Factoring by
Greatest Common
Monomial Factor
Subject Teacher: Saturnino P. Odon Jr.
Activity: Pieces of My Life
Directions: Find the possible factors of
the given number or expression below.
Choose you answers from the box and
write it your answer sheet.
2
x
z
a
y
5
10
6
4
b
3
Number/Expression
1.8
2.2x
3.5ab
4.12z
5.20xy
Factors
1.
2.
3.
4.
5.
2, 4
2, x
5, a, b
2, 3, 4, 6, z
2, 10, 5, 4, x, y
Questions:
The area of a rectangle is the product
of the length and the width, or
𝐴𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 = 𝐿 ⋅ 𝑊
1.
What is the area of the
rectangle?
2.
Is the area of the rectangle a
polynomial?
3.
What is the relationship
between the area of the
rectangle and its sides?
4.
What can you say about the
width of the rectangle
comparing it to the area?
5.
What do you call the process of
rewriting the polynomial as a
product of polynomial factors?
FACTORING OUT GCMF
Factor Completely
– to express as the product
of prime factors
Factor the following
completely:
1) 5x2
2) 14x2y3
1) 5x2
5
Example: Factor completely 24
24
x x
14x2y3
2)
6 4
2 3
2 2
Therefore the factors of 24
are 2  2  2  3
5xx
x2
14
2
7
x
x2
y3
y
x
y
y
27xxyyy
GCF of 12 and 18
GCMF : Greatest
Common Monomial
Factor – The greatest
monomial that is a
factor (will divide
EVENLY into) of all the
given monomials.
12 = 1, 2, 3, 4, 6, 12
18 = 1, 2, 3, 4, 6, 9, 18
GCF = 6
GCF of 12x4y3 and 18xy5
12x4y3 = 1, 2, 3, 4, 6, 12, x, x,
x, x, y, y, y
18xy5 = 1, 2, 3, 4, 6, 9, 18, x,
y, y, y, y, y
GCF = 6xy3
To find the GCMF of two or
more Monomials
•First find the GCF of the
coefficients
•Find the largest power of
each variable that is
COMMON to all the
monomials
•The GCMF = product of GCF
of coefficients and common
variable factors
Example:
Find the GCF of each
pair of monomials.
a. 4x3 and 8x2
b. 15y6 and 9z
Solutions:
a. 4x3 and 8x2
Step 1:Factor each monomial.
4x3 = 2 ⦁ 2 ⦁ x ⦁ x ⦁ x
8x2 = 2 ⦁ 2 ⦁ 2 ⦁ x ⦁ x
Step 2. Identify the common factors.
4x3 = 2 ⦁ 2 ⦁ x ⦁ x ⦁ x
8x2 = 2 ⦁ 2 ⦁ 2 ⦁ x ⦁ x
Step 3. Find the product of the
common factors.
2 ⦁ 2 ⦁ x ⦁ x = 4x2
Therefore the GCMF of 4x3 and 8x2 is 4x2
b. 15y6 and 9z
Activity 1:
1. Find GCMF of 12x2 and 18x
2. Find GCMF of 21x2 and 35x5
3. Find GCMF of 24x2y3 and 36x3y
Solutions:
1. Find GCMF of 12x2 and 18x
GCF of coefficients = 6
Common variable(s) : only have
one x in common
GCMF = 6x
2. Find GCMF of 21x2 and 35x5
GCF of Coefficients = 7
Common variable factors : two x’s
GCMF =
7x2
3. Find GCMF of 24x2y3 and 36x3y
GCF of coefficients = 12
Common variable factors : two
x’s and one y
GCMF = 12x2y
Notice that in the examples above, prime
factorization is used to find the GCMF of the
given pair of monomials. The next examples
illustrate how the GCMF is used to factor
polynomials.
NOW, we are going to use GCMF’s to
Factor Quadratic Expressions. Factoring
Out the GCMF is the inverse (un-doing) of
the Distributive Property
To factor – undoing
distributive property
1) Perform Distributive Property
6(2 + 3) = 12 + 18
Factor : 12 + 18 = 6(2 + 3)
2) Use Distributive Property to simplify
3(x + 7) = 3x + 21
Factor: 3x + 21 = 3(x + 7)
3) Factor: 12x2y – 14xy3
= 2xy(6x – 7y2)
Ex.Distribute 3x(x + 5) Means to multiply the 3x
through the (x + 5)
3x(x) + 3x(5)
3x2 + 15x
Ex. Factor 3x2 + 15x Means to Divide the GCMF
out of the polynomial (divide each term by GCMF)
GCMF = 3x
Recall how to divide by monomial
Divide (3x2 + 15x) by GCMF (3x)
2
3
x
15 x 3  x  x 3  5  x
3 x  15 x
 x5



3x 3x
3 x
3 x
3x
2
Factored form is 3x(x + 5)
To factor a polynomial by
factoring out the GCMF:
Example: Factor 15x2 – 9
1) Find the GCMF
Step 1) GCMF = 3
2) Divide the polynomial
(each term of the
polynomial) by the GCMF
Step 2) Divide 15x2 – 9 by the GCMF
3) Write the polynomial as
the product of the GCMF
and the result from step
#2
Solution:
15 x  9 15 x 9

  5x 2  3
3
3
3
2
2
Step 3) Write as a product of
GCMF and result of step 2
3(5x2 – 3)
Another Example!!!
1. Write 6𝑥+ 3x2 in factored form.
Step 1
Determine the number of terms.
In the given expression, we have 2 terms: 6𝑥 and 3x2.
Determine the GCF of the numerical coefficients.
Step 2
Coefficient
3
Factors
1, 3
6
1, 2, 3
Step 3
Common Factors
GCF
1 and 3
3
Determine the GCF of the variables. The GCF of the
variables is the one with the least exponent.
GCF (x, x2) = x
Another Example!!!
1. Write 6𝑥+ 3x2 in factored form.
Step 4
Find the product of GCF of the numerical coefficient and
the variables.
(3)(x) = 3x
Hence, 3𝑥 is the GCMF of 6𝑥 and 3x2.
Step 5
Find the other factor, by dividing each term of the polynomial 6𝑥+
3x2 by the GCMF 3𝑥.
Step 6
Write the complete factored form
6𝑥 + 3x2 = 3x(2 + x)
Activity 2:
Factor the following expressions.
1. 28a3-12a2
2. 15a – 25b + 20
3. 16x5 – 14x3 + 26x2
Factor
1) 28a3-12a2
2)
GCMF = 4a2
Factored Form
4a2(7a – 3)
15a – 25b + 20
GCMF = 5
Factored Form
5(3a-5b+4)
3) 16x5 – 14x3 + 26x2
GCMF = 2x2
Factored Form
2x2(8x3 – 7x + 13)