Download Section 01

10-1: Factors and Greatest Common Factors
OBJECTIVES:
You must find prime factorizations of integers and find greatest common
factors (GCF) for sets of monomials.
Factoring is the opposite of multiplying.
A way to think of it is algebraic dividing with special rules.
To factor monomials, you will need to be able to find all the factor pairs
that go into the number.
EXAMPLE 1: Find the factors of 72.
To find all the factors of a number, start with one and work your way upwards, listing all numbers that go
evenly into 72 along with the remainder. Stop when the remainder is less than the original number.
1 x 72
2 x 36
3 x 24
4 x 18
5 x 14.4
14.4 is not a
whole number
6 x 12
7 x 10.28...
8x9
10.28 is not a
whole number
© William James Calhoun, 2001
10-1: Factors and Greatest Common Factors
10.1.1: Definition of Prime and Composite Numbers
A Prime number is a whole number, greater than 1, whose only factors are 1 and
itself. A composite number is a whole number, greater than 1, that is not prime.
Using these definitions, you can find the prime factorization of any
number.
EXAMPLE 2: Find the prime factorization of 140.
Quick rules to remember:
140
Even numbers can be divided by 2.
Numbers ending 5 or 0 can be divided by 5.
If you add all digits of the number and that
sum is divisible by 3 or 9, the number is
divisible by 3 or 9, respectively.
Start breaking the number down into its
factors using a factor tree.
2
2
2
70
2
2
35
5
7
Now, rewrite the factors in increasing order using
exponents to represent multiple prime factors and dots
for the multiply signs.
22 · 5 · 7
© William James Calhoun, 2001
10-1: Factors and Greatest Common Factors
If the number to be prime factored is negative, automatically take out a
“-1”.
EXAMPLE 3: Factor -150 completely.
-150
-1
150
-1 2
-1
-1
2
2
75
5
5
15
5
3
Now, rewrite the factors in increasing order using
exponents to represent multiple prime factors and dots
for the multiply signs.
-1 · 2 · 3 · 52
© William James Calhoun, 2001
10-1: Factors and Greatest Common Factors
If you are asked to factor a monomial with variables in it, break down the
number and all the letters. The answer should be left in expanded form.
EXAMPLE 4: Factor 45x3y2.
3 15
3
3
5 x x xy y
Now just write everything out without exponents.
Still make sure the numbers are in increasing order.
3·3·5·x·x·x·y·y
© William James Calhoun, 2001
10-1: Factors and Greatest Common Factors
10.1.2: Definition of Greatest Common Factor
The greatest common factor (GCF) of two or more integers is the
greatest number that is a factor of all of the integers.
To find the GCF, you find the biggest number that will go into the set of
numbers you are given.
An easy way to do this is to find the prime factorization of each number and find all the
factors the numbers have in common and multiply them.
EXAMPLE 5: Find the GCF of 54, 63, and 180.
Break 54 down into expanded form.
54 = 2 · 3 · 3 · 3
Break 63 down into expanded form.
Make a list of all the factors that are shared by each number.
3 x 3
Multiply the shared factors to find the GCF.
63 = 3 · 3 · 7
Break 180 down into expanded form.
The GCF of 54, 63, and 180 is 9.
54 = 2 · 2 · 3 · 3 · 5
© William James Calhoun, 2001
10-1: Factors and Greatest Common Factors
To find the GCF of monomials with variables, use the same process to
find the numeric GCF, then find the letters that are shared by each
monomial.
EXAMPLE 6: Find the GCF of 12a2b and 90a2b2c.
Break 12a2b down into expanded form.
12a2b = 2 · 2 · 3 · a · a · b
Break 90a2b2c down into expanded form.
90a2b2c = 2 · 3 · 3 · 5 · a · a · b · b · c
Make a list of all the factors that are shared by each number.
2 x 3x a x a x b
Multiply the shared factors to find the GCF.
The GCF of 12a2b and 90a2b2c is 6a2b.
© William James Calhoun, 2001
10-1: Factors and Greatest Common Factors
HOMEWORK
Page 561
#19 - 47 odd
© William James Calhoun, 2001