says? No, it s not part of the learning activity. It s just there to give you a breather. It s the same for the math whiz pages, which are designed especially for those who love math. They are so stimulating that even if you don t have to do them, you ll still want to. And the whole module has been arranged to make learning easier. For example. words in boldface italics appear in the glossary at the end of the module... Great!... statements in boxes are important points to remember, like definitions, formulas and rules. I m telling you, the format makes everything much easier. The third section contains the final review, which interrelates the different parts of the module. There is also a self-evaluation test and answer key. They tell you if you re ready for the final evaluation. Thanks, Monica, you ve been a big help. I m glad! Now, I ve got to run. See you! Later... This is great! I never thought that I would like mathematics as much as this! 0.8
GENERAL INTRODUCTION In this module, we will look at factoring and the four operations (+,,, ) on algebraic fractions. In the first part of the module, you will learn the five factoring methods: 1. factoring by removing the common factor; 2. factoring by grouping; 3. factoring trinomials of the form x 2 + bx + c or x 2 + bxy + cy 2 ; 3. factoring trinomials of the form ax 2 + bx + c or ax 2 + bxy + cy 2 ; 5. factoring differences of squares. To factor a polynomial is to write the polynomial as a product of two or more polynomials. In other words, to factor a polynomial is to find the factors of the polynomial. Each of these methods will be examined in a separate unit. Factoring is a precious mathematical tool for solving second-degree equations, i.e., equations in which the highest exponent is 2. Unfortunately, certain polynomials are not factorable. To solve equations containing this type of polynomial, it is necessary to resort to more advanced techniques which will be covered in a subsequent module. In the second part of the module, you will learn how to perform various operations on algebraic fractions. You will first learn how to simplify them (factoring the numerator and the denominator). It is important to master this skill before going on, for you will use it in all the units in the second part of the module. Indeed, all your results will have to be reduced to lowest terms. 0.9
In the units that follow the unit on simplification, you will learn how to multiply, divide, add and subtract algebraic fractions. What you have already learned about the operations on numerical fractions will help you a great deal here. Furthermore, a keen sense of observation, order and method will definitely come in handy. These are the main concepts that will be covered in this module on factoring and the four operations on algebraic fractions. 0.10
INTERMEDIATE AND TERMINAL OBJECTIVES OF THE MODULE Module MTH-4106-1 consists of nine units and requires 25 hours of study distributed as shown below. The terminal objectives appear in boldface. Objectives Number of hours** % (evaluation) 1 to 6 11 35% 7 to 9 13 65% * One hour is allotted for the final evaluation. 1. Factoring by removing the common factor Find the common factor of all the terms of a polynomial containing up to six terms linked by + or signs. The result must be expressed as the product of a monomial and a polynomial, which is placed in parentheses. The numerical coefficients of the terms of the polynomial are rational numbers, and the exponents of the variables are natural numbers. 2. Factoring by grouping Factor a polynomial of up to six terms linked by + or signs by applying the method of grouping. The result must be expressed as the product of two binomials or as the product of a binomial and a trinomial. The terms of the polynomial may have to be rearranged before being grouped and factored. The numerical coefficients of the terms of the polynomial are rational numbers, and the exponents of the variables are natural numbers. The steps in the solution must be shown. 0.11
