MATHEMATICS 905
FACTORS
CONTENTS
I.
FINDING THE GREATEST COMMON FACTOR
2
Numerical Terms
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2
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6
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9
Literal Terms
Polynomials
II.
FINDING BINOMIAL FACTORS
Trinomials
Binomials
III.
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FINDING COMPLETE FACTORIZATIONS
Second-Degree Polynomials
Higher-Degree Polynomials
Four-Term Polynomials
Word Problems
Author:
Editor-in-Chief:
Editor:
Consulting Editor:
Revision Editor:
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Arthur C. Landrey, M.A.Ed.
Richard W. Wheeler, M.A.Ed.
Robin Hintze Kreutzberg, M.B.A
Robert L. Zenor, M.A., M.S.
Alan Christopherson, M.S.
804 N. 2nd Ave. E., Rock Rapids, IA 51246-1759
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FACTORS
are called factors of that product. You will
need a solid foundation in factoring methods
for success in your subsequent study in
algebra. In this LIFEPAC you will learn to
find greatest common factors, binomial
factors, and complete factorizations of
polynomials.
In this LIFEPAC® you will continue your
study in the mathematical system known as
algebra
by
learning
an
important
procedure—factoring. You have already
learned that the answer obtained from
multiplying numbers or polynomials is called
product; these numbers or polynomials then
OBJECTIVES
Read these objectives. The objectives tell you what you will be able to do when
you have successfully completed this LIFEPAC.
When you have finished this LIFEPAC, you should be able to:
1.
2.
3.
4.
5.
Factor numerical terms and literal terms.
Find the greatest common factor of monomials.
Factor polynomials by finding the greatest common factor.
Multiply binomials mentally.
Factor trinomials of the form ax 2 + bx + c.
6.
7.
8.
9.
Factor differences of two squares.
Factor binomials and trinomials completely.
Factor four-term polynomials.
Solve verbal problems involving factoring.
Survey the LIFEPAC. Ask yourself some questions about this study. Write your
questions here.
1
I.
FINDING
THE GREATEST
COMMON FACTOR
OBJECTIVES
When you have completed this section, you
should be able to:
1. Factor numerical terms and literal terms.
2. Find the greatest common factor of
monomials.
3. Factor polynomials by finding the
greatest common factor.
Any expression has 1 and itself as factors, of course; if these numbers are its only
factors, then the expression is called prime. However, if a number or a polynomial is not
prime, you need to be able to find its other factors. In this section you will work first with
numerical terms, then with literal terms, and finally with polynomials. The key definition for
this section is the definition of the greatest common factor.
DEFINITION
The greatest common factor (GCF) of two or more expressions is the largest
value that will divide each expression exactly.
Models:
The GCF of 8 and 12 is 4 (since 4
is the largest value that will divide
both 8 and 12 exactly).
The GCF of 12 and 30 is 6.
The GCF of 8, 12, and 30 is 2.
The GCF of ab and bc is b.
The GCF of 9a, 12b, and 16c is 1.
NUMERICAL TERMS
The GCF for numerical terms may be found in two ways. All the factors may be
listed; or prime factorizations may be used.
LISTS OF FACTORS
One method for finding the greatest common factor of numerical terms is to
list all the factors (positive integral divisors) of each number and then to choose the
largest factor that is common to every list.
2
Model:
Find the GCF of 28 and 42 by listing
all the factors.
Solution:
The factors of 28 are 1, 2, 4, 7, 14,
and 28.
The factors of 42 are 1, 2, 3, 6, 7, 14,
21, and 42.
The common factors are 1, 2, 7, and
14.
∴ The GCF of 28 and 42 is 14.
List all the factors of each number.
1.1
20
1.2
100
1.3
36
1.4
23
1.5
30
Using the lists you made in Problems 1.1 through 1.5, find the GCF of
the following sets of numbers.
1.6
20 and 100
1.7
100 and 36
1.8
36 and 23
1.9
36 and 30
1.10
20, 100, 36, and 30
PRIME FACTORIZATIONS
Another method for finding the greatest common factor of numerical terms
is to obtain the prime factorization of each number and then to use the product of any
common prime factors. If no common prime factors exist, then the GCF is 1.
3
Model 1:
Find the GCF of 28 and 42 by prime factorizations.
Solution:
28 = 2 • 14
= 2 • 2 • 7
42 = 2 • 21
= 2 • 3 • 7
The common prime factors are 2 and 7.
∴ The GCF of 28 and 42 is 2 • 7 = 14.
Model 2:
Find the GCF of 72 and 90.
Solution:
72 =
=
=
=
2
2
2
2
•
•
•
•
36
2 • 18
2 • 2 • 9
2 • 2 • 3 • 3
90 = 2 • 45
= 2 • 3 • 15
= 2 • 3 • 3 • 5
The product of the common prime
factors is 2 • 3 • 3.
∴ The GCF of 72 and 90 is 18.
Factor each number into the product of primes.
1.11
210
1.14
170
1.12
27
1.15
325
1.13
198
Using the prime factorizations from Problems 1.11 - 1.15, find the GCF
of the following sets of numbers.
1.16
210 and 27
1.17
27 and 198
1.18
210 and 170
4