MATH 8 - FACTORING POLYNOMIALS WITH COMMON MONOMIAL FACTOR

1. FACTORING OF
POLYNOMIALS
MODULE 1

2. FACTORING TECHNIQUES:
Lesson 1:COMMON MONOMIAL
factoring
Lesson 2: DIFFERENCE OF TWO
SQUARES
Lesson 3: SUM AND DIFFERENCE OF
TWO CUBES

3. Lesson 4: PERFECT SQUARE
TRINOMIAL
Lesson 5: GENERAL QUADRATIC
TRINOMIAL, where a = 1
Lesson 6: GENERAL QUADRATIC
TRINOMIAL, where a >1
Lesson 7: FACTORING BY
GROUPING

4. MELC (MOST ESSENTIAL LEARNING COMPETENCY)
factors completely different types of polynomials
(polynomials with common monomial factor,
difference of two squares,
sum and difference of two cubes,
perfect square trinomials, and
general trinomials).
LC Code: M8AL-Ia-b-1

5. A. FACTORING POLYNOMIAL WITH
GREATEST COMMON MONOMIAL
Objectives:
1.find the greatest common monomial
factor (GCMF) of polynomials
2.factor polynomials with greatest common
monomial factor (GCMF) completely.

6. DEFINITION OF TERMS
Factoring is the process of
finding factors of a given
product. It is the reverse of
multiplication.
6 = 1 ● 6
6 = Product 2, 3 = Factors
6 = 2 ● 3
1, 6 = Factors
6 = Product

7. THEREARE THREE TECHNIQUES YOU CAN
USE FOR MULTIPLYING POLYNOMIALS.
1.Distributive Property
2. FOIL Method
3. Box Method

8. RECALL:::
1. DISTRIBUTIVE PROPERTY
Example 1: 4(3a + 2)
4(3a + 2) = 12a + 8
Example 2: -2x2(2x – 1)
-2x2(2x – 1) = -4x3 + 2x2
Example 3: 3x2y(4x3y – 5)
3x2y(4x3y – 5)= 12x5y2 – 15x2y

9. VERTICAL METHOD IN
MULTIPLYING BINOMIAL
𝒙 + 𝟔
𝒙 + 𝟒
𝒙 + 𝟔
𝒙 + 𝟒
𝒙 + 𝟔
𝒙 + 𝟒
𝒙 + 𝟔
𝒙 + 𝟒
𝒙 + 𝟔
𝒙 + 𝟒
24 6x + 24 6x + 24
4x
6x + 24
𝒙𝟐
+4x
6x + 24
𝒙𝟐
+ 4x
𝒙𝟐
+ 10x + 24

10. F
O
I
L
Made of four
multiplication
steps
FIRST
OUTER
INNER
LAST
T
E
R
M
S
This method can
only be used when
we are multiplying
a binomial by a
binomial.

11. RECALL ON SQUARE OF BINOMIAL
Example 1: 𝒂 + 𝟒 𝒂 + 𝟒
2. USING FOIL METHOD
𝒂 + 𝟒 𝒂 + 𝟒 = 𝒂𝟐
+ 𝟖𝒂 + 𝟏𝟔
FIRST: a ▪ a = 𝒂𝟐
OUTER : a ▪ 4 = 4a
F = FIRST
O = OUTER
I = INNER
L = LAST
INNER : 4 ▪ a = 4a
LAST : 4 ▪ 4 = 16
8a

12. EXAMPLE
Given:
Solution
)
2
)(
3
( 
 x
x
)
(x
x
)
2
)(
3
( 
 x
x
6
5
2


 x
x
First
First
Last
Outer
Inner Inner
Outer
Last
)
2
(
x
)
(
3 x
)
2
(
3
2
x

x
2

6

x
3

2
x
 x
2
 x
3
 6


13. 3. BOXMETHOD
 By using this method we can multiply any two
polynomials.
 In this method we need to draw a box which contains
some rows and columns.
 the number of rows is up to the number of terms of the
first polynomial and the number of columns is up to the
number of terms of the second polynomial.

14. 1
4 
x
EXAMPLE
Given:
)
2
3
)(
1
4
( 
 x
x
The 1st and 2nd polynomial is containing two terms, so the
number of rows and number of columns in the box must be 2.
2
12x x
8
2
3 
x
2
3 
x
x
3 2
Step 1
Step 2 Combine Like Terms
2
3
8
12 2


 x
x
x
12𝑥2
+ 11𝑥 + 2

15. 1

x
EXAMPLE
Given:
)
2
3
)(
1
( 2


 x
x
x
The 1st polynomial is containing two terms and 2nd polynomial is containing three
terms, so the number of rows in the box is 2 and the number of columns in the box
must be 3.
3
x 2
3x
2
3
2

 x
x
2
3
2

 x
x
2
x
 x
3

Step 1
Step 2 Combine Like Terms
2
3
2
3 2
2
3




 x
x
x
x
x
𝑥3
+ 2𝑥2
− 𝑥 − 2
x
2
2


16. FACTORING
is the process of finding the
factors of a mathematical
expression or is the reverse
process of multiplication.
ab + ac = a(b+c)

17. GREATEST COMMON MONOMIAL
FACTOR
Formula :
ax + bx + cx = x (a + b+ c)
The Greatest Common Factor
(GCF) is the largest value of a
number, a variable, or a
combination of numbers and
variables which is common in
each term of a given polynomial.