3. Factoring trinomials of the form x 2 + bx + c or x 2 + bxy + cy 2 Factor a trinomial of the form x 2 + bx + c or x 2 + bxy + cy 2, where b and c are integers. The result must be expressed as the product of two binomials of the form (x + d)(x + e) or (x + dy)(x + ey), where d and e are integers. The steps in the solution must be shown. 4. Factoring trinomials of the form ax 2 + bx + c or ax 2 + bxy + cy 2 Factor a trinomial of the form ax 2 + bx + c or ax 2 + bxy + cy 2, where a, b and c are integers. The result must be expressed as the product of two binomials of the form (kx + l)(mx + n) or (kx + ly)(mx + ny), where k, l, m and n are integers. The steps in the solution must be shown. 5. Factoring differences of two squares Factor the difference of two squares as the product of two binomials consisting of the sum and the difference of the square roots of each term of the initial algebraic expression. The difference of squares is of the form (ax 2n by 2m ), where a and b are squares of rational numbers, x and y are variables, and n and m are natural numbers equal to or greater than 1 and less than or equal to 4. 6. Factoring polynomials Factor a polynomial containing up to six terms as the product of no more than three prime factors by removing the common factor and applying one other factoring method selected from the list below: factoring by grouping; factoring trinomials of the form x 2 + bx + c or x 2 + bxy + cy 2 ; factoring trinomials of the form ax 2 + bx + c or ax 2 + bxy + cy 2 ; factoring differences of squares. The steps in the solution must be shown. 0.12
7. Simplifying algebraic fractions To simplify a rational algebraic fraction whose numerator and denominator are factorable polynomials containing up to three terms each. Each term contains no more than two variables. The operation must be factored a maximum of four times, including no more than two per polynomial. If a polynomial must be factored twice, one of the factoring methods must involve removing the common factor. The steps in the solution must be shown. 8. Multiplying and dividing two algebraic fractions Find the product and quotient of two rational algebraic fractions. The polynomials in the numerators and denominators are factorable and contain at most three terms. Each term contains no more than two variables. The solution must be factored a maximum of four times, including no more than two per polynomial. If a polynomial must be factored twice, one of the factoring methods must involve removing the common factor. The product must be reduced to lowest terms and the steps in the solution must be shown. 9. Adding and subtracting two algebraic fractions and comparing algebraic expressions Reduce to lowest terms an algebraic expression containing two rational algebraic fractions joined by addition or subtraction. The numerators and the denominators are factorable or non-factorable polynomials, containing at most three terms. Each term contains no more than two variables. If a polynomial must be factored twice, one of the factoring methods must involve removing the common factor. The common denominator must contain at most two binomials and one monomial. The steps in the solution must be shown. 0.13
Determine the equivalence of algebraic expressions by reducing them to lowest terms. The expressions are made up by the sum or difference of two algebraic fractions. The polynomials in the numerators and denominators contain at most three terms. Each term contains no more than two variables. 0.14
