4-4: (GCF)
Greatest Common Factor
And
4-3 review of Factoring
What you will learn…
 To find prime factorizations of monomials.
 To find the greatest common factors of
monomials.
… and why.
 In order to build toward factoring greater
polynomials.
Factoring Monomials
Factoring is the reverse of multiplying. To factor
means to break down an expression into a product
of two or more expressions called factorizations.
Ex. (4x)(5x) = 20x2
Multiplying
Ex. 2(10x2) = 20x2
(2x)(10x) = 20x2
x(20x) = 20x2
(-4x)(-5x) = 20x2
Many ways to multiply and
end with a product of 20x2
20x2 = (4x)(5x)
Factoring
20x2 = 2(10x2)
20x2 = (2x)(10x)
20x2 = x(20x)
20x2 = (-4x)(-5x)
Many ways to factor 20x2
Factoring Completely
Factoring completely (finding the Prime Factorization)
means breaking down a product or whole number into a
product of prime numbers
(numbers that cannot be broken down any further).
Factoring Completely can be done using a Factor Tree…
Ex. Factor each monomial completely.
50g2h
2
•
25
5 • 5
2•5•5•g•g•h
Factoring Completely
Factoring completely (finding the Prime Factorization)
means breaking down a product or whole number into a
product of prime numbers
(numbers that cannot be broken down any further).
Factoring Completely can be done using a Factor Tree…
Ex. Factor each monomial completely.
–49a3b2
–1
•
49
7 • 7
–1 • 7 • 7 • a • a • a • b • b
Factoring Completely
Factoring Completely can be done using a Factor Tree…
Ex. Factor each monomial completely.
14m2n
27a2b2
70rs2
2•7•m•m•n
3•3•3•a•a•b•b
2•5•7•r•s•s
Factoring Completely
Factoring Completely can be done using a Factor Tree…
Ex. Factor each monomial completely.
66d4
2 • 3 • 11 • d • d • d • d
–30c2d
–1 • 2 • 3 • 5 • c • c • d
243n2
3•3•3•3•3•n•n
Finding the GCF
The GCF (greatest common factor) means finding the
greatest number that can “go into” two or more numbers.
Finding the GCF can be found by breaking down
numbers into their prime factorization and finding which
factors they share…
Ex. Find the GCF of each set of monomials.
27
3•3•3
72
2•2•2•3•3
In other words, what is the
biggest number that can go
into both 27 and 72??
Since they both
share (3 • 3) their
GCF is equal to 9
Determine what both prime factorizations share…
Finding the GCF
Greatest Common Factor
Finding the GCF can be found by breaking down
numbers into their prime factorization and finding which
factors they share…
Ex. Find the GCF of each set of monomials.
24d2
2•2•2•3•d•d
30c2d
In other words, what is the
biggest number that can go
into both 24d2 and 30c2d??
2•3•5•c•c•d
They both share
2 • 3 • d so their
GCF is 6d
Determine what both prime factorizations share…
Finding the GCF
Greatest Common Factor
Finding the GCF can be found by breaking down
numbers into their prime factorization and finding which
factors they share…
Ex. Find the GCF of each set of monomials.
42a2b
6a5
18a3
GCF is 6a2
Determine what both prime factorizations share…
Finding the GCF
Greatest Common Factor
Finding the GCF can be found by breaking down
numbers into their prime factorization and finding which
factors they share…
Ex. Find the GCF of each set of monomials.
28a2b2
63a3b2
91b3
GCF is 7b2
Determine what both prime factorizations share…