This document teaches how to factor polynomials by finding the greatest common monomial factor (gcmf) through specific steps, including determining the greatest common factor (gcf) of coefficients and variable factors. Several examples demonstrate the process of factoring out the gcmf and rewriting polynomials as products. The lesson ends with a homework assignment involving practice worksheets.

1. Lesson 10.4B : Factoring out GCMF
Factor Completely – to express as the product
of prime factors
Ex. Factor completely : 24
6 4
2 3 2 2
2  2  2  3
Factor the following completely:
1) 5x2
2) 14x2
y3

2. GCF of 12 and 18 GCF = 6
GCMF : Greatest Common Monomial Factor – The
greatest monomial that is a factor (will divide EVENLY
into) of all the given monomials.
GCF of 12x4
y3
and 18xy5
GCF = 6xy3

3. 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

4. Ex. Find GCMF of 12x2
and 18x
GCF of coefficients = 6
Common variable(s) : only have one x in common
GCMF = 6x
Find GCMF of 21x2
and 35x5
GCF of Coefficients = 7
Common variable factors : two x’s
GCMF = 7x2

5. Find GCMF of 24x2
y3
and 36x3
y
GCF of coefficients = 12
Common variable factors : two x’s and one y
GCMF = 12x2
y
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

6. 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: 12x2
y – 14xy3
2xy(6x – 7y2
)

7. 3x(x) + 3x(5)
3x2
+ 15x
Ex.Distribute 3x(x + 5) Means to multiply the 3x
through the (x + 5)
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
x
x
x
3
15
3 2

x
x
x
x
x








3
5
3
3
3
Divide (3x2
+ 15x) by GCMF (3x)
Factored form is 3x(x + 5)
x
x
x
x
3
15
3
3 2

 5

x

8. To factor a polynomial by factoring out the GCMF:
1) Find the GCMF
2) Divide the polynomial (each term of the
polynomial) by the GCMF
3) Write the polynomial as the product of the
GCMF and the result from step #2

9. Example: Factor
1) 15x2
– 9
Step 1) GCMF = 3
Step 2) Divide 15x2
– 9 by the GCMF
3
5
3
9
3
15
3
9
15 2
2
2





x
x
x
Step 3) Write as a product of GCMF and result of step 2
3(5x2
– 3)

10. Factor
1) 28a3
-12a2
GCMF = 4a2
Factored Form
4a2
(7a – 3)
2) 15a – 25b + 20
GCMF = 5
Factored Form
5(3a-5b+4)
3) 16x5
– 14x3
+ 26x2
GCMF = 2x2
Factored Form
2x2
(8x3
– 7x + 13)

11. Homework : Worksheet