18. GREATEST COMMON FACTOR
Find the GCF of the following numbers.
Write the prime factorization of each numbers.
a. 16 and 28 =
16 = 4 ▪ 4 = 2 ▪ 2 ▪ 2 ▪ 2
28 = 7 ▪ 4 = 7 ▪ 2 ▪ 2
Greatest Common Factor = 2 ▪ 2 = 4

19. b. 𝒂𝟐
𝒃𝒄𝟐
, 𝒂𝟐
𝒃𝟐
𝒄𝟑
𝒂𝟐
𝒃𝒄𝟐
= a ▪ a ▪ b ▪ c ▪c
𝒂𝟐
𝒃𝟐
𝒄𝟑
= a ▪ a ▪ b ▪ b ▪ c▪ c ▪ c
Greatest Common Factor =
a ▪ a ▪ b ▪ c ▪ c = 𝒂𝟐
𝒃𝒄𝟐

20. C. 𝟏𝟐𝒙𝟐
𝒚𝟑
𝒛𝟐
, 𝟑𝟔𝒙𝟐
𝒚𝒛
𝟏𝟐𝒙𝟐
𝒚𝟑
𝒛𝟐
= 3 ▪ 2 ▪ 2 ▪ x ▪ x ▪ y ▪ y ▪ y ▪ z ▪ z
36𝒙𝟐
𝒚𝒛 = 3 ▪ 2 ▪ 2 ▪ 3 ▪ x ▪ x ▪ y ▪ z
Greatest Common Factor
3 ▪ 2 ▪ 2 ▪ x ▪ x ▪ y ▪ z = 12𝒙𝟐
yz
NOTE :
In choosing the Greatest Common Factor,
use the following rules.
1. Solve for the GCF for the constants.
2. Get the common variable with the
lowest exponent.

21. Steps in factoring polynomials with Greatest
Common Monomial Factor (GCMF):
1.Find the GCF.
2.Divide each term in the polynomial by
its GCMF.
3.Combine the answers in Steps 1
and 2 as a product.
STEPS :

22. EXAMPLE 1
a. Find the GCF of 6x and 4x²
GCMF = 2x
b. Divide each term in the polynomial by its GCMF.
6x + 4x² = 3 + 2x
2x 2x
c. Combine the answers in Steps 1 and 2 as a product.
Factor 6x + 4x²
2x (3 + 2x)
To check, apply the distributive property.
2x (3 + 2x) = 6x + 4x²
Therefore, the factors are 2x (3 + 2x).

23. EXAMPLE 2
a. Find the GCF of 12x²y3z² and 36x²yz
GCMF = 12x²yz
b. Divide each term in the polynomial by its GCMF.
12x²y3z² + 36x²yz = y2z + 3
12x²yz 12x²yz
c. Combine the answers in Steps 1 and 2 as a product.
Factor 12x²y3z² + 36x²yz
12x²yz (y2z + 3)
To check, apply the distributive property.
12x²yz (y2z + 3) = 12x²y3z² + 36x²yz
Therefore, the factors are 12x²yz (y2z + 3)

24. EXAMPLE 3
a. Find the GCF of 3a and 7b
GCMF = 1
b. Divide each term in the polynomial by its GCMF.
3a + 7b =3a + 7b
1 1
c. Combine the answers in Steps 1 and 2 as a product.
Factor 3a + 7b
Prime polynomials are polynomials cannot be factored.
Other examples of prime polynomials are 5+3𝑏, 2𝑥−7𝑦 and
𝑎+2𝑏+3𝑐.
This polynomial cannot be factored by removing the
common factor since the GCF in each term is 1.
Polynomial of this type is called Prime Polynomial.

25. EXAMPLE 4
Factor x²yz + xy²z + xyz²
Solution:
GCF = xyz
x²yz + xy²z + xyz² = x + y+ z
xyz
x²yz +xy²z +xyz² = xyz (x + y + z)

26. EXAMPLE 5
Factor 14m²n² - 4mn³
Solution:
GCF = 2mn²
14m²n² - 4mn³ = 2mn² (7m – 2n)
Note: remember to write the
sign for addition (+) or
subtraction (-) when dividing.
14m²n² – 4mn³ = 7m – 2n
2mn² 2mn²