DIAGNOSTIC TEST ON THE PREREQUISITES Instructions 1. Answer as many questions as you can. 2. Do not use your calculator. 3. Write your answers on the test paper. 4. Do not waste any time. If you cannot answer a question, go on to the next one immediately. 5. When you have answered as many questions as you can, correct your answers using the answer key which follows the diagnostic test. 6. To be considered correct, your answers must be identical to those in the answer key. In addition, the various steps in your solution should be equivalent to those shown in the answer key. 7. Copy your results onto the chart which follows the answer key. This chart gives an analysis of the diagnostic test results. 8. Do the review activities that apply to your incorrect answers. 9. If all your answers are correct, you may begin working on this module. 0.15
1. Find all the factors of the following numbers. a) 24:... b) 54:... c) 100:... 2. Reduce the following fractions to lowest terms. a) c) 35 75 = b) 81 27 = 12 64 = 3. Perform the following multiplications and divisions. Your results should be reduced to lowest terms. a) 8 14 7 12 =... b) 14 15 5 8 =... c) 6 11 5 22 =... d) 2 9 2 3 =... 4. Perform the following additions and subtractions. Your results should be reduced to lowest terms. a) b) c) d) 5 12 + 21 8 =... 3 8 + 7 32 =... 18 7 2 3 =... 5 6 7 15 =... 0.16
5. Perform the following operations. a) 4a 2 b + 6ab 3a 2 b a 2 b + 5ab =... b) (7yz + 2z 3y) (4z 3yz + y) =... c) 3cd(4d 8c 2 + cd 2 2) =... d) m 4 n 2 3 2m2 5 + mn2 2 =... e) (2u + 3)(u 4) =... f) (20p 3 q 2 12p 2 q 3 4p 3 q) 4pq =... g) (3s + 4) 2 =... h) 2r 2 t 2 3 r 2 + t 4 5rt 7 3rt2 4 =... 0.17
ANSWER KEY FOR THE DIAGNOSTIC TEST ON THE PREREQUISITES 1. a) The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. b) The factors of 54 are 1, 2, 3, 6, 9, 18, 27 and 54. c) The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100. 2. a) 35 75 = 35 5 75 5 = 7 15 b) 12 64 = 12 4 64 4 = 3 16 c) 81 27 = 81 27 27 27 = 3 1 = 3 3. a) 1 2 8 14 7 12 = 1 3 2 1 1 3 b) 7 14 15 5 8 = 7 3 1 4 = 7 12 3 1 4 c) 6 11 5 22 = 6 11 22 5 = 6 1 2 5 = 12 5 1 2 d) 1 2 9 2 3 = 2 9 3 2 = 1 3 3 1 1 4. a) 5 12 + 21 8 = 10 24 + 63 24 = 73 24 b) 3 8 + 7 32 = 12 32 + 7 32 = 19 32 c) 18 7 2 3 = 54 21 14 21 = 40 21 d) 5 6 7 15 = 25 30 14 30 = 11 30 5. a) 4a 2 b + 6ab 3a 2 b a 2 b + 5ab = 11ab b) (7yz + 2z 3y) (4z 3yz + y) = 7yz + 2z 3y 4z + 3yz y = 10yz 4y 2z 0.19
c) 3cd(4d 8c 2 + cd 2 2) = 12cd 2 24c 3 d + 3c 2 d 3 6cd d) m 4 n 2 3 2m2 5 + mn2 2 = mn2 12 m3 10 + m2 n 2 8 e) (2u + 3)(u 4) = 2u 2 8u + 3u 12 = 2u 2 5u 12 f) (20p 3 q 2 12p 2 q 3 4p 3 q) 4pq = 5p 2 q 3pq 2 p 2 g) (3s + 4) 2 = (3s + 4)(3s + 4) = 9s 2 + 12s + 12s + 16 = 9s 2 + 24s + 16 h) 2r 2 t 2 3 r 2 + t 4 5rt 7 3rt2 4 = r 3 t 2 3 + r 2 t 3 6 10r 3 t 3 21 3rt2 4 = r 3 t 2 3 + r 2 t 3 6 10r 3 t 3 21 4 = 4r 2 3rt 2 9 2rt 9 + 40r 2 t 63 0.20
ANALYSIS OF THE DIAGNOSTIC TEST RESULTS Questions Answers Review Before going on to Corrects Incorrects Section Page unit 1.a) 11.1 11.4 1 b) 11.1 11.4 1 c) 11.1 11.4 1 2.a) 11.2 11.9 7 b) 11.2 11.9 7 c) 11.2 11.9 7 3.a) 11.3 11.12 8 b) 11.3 11.12 8 c) 11.3 11.12 8 d) 11.3 11.12 8 4.a) 11.4 11.17 9 b) 11.4 11.17 9 c) 11.4 11.17 9 d) 11.4 11.17 9 5.a) 11.5 11.25 1 b) 11.5 11.25 1 c) 11.5 11.25 1 d) 11.5 11.25 1 e) 11.5 11.25 1 f) 11.5 11.25 1 g) 11.5 11.25 1 h) 11.5 11.25 1 If all your answers are correct, you may begin working on this module. For each incorrect answer, find the related section listed in the Review column. Complete this section before beginning the unit listed in the righthand column under the heading Before going on to unit. 0.21
