MTH-4106-1 C1-C4 Factorization
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F
MTH-4106-1
actoring
and
Algebraic Fractions
MTH-4106-1
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
Answer Key
MTH-4106-1
Factoring and Algebraic Fractions
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 x2 + bx + c
or x2 + bxy + cy2 ......................................................................................... 3.1
4. Factoring Trinomials of the Form ax2 + bx + c
or ax2 + bxy + cy2 ....................................................................................... 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
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MTH-4106-1
Factoring and Algebraic Fractions
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!
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MTH-4106-1
Factoring and Algebraic Fractions
PROGRAM FLOWCHART
CEGEP
MTH-5112-1
MTH-5111-2
MTH-536
MTH-5104-1
MTH-5103-1
Optimization II
MTH-5106-1
Real Functions and Equations
Probability II
MTH-5105-1
Conics
Statistics III
Optimization I
MTH-436
MTH-4110-1
The Four Operations on
Algebraic Fractions
Sets, Relations and Functions
MTH-4108-1
Quadratic Functions
MTH-4107-1
Straight Lines II
MTH-4106-1
Factoring and Algebraic Functions
MTH-4105-1
Exponents and Radicals
MTH-4103-1
MTH-4102-1
MTH-4101-2
Complement and Synthesis I
MTH-4109-1
MTH-4104-2
© SOFAD
Geometry IV
MTH-5107-1
MTH-426
MTH-116
MTH-5109-1
MTH-4111-2
Trades
DVS
MTH-216
Introduction to Vectors
Trigonometric Functions and Equations
MTH-5101-1
MTH-314
MTH-5110-1
Exponential and Logarithmic Functions
and Equations
MTH-5102-1
MTH-416
Complement and Synthesis II
MTH-5108-1
MTH-526
MTH-514
Logic
You ar e h er e
Statistics II
Trigonometry I
Geometry III
Equations and Inequalities II
MTH-3003-2
Straight Lines I
MTH-3002-2
Geometry II
MTH-3001-2
The Four Operations on Polynomials
MTH-2008-2
Statistics and Probabilities I
MTH-2007-2
Geometry I
MTH-2006-2
Equations and Inequalities I
MTH-1007-2
Decimals and Percent
MTH-1006-2
The Four Operations on Fractions
MTH-1005-2
The Four Operations on Integers
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25 hours
= 1 credit
50 hours
= 2 credits
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MTH-4106-1
Factoring and Algebraic Fractions
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?
Whether you are
registered at an
adult education
center or pursuing distance
education, ...
Now, the module you have in your
hand is divided into three
sections. The first section is...
You’ll see that with this method, math is
a real breeze!
I’m Andy.
... you have probably taken a
placement test which tells you
exactly which module you
should start with.
... the entry activity, which
contains the test on the
prerequisites.
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My results on the test
indicate that I should begin
with this module.
By carefully correcting this test using the
corresponding answer key, and recording your results on the analysis sheet ...
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MTH-4106-1
Factoring and Algebraic Fractions
... 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.
START
The starting line
shows where the
learning activities
begin.
Exactly! The second section
contains the learning activities. It’s
the main part of the module.
?
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.
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Factoring and Algebraic Fractions
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...”
It’s the same for the “math whiz”
pages, which are designed especially for those who love math.
For example. words in boldface italics appear in the
glossary at the end of the
module...
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.
And the whole module has
been arranged to make
learning easier.
They are so stimulating that
even if you don’t have to do
them, you’ll still want to.
... 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.
Great!
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!
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Later ...
This is great! I never thought that I would
like mathematics as much as this!
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MTH-4106-1
Factoring and Algebraic Fractions
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 x2 + bx + c or x2 + bxy + cy2;
3. factoring trinomials of the form ax2 + bx + c or ax2 + bxy + cy2;
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.
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MTH-4106-1
Factoring and Algebraic Fractions
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.
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MTH-4106-1
Factoring and Algebraic Fractions
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.
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Factoring and Algebraic Fractions
3. Factoring trinomials of the form x2 + bx + c or x2 + bxy + cy2
Factor a trinomial of the form x2 + bx + c or x2 + bxy + cy2, 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 ax2 + bx + c or ax2 + bxy + cy2
Factor a trinomial of the form ax2 + bx + c or ax2 + bxy + cy2, 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
(ax2n – by2m), 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 x2 + bx + c or x2 + bxy + cy2;
• factoring trinomials of the form ax2 + bx + c or ax2 + bxy + cy2;
• factoring differences of squares.
The steps in the solution must be shown.
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MTH-4106-1
Factoring and Algebraic Fractions
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.