INFORMATION FOR DISTANCE EDUCATION STUDENTS You now have the learning material for MTH-4106-1 and the relevant homework assignments. Enclosed with this package is a letter of introduction from your tutor, indicating the various ways in which you can communicate with him or her (e.g. by letter or telephone), as well as the times when he or she is available. Your tutor will correct your work and help you with your studies. Do not hesitate to make use of his or her services if you have any questions. DEVELOPING EFFECTIVE STUDY HABITS Learning by correspondence is a process which offers considerable flexibility, but which also requires active involvement on your part. It demands regular study and sustained effort. Efficient study habits will simplify your task. To ensure effective and continuous progress in your studies, it is strongly recommended that you: draw up a study timetable that takes your work habits into account and is compatible with your leisure and other activities; develop a habit of regular and concentrated study. 0.23
The following guidelines concerning theory, examples, exercises and assignments are designed to help you succeed in this mathematics course. Theory To make sure you grasp the theoretical concepts thoroughly: 1. Read the lesson carefully and underline the important points. 2. Memorize the definitions, formulas and procedures used to solve a given problem; this will make the lesson much easier to understand. 3. At the end of the assignment, make a note of any points that you do not understand using the sheets provided for this purpose. Your tutor will then be able to give you pertinent explanations. 4. Try to continue studying even if you run into a problem. However, if a major difficulty hinders your progress, contact your tutor before handing in your assignment, using the procedures outlined in the letter of introduction. Examples The examples given throughout the course are applications of the theory you are studying. They illustrate the steps involved in doing the exercises. Carefully study the solutions given in the examples and redo the examples yourself before starting the exercises. 0.24
Exercises The exercises in each unit are generally modeled on the examples provided. Here are a few suggestions to help you complete these exercises. 1. Write up your solutions, using the examples in the unit as models. It is important not to refer to the answer key found on the coloured pages at the back of the module until you have completed the exercises. 2. Compare your solutions with those in the answer key only after having done all the exercises. Careful! Examine the steps in your solutions carefully, even if your answers are correct. 3. If you find a mistake in your answer or solution, review the concepts that you did not understand, as well as the pertinent examples. Then redo the exercise. 4. Make sure you have successfully completed all the exercises in a unit before moving on to the next one. Homework Assignments Module MTH-4106-1 comprises three homework assignments. The first page of each assignment indicates the units to which the questions refer. The assignments are designed to evaluate how well you have understood the material studied. They also provide a means of communicating with your tutor. When you have understood the material and have successfully completed the pertinent exercises, do the corresponding assignment right away. Here are a few suggestions: 0.25
1. Do a rough draft first, and then, if necessary, revise your solutions before writing out a clean copy of your answer. 2. Copy out your final answers or solutions in the blank spaces of the document to be sent to your tutor. It is best to use a pencil. 3. Include a clear and detailed solution with the answer if the problem involves several steps. 4. Mail only one homework assignment at a time. After correcting the assignment, your tutor will return it to you. In the section Student s Questions, write any questions which you wish to have answered by your tutor. He or she will give you advice and guide you in your studies, if necessary. In this course Homework Assignment 1 is based on units 1 and 6. Homework Assignment 2 is based on units 7 to 9. Homework Assignment 3 is based on units 1 to 9. CERTIFICATION When you have completed all your work, and provided you have maintained an average of at least 60%, you will be eligible to write the examination for this course. 0.26