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Factoring and Algebraic Fractions
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.
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Factoring and Algebraic Fractions
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.
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Factoring and Algebraic Fractions
1. Find all the factors of the following numbers.
a) 24: ............................................... b) 54: ..................................................
c) 100: .............................................
2. Reduce the following fractions to lowest terms.
a) 35 =
75
b) 12 =
64
c) 81 =
27
3. Perform the following multiplications and divisions. Your results should be
reduced to lowest terms.
a)
8 × 7 = .................................. b) 14 × 5 = .......................................
15 8
14 12
c)
6 ÷ 5 = .................................. d) 2 ÷ 2 = .........................................
9 3
11 22
4. Perform the following additions and subtractions. Your results should be
reduced to lowest terms.
a)
5 + 21 = ...................................................................................................
8
12
b) 3 + 7 = .....................................................................................................
8 32
c) 18 – 2 = .....................................................................................................
7
3
d) 5 – 7 = .....................................................................................................
6 15
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Factoring and Algebraic Fractions
5. Perform the following operations.
a) 4a2b + 6ab – 3a2b – a2b + 5ab = ..................................................................
b) (7yz + 2z – 3y) – (4z – 3yz + y) = .................................................................
c) 3cd(4d – 8c2 + cd2 – 2) = ..............................................................................
2
2
2
d) m n – 2m + mn
4 3
5
2
= .............................................................................
e) (2u + 3)(u – 4) = ...........................................................................................
f) (20p3q2 – 12p2q3 – 4p3q) ÷ 4pq =..................................................................
g) (3s + 4)2 = .....................................................................................................
2 2
2
h) 2r t r + t – 5rt ÷ – 3rt
2 4
3
4
7
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Factoring and Algebraic Fractions
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 = 35 ÷ 5 = 7
75 75 ÷ 5 15
b) 12 = 12 ÷ 4 = 3
64 64 ÷ 4 16
c) 81 = 81 ÷ 27 = 3 = 3
27 27 ÷ 27 1
1
2
1
7
3. a) 8 × 7 = 1
14 12 3
2
1
3
3
c) 18 – 2 = 54 – 14 = 40
3 21 21 21
7
1
d) 2 ÷ 2 = 2 × 3 = 1
9 3 9 2 3
3
1
5 + 21 = 10 + 63 = 73
8
24 24 24
12
4
1
2
c) 6 ÷ 5 = 6 × 22 = 6 × 2 = 12
11 22 11
5
1 5
5
4. a)
1
b) 14 × 5 = 7 × 1 = 7
15 8 3 4 12
1
b) 3 + 7 = 12 + 7 = 19
8 32 32 32 32
d) 5 – 7 = 25 – 14 = 11
6 15 30 30 30
5. a) 4a2b + 6ab – 3a2b – a2b + 5ab = 11ab
b) (7yz + 2z – 3y) – (4z – 3yz + y) = 7yz + 2z – 3y – 4z + 3yz – y =
10yz – 4y – 2z
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Factoring and Algebraic Fractions
c) 3cd(4d – 8c2 + cd2 – 2) = 12cd2 – 24c3d + 3c2d3 – 6cd
2
2
2
2
3
2 2
d) m n – 2m + mn = mn – m + m n
4 3
5
2
12
10
8
e) (2u + 3)(u – 4) = 2u2 – 8u + 3u – 12 = 2u2 – 5u – 12
f) (20p3q2 – 12p2q3 – 4p3q) ÷ 4pq = 5p2q – 3pq2 – p2
g) (3s + 4)2 = (3s + 4)(3s + 4) = 9s2 + 12s + 12s + 16 = 9s2 + 24s + 16
2 2
2
3 3
2
3 2
2 3
h) 2r t r + t – 5rt ÷ – 3rt = r t + r t – 10r t ÷ – 3rt
3
2 4
4
3
6
21
4
7
=
r 3t 2 + r 2t 3 – 10r 3t 3 × – 4 = – 4r 2 – 2rt + 40r 2t
3
6
21
9
63
9
3rt 2
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Factoring and Algebraic Fractions
ANALYSIS OF THE DIAGNOSTIC
TEST RESULTS
Answers
Questions Corrects Incorrects
1.a)
b)
c)
2.a)
b)
c)
3.a)
b)
c)
d)
4.a)
b)
c)
d)
5.a)
b)
c)
d)
e)
f)
g)
h)
Review
Section
Page
Before going on to
unit
11.1
11.1
11.1
11.2
11.2
11.2
11.3
11.3
11.3
11.3
11.4
11.4
11.4
11.4
11.5
11.5
11.5
11.5
11.5
11.5
11.5
11.5
11.4
11.4
11.4
11.9
11.9
11.9
11.12
11.12
11.12
11.12
11.17
11.17
11.17
11.17
11.25
11.25
11.25
11.25
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• 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”.