START UNIT 1 FACTORING BY REMOVING THE COMMON FACTOR 1.1 SETTING THE CONTEXT It s Common Knowledge! Cindy and Jeff have just finished solving a mathematical problem. When they compare results, they find that Jeff obtained the algebraic expression 4x 5 + 12x 4 + 8x 3 for an answer, while Cindy got 4x 3 (x 2 + 3x + 2). At first glance, these results seem different. These two algebraic expressions are, however, equivalent. In other words, they have the same value. The expression 4x 5 + 12x 4 + 8x 3 is a polynomial. Because this expression contains three terms, it is called a trinomial. 1.1
A polynomial is an algebraic expression made up of one or more terms linked by plus and/or minus signs. If this expression contains only one term, it is called a monomial. If it contains two terms, it is a binomial; and if it is made up of three terms, it is a trinomial. What relationship can we find between the expressions 4x 5 + 12x 4 + 8x 3 and 4x 3 (x 2 + 3x + 2)? To answer this question, we have to factor a polynomial. Factoring a polynomial means writing it in the form of the product of two or more polynomials. The algebraic expression 4x 3 (x 2 + 3x + 2) that Cindy obtained is the product of the monomial 4x 3 and the trinomial x 2 + 3x + 2. This expression is equivalent to Jeff s answer because it is obtained by factoring the polynomial 4x 5 + 12x 4 + 8x 3. How this is done is what this unit is all about. To reach the objective of this unit, you should be able to factor polynomials of up to six terms by removing the common factor. There are several methods of factoring. The method that can be used to solve Cindy and Jeff s problem is called removing the common factor. To apply it, we must first find the greatest common factor of all the terms of the polynomial to be factored. 1.2
A common factor, also known as a common divisor, is a number or term that can divide several numbers or terms without a remainder. For example, 5 is a common factor of 10 and 15 because 5 divides 10 and 15 without a remainder (10 5 = 2 and 15 5 = 3); 3x is a common factor of 6x and 9x 2 because 3x divides 6x and 9x 2 2 without a remainder 6x =2et 9x 3x 3x =3x Example 1 Find the greatest common factor of the binomial 3x 2 + 6xy. 1. Find the greatest common factor of the numerical coefficients of both terms of the binomial: the factors of 3 are 1, 3 ; the factors of 6 are 1, 2, 3, 6. The greatest common factor of the numerical coefficients is 3. The numerical coefficient is the number that multiplies the variable or variables of a term. It is the numerical part of an expression. Thus, the numerical coefficients of the expressions 5x 3 ; 2y 5 ; 1 3 ab 2 ; 0.5a 3 b 5 c 4 are, respectively, 5; 2; 1 and 0.5. 3 2. Find the greatest common factor of the algebraic part of both terms of the binomial. To do this: a) find the variable or variables common to both terms: the variable x is common to both terms; 1.3
b) assign each common variable the smallest exponent in the original polynomial: in the binomial 3x 2 + 6xy, 1 is the smallest exponent of the variable x. The greatest common factor of the algebraic part is x. 3. Multiply each of the common elements: 3 x = 3x. The greatest common factor of the binomial 3x 2 + 6xy is 3x. Example 2 Find the greatest common factor of the trinomial 10x 4 y 3 + 4x 3 y 2x 2 y 2. 1. Find the greatest common factor of the numerical coefficients of all the terms of the trinomial: the factors of 10 are 1, 2, 5, 10; the factors of 4 are 1, 2, 4; the factors of 2 are 1, 2. The greatest common factor of the numerical coefficients of the trinomial is 2. 2. Find the greatest common factor of the algebraic part of all the terms in the trinomial. To do this: a) find the variable or variables common to the three terms: the variables x and y are common to all three terms; 1.4
b) assign each common variable the smallest exponent in the original polynomial: in the trinomial 10x 4 y 3 + 4x 3 y 2x 2 y 2, 2 is the smallest exponent of the variable x and 1 is the smallest exponent of the variable y. The greatest common factor of the algebraic part is x 2 y. 3. Multiply each of the common elements: 2 x 2 y = 2x 2 y. The greatest common factor of the expression 10x 4 y 3 + 4x 3 y 2x 2 y 2 is 2x 2 y. It s as simple as that! Now let s summarize the steps involved in finding the greatest common factor of all the terms of a polynomial. To find the greatest common factor of the terms of a polynomial: 1. Find the greatest common factor of the numerical coefficients of all the terms of the polynomial. 2. Find the greatest common factor of the algebraic part of all the terms of the polynomial by: a) identifying the variable(s) common to all the terms of the polynomial, b) assigning each common variable the smallest exponent in the original polynomial. 3. Multiply each of the common elements. 1.5
? What is the greatest common factor of the expression 4a 3 b 2 c 4 8a 2 b 3 c 3 + 6b 3 c 12a 5 b 2 c 4? 1.... 2.... 3.... If your answer was 2b 2 c, good for you! If not, read the following solution carefully and redo the exercise. 1. The factors of 4 are 1, 2, 4; The factors of 8 are 1, 2, 4, 8; The factors of 6 are 1, 2, 3, 6; The factors of 12 are 1, 2, 3, 4, 6, 12. The greatest common factor of the numerical coefficients is 2. 2. a) The variables common to all the terms of the polynomial are b and c; b) 2 is the smallest exponent of the variable b and 1 is the smallest exponent of the variable c. The greatest common factor of the algebraic part is b 2 c. 3. The greatest common factor of the algebraic expression 4a 3 b 2 c 4 8a 2 b 3 c 3 + 6b 3 c 12a 5 b 2 c 4 is 2 b 2 c = 2b 2 c. Let s do some more exercises of this type. Knowing how to find the greatest common factor of a polynomial is essential to understanding the upcoming concepts. 1.6
Exercise 1.1 Find the greatest common factor of the following polynomials. 1. 8x 3 + 12x 4 + 4x 5 1.... 2.... 3.... 2. 6a 2 bc + 12a 2 b 3 c 2 18a 2 b 2 1.... 2.... 3.... 3. 16x 3 y + 12x 3 y 2 8x 2 y 4 + 20x 2 y 2 1.... 2.... 3.... 4. 2m 2 np 3mn 2 p 2 + 5m 3 n 3 p 1.... 2.... 3.... 5. 3k 3 l 4 6k 2 l 3 + 18k 4 l 2 12k 5 l 3 + 3k 3 l 2 1.... 2.... 3.... 1.7
Having completed these mental gymnastics, you can now solve Cindy and Jeff s problem, using the method of factoring by removing the common factor. The example below shows you the procedure to follow. Example 3 Factor the trinomial 4x 5 + 12x 4 + 8x 3 obtained by Jeff. 1. Find the greatest common factor of all the terms of the trinomial: the greatest common factor of the numerical coefficients of all the terms is 4; the greatest common factor of the algebraic part of all the terms is x 3. The greatest common factor of the trinomial 4x 5 + 12x 4 + 8x 3 is 4x 3. 2. Divide each term of the trinomial by this common factor. 1 4x 5 4x + 12x4 + 8x3 3 4x 3 4x = 3 x5 3 + 3x 4 3 + 2x 3 3 = x 2 + 3x + 2. 1 3 2 1 1 To perform the division of two monomials, divide the numerical coefficients of the two monomials and subtract the exponents of the same variable: a m = a a n m n. Any variable raised to the power of 0 is equal to 1: a 0 = 1. 3. Put the new trinomial in parentheses and write the greatest common factor in front of the parentheses. We thereby remove the common factor: 4x 3 (x 2 + 3x + 2). 1.8