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MTH-4106-1
Factoring and Algebraic Fractions
INFORMATION
FOR
EDUCATION STUDENTS
DISTANCE
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.
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Factoring and Algebraic Fractions
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.
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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:
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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.
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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
4x5 + 12x4 + 8x3 for an answer, while Cindy got 4x3(x2 + 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 4x5 + 12x4 + 8x3 is a polynomial. Because this expression
contains three terms, it is called a trinomial.
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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 4x5 + 12x4 + 8x3 and
4x3(x2 + 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 4x3(x2 + 3x + 2) that Cindy obtained is the product of the
monomial 4x3 and the trinomial x2 + 3x + 2. This expression is equivalent to
Jeff ’s answer because it is obtained by factoring the polynomial 4x5 + 12x4 + 8x3.
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
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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 9x2 because 3x divides
2
6x and 9x2 without a remainder 6x = 2 et 9x = 3x
3x
3x
Example 1
Find the greatest common factor of the binomial 3x2 + 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 expres2
sions 5x 3 ; – 2y 5 ; 1 ab ; 0.5a 3b 5 c 4 are, respectively,
3
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;
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b) assign each common variable the smallest exponent in the original
polynomial:
• in the binomial 3x2 + 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 3x2 + 6xy is 3x.
Example 2
Find the greatest common factor of the trinomial 10x4y3 + 4x3y – 2x2y2.
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;
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b) assign each common variable the smallest exponent in the original
polynomial:
• in the trinomial 10x4y3 + 4x3y – 2x2y2, 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 x2y.
3. Multiply each of the common elements: 2 × x2y = 2x2y.
∴ The greatest common factor of the expression 10x4y3 + 4x3y – 2x2y2 is
2x2y.
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.
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What is the greatest common factor of the expression
4a3b2c4 – 8a2b3c3 + 6b3c – 12a5b2c4?
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
If your answer was 2b2c, 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 b2c.
3. The greatest common factor of the algebraic expression
4a3b2c4 – 8a2b3c3 + 6b3c – 12a5b2c4 is 2 × b2c = 2b2c.
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
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Exercise 1.1
Find the greatest common factor of the following polynomials.
1. 8x3 + 12x4 + 4x5
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
2. 6a2bc + 12a2b3c2 – 18a2b2
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
3. 16x3y + 12x3y2 – 8x2y4 + 20x2y2
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
4. 2m2np – 3mn2p2 + 5m3n3p
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
5. 3k3l4 – 6k2l3 + 18k4l2 – 12k5l3 + 3k3l2
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
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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 4x5 + 12x4 + 8x3 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 x3.
∴ The greatest common factor of the trinomial 4x5 + 12x4 + 8x3 is 4x3 .
2. Divide each term of the trinomial by this common factor.
1
3
2
1
1
1
4x 5 + 12x 4 + 8x 3 = x5 – 3 + 3x4 – 3 + 2x3 – 3 = x2 + 3x + 2.
4x 3
4x 3
4x 3
• 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 = am – n.
an
• Any variable raised to the power of 0 is equal to 1: a0 = 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: 4x3(x2 + 3x + 2).
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4. Check the answer.
To do this, multiply each of the terms in parentheses by the monomial in
front of the parentheses.
4x3(x2 + 3x + 2) = 4x3(x2) + 4x3(3x) + 4x3(2) = 4x3 + 2 + 12x3 + 1 + 8x3 =
4x5 + 12x4 + 8x3
Since the polynomial obtained is equal to the original polynomial, the
polynomial has been factored. Thus:
4x3(x2 + 3x + 2) = 4x5 + 12x4 + 8x3
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: am × an = am + 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 4x2 in Step 1, you will obtain
4x2(x3 + 3x2 + 2x) = 4x5 + 12x4 + 8x3, 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.
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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:
– 3m3n – 7m3r + 8m3rt
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 m3.
∴ The greatest common factor of the polynomial is m3.