4. Check the answer. To do this, multiply each of the terms in parentheses by the monomial in front of the parentheses. 4x 3 (x 2 + 3x + 2) = 4x 3 (x 2 ) + 4x 3 (3x) + 4x 3 (2) = 4x 3 + 2 + 12x 3 + 1 + 8x 3 = 4x 5 + 12x 4 + 8x 3 Since the polynomial obtained is equal to the original polynomial, the polynomial has been factored. Thus: 4x 3 (x 2 + 3x + 2) = 4x 5 + 12x 4 + 8x 3 To multiply a monomial by a polynomial, multiply the numerical coefficient of the monomial by each numerical coefficient in the polynomial and add the exponents of the same variable: a m a n = a m + n. Checking your answer is an important step, since it allows you to make sure that you performed the division in Step 2 correctly. It also allows you to identify incorrect signs or an incorrect greatest common factor. It does not, however, allow you to determine with certainty that you correctly identified the greatest common factor. For example, if in Example 3, you found 4x 2 in Step 1, you will obtain 4x 2 (x 3 + 3x 2 + 2x) = 4x 5 + 12x 4 + 8x 3, which appears to be correct. However, the initial expression has not been factored completely since, in the parentheses, the factor x is still common to the three terms of the polynomial. It is therefore crucial to make sure that you correctly determined the greatest common factor of all the terms of the polynomial from which you want to remove the common factor. 1.9
Jeff and Cindy were therefore both right. Cindy arrived at her solution by factoring the expression that Jeff obtained. Let s go on to another example. Example 4 Factor the following polynomial by removing the common factor: 3m 3 n 7m 3 r + 8m 3 rt 1. Find the greatest common factor of all the terms of the polynomial: the greatest common factor of the numerical coefficients of all the terms is 1; the greatest common factor of the algebraic part is m 3. The greatest common factor of the polynomial is m 3. 2. Divide each term of the polynomial by this common factor. 3m 3 n m 3 7m3 r m 3 + 8m3 rt m 3 = 3m 3 3 n 7m 3 3 r + 8m 3 3 rt = 3n 7r + 8rt 3. Put the new polynomial in parentheses and write the greatest common factor in front of the parentheses. m 3 ( 3n 7r + 8rt) 4. Check the answer: m 3 ( 3n 7r + 8rt ) = 3m 3 n 7m 3 r + 8m 3 rt Since the product is equal to the original polynomial, the polynomial has been factored. Thus: 3m 3 n 7m 3 r + 8m 3 rt = m 3 ( 3n 7r + 8rt) 1.10
N.B. In the previous example, we can also remove the factor m 3. In such a case, it is important to pay attention to the signs joining each term of the trinomial in parentheses. After factoring, we obtain m 3 (3n + 7r 8rt) because: m 3 (3n + 7r 8rt) = 3m 3 n 7m 3 r + 8m 3 rt Law of signs for multiplication or division + times + = + times = + + times = times + = Factoring by removing the common factor is a snap! Let s summarize the steps involved in applying this method. To factor a polynomial by removing the common factor: 1. Find the greatest common factor of all the terms of the polynomial. 2. Divide each term of the polynomial by this common factor. 3. Put the new polynomial in parentheses and write the greatest common factor in front of the parentheses. 4. Check the answer by multiplying the isolated factor by each of the terms of the polynomial in parentheses. 1.11
? Factor the polynomial 2ab 3 4b 3 c 12b 3 d by removing the common factor. 1.... 2.... 3.... 4.... To factor this polynomial, you must remove the common factor 2b 3 or the common factor 2b 3. 1. The greatest common factor of all the terms of the polynomial is 2b 3 or 2b 3. 2. 1 2ab 3 4b3 c 2b 3 2b 12b3 d = a + 2c + 6d 3 2b 3 1 2 6 1 1 1 2ab3 2b 3 1 2 4b3 c 2b 3 1 or 6 12b3 d = a 2c 6d 2b 3 1 3. 2b 3 (a + 2c + 6d) or 2b 3 ( a 2c 6d) 4. 2b 3 (a + 2c + 6d) = 2ab 3 4b 3 c 12b 3 d or 2b 3 ( a 2c 6d) = 2ab 3 4b 3 c 12b 3 d 1.12
Since the product is equal to the original polynomial, the polynomial has been factored completely. Thus: 2b 3 (a + 2c + 6d) = 2ab 3 4b 3 c 12b 3 d or 2b 3 ( a 2c 6d) = 2ab 3 4b 3 c 12b 3 d N.B. If the first term of the polynomial to be factored is negative, it is preferable to remove a common factor with a negative sign. To become an ace at factoring, nothing beats practice! The following exercises will help you improve your skills. Exercise 1.2 Factor the following polynomials by removing the common factor, following the steps described above. 1. a 3 ax 1.... 2.... 3.... 4.... 1.13
2. 5ab 5a 3 b 2 1.... 2.... 3.... 4.... 3. a 2 bc + ab 2 c + abc 2 1.... 2.... 3.... 4.... 4. 18a 3 24a 3 b + 12ab 2 1.... 2.... 3.... 4.... 1.14
5. 12x 3 y 2 8x 2 y 3 4xy 1.... 2.... 3.... 4.... 6. 8m 3 n 3 12m 2 n 2 p 5 + 20m 5 np 4 1.... 2.... 3.... 4.... 7. 11a 2 x 3 y 22b 2 x 2 y 2 + 33c 2 x 2 yz 1.... 2.... 3.... 4.... 1.15
8. 12a 2 x 3 y 2 + 9ax 2 y 3 15a 4 x 4 y 5 1.... 2.... 3.... 4.... 9. h 2 k k 2 k 3 1.... 2.... 3.... 4.... 10. 7r 2 st + 14rs 2 t 2 u 21r 2 s 2 t 2 7r 3 s 2 t 1.... 2.... 3.... 4.... 1.16
To factor a polynomial by removing the common factor, you simply have to know how to find the greatest common factor of a polynomial and to divide two monomials. But watch the signs when you remove a negative factor! Checking your answer becomes very important in this instance, for it allows you to make sure that the sign of each term obtained after multiplying is the same as in the original polynomial. Now before going on to the practice exercises, let s have some fun! 1.17
Did you know that... in the following addition exercise, if you replace each of the letters A, B, C, D, E, F, G with a number from 1 to 7, you can get the sum of 9 999 999? Careful! Each letter corresponds to a specific number. 2 F C 8 E E 0 D 5 9 D 4 9 A G D 6 1 A E G + B A 7 C G C 3 9 9 9 9 9 9 9 Solution 2 5 6 8 7 7 0 3 5 9 3 4 9 4 2 3 6 1 4 7 2 +14 7 6 26 3 9 9 9 9 9 9 9 A = 4, B = 1, C = 6, D = 3, E = 7, F = 5 et G = 2 1.18
? 1.2 PRACTICE EXERCISES Factor the following polynomials by removing the common factor. 1. x 2 y 3 x 2 y 2 + x 2 y = 2. 10x 3 25x 4 y = 3. 16c + 64c 2 d = 4. 6a 2 b 3 + 14a 4 b 3 c = 5. 38x 3 y 5 + 57x 4 y 2 = 1.19
6. 1 2 m2 n + 1 4 mn2 = 7. 5ax 5 10a 2 x 3 15a 3 x 3 = 8. 48a 3 b 2 c + 24a 3 bc 3 16ab 3 c 3 + 32ab 2 c 4 = 9. 3a 4 b 2 3a 3 b + 6a 2 b 9ab 3 + 9a 3 b 2 12a 2 b = 10. 5.2m 3 n 2 + 10.4m 3 n 3 + 15.6m 2 n 3 = 1.20
11. 12mx 4 y 3 18nx 3 y 21x 2 y 4 + 6x 2 yz = 12. 9b 2 81b = 13. 8x 3 yz 3 12x 2 y 3 z 5 + 20x 5 yz 4 = 14. 15x 3 y 12x 4 y 3 z + 7xy 2 = 15. 8m 2 n 3 4m 4 n 2 16m 3 n 2 = 1.21
1.3 REVIEW EXERCISES 1. Complete the following sentences by filling in the blanks with the missing term or expression. To find the greatest common factor of the terms of a polynomial, find the greatest common factor of the...... of all the... of the polynomial. Then find the greatest common factor of the... part of all the terms of the polynomial. To do this, identify the...(s) common to all the terms of the polynomial and assign each of them the... exponent in the original polynomial. Lastly, multiply each of the... elements. 2. Explain in your own words what factoring a polynomial means....... 3. List the four steps in factoring a polynomial by removing the common factor................... 1.22
1.4 THE MATH WHIZ PAGE Uncommon Common Factors The greatest common factor of a polynomial may be a binomial. For example, in the expression 3x(2a + b) 5y(2a + b), the binomial (2a + b) is the common factor of both terms of the algebraic expression. To factor this type of expression, you need only follow the procedure shown in this unit. Thus, by dividing each term of the algebraic expression by the common factor, we get: 3x(2a + b) (2a + b) 5y(2a + b) (2a + b) = 3x 5y You then simply put the new polynomial in parentheses and write the common factor in front of the parentheses. The result of the factoring is therefore: (2a + b)(3x 5y) Following the same reasoning, factor the algebraic expressions shown below. Careful! In some cases, one of the two terms obtained may need to be factored a second time. 1.23
1. y(y 1) + 2(y 1) = 2. (a 5)a 3(a 5) = 3. 5x(a + b) + 15y(a + b) 10z(a + b) = 4. 6x 2 (3a b) 15x(3a b) = 1.24