2. Divide each term of the polynomial by this common factor.
– 3m3n – 7m 3r + 8m 3rt = – 3m3 – 3n – 7m3 – 3r + 8m3 – 3rt
m3
m3
m3
= – 3n – 7r + 8rt
3. Put the new polynomial in parentheses and write the greatest common
factor in front of the parentheses.
m3 (– 3n – 7r + 8rt)
4. Check the answer:
m3(– 3n – 7r + 8rt ) = – 3m3n – 7m3r + 8m3rt
Since the product is equal to the original polynomial, the polynomial has
been factored. Thus:
– 3m3n – 7m3r + 8m3rt = m3 (– 3n – 7r + 8rt)
1.10
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N.B. In the previous example, we can also remove the factor –m3. 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 – m3(3n + 7r – 8rt) because:
– m3(3n + 7r – 8rt) = – 3m3n – 7m3r + 8m3rt
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.
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Factor the polynomial – 2ab3 – 4b3c – 12b3d by removing the common factor.
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
4. .....................................................................................................................
To factor this polynomial, you must remove the common factor – 2b3 or the
common factor 2b3.
1. The greatest common factor of all the terms of the polynomial is – 2b3 or 2b3.
1
2
6
3
3
3
2. – 2ab3 – 4b c3 – 12b d
= a + 2c + 6d
3
– 2b
– 2b
– 2b
1
1
1
or
1
2
6
3
3
3
– 2ab3 – 4b 3c – 12b 3d = – a – 2c – 6d
2b
2b
2b
1
1
1
3. – 2b3(a + 2c + 6d)
or
2b3(– a – 2c – 6d)
4. – 2b3(a + 2c + 6d) = – 2ab3 – 4b3c – 12b3d
or
2b3(– a – 2c – 6d) = – 2ab3 – 4b3c – 12b3d
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Since the product is equal to the original polynomial, the polynomial has been
factored completely. Thus:
– 2b3(a + 2c + 6d) = – 2ab3 – 4b3c – 12b3d
or
2b3(– a – 2c – 6d) = – 2ab3 – 4b3c – 12b3d
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. a3 – ax
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
4. .....................................................................................................................
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2. 5ab – 5a3b2
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
4. .....................................................................................................................
3. a2bc + ab2c + abc2
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
4. .....................................................................................................................
4. 18a3 – 24a3b + 12ab2
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
4. .....................................................................................................................
1.14
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5. 12x3y2 – 8x2y3 – 4xy
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
4. .....................................................................................................................
6. 8m3n3 – 12m2n2p5 + 20m5np4
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
4. .....................................................................................................................
7. 11a2x3y – 22b2x2y2 + 33c2x2yz
1. .....................................................................................................................
2. .....................................................................................................................
3. .....................................................................................................................
4. .....................................................................................................................
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8. 12a2x3y2 + 9ax2y3 – 15a4x4y5
1. ....................................................................................................................
2. ....................................................................................................................
3. ....................................................................................................................
4. ....................................................................................................................
9. – h2k – k2 – k3
1. ....................................................................................................................
2. ....................................................................................................................
3. ....................................................................................................................
4. ....................................................................................................................
10. – 7r2st + 14rs2t2u – 21r2s2t2 – 7r3s2t
1. ....................................................................................................................
2. ....................................................................................................................
3. ....................................................................................................................
4. ....................................................................................................................
1.16
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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!
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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
3
2
+ 1
9
5
5
3
4
9
6
9
6
7
9
8
3
1
6
9
7
4
4
2
9
7
9
7
6
9
0
4
2
3
9
A = 4, B = 1, C = 6, D = 3, E = 7, F = 5 et G = 2
1.18
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PRACTICE EXERCISES
Factor the following polynomials by removing the common factor.
1. x2y3 – x2y2 + x2y =
2. 10x3 – 25x4y =
3. – 16c + 64c2d =
4. 6a2b3 + 14a4b3c =
5. 38x3y5 + 57x4y2 =
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6. 1 m 2n + 1 mn 2 =
4
2
7. 5ax5 – 10a2x3 – 15a3x3 =
8. 48a3b2c + 24a3bc3 – 16ab3c3 + 32ab2c4 =
9. 3a4b2 – 3a3b + 6a2b – 9ab3 + 9a3b2 – 12a2b =
10. 5.2m3n2 + 10.4m3n3 + 15.6m2n3 =
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11. 12mx4y3 – 18nx3y – 21x2y4 + 6x2yz =
12. – 9b2 – 81b =
13. – 8x3yz3 – 12x2y3z5 + 20x5yz4 =
14. 15x3y – 12x4y3z + 7xy2 =
15. – 8m2n3 – 4m4n2 – 16m3n2 =
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Factoring and Algebraic Fractions
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.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
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Factoring and Algebraic Fractions
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) 5y(2a + b)
–
= 3x – 5y
(2a + b)
(2a + b)
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.
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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. 6x2(3a – b) – 15x(3a – b) =
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