Math Explorer with scaffolding approach toward independent learning Module : Factoring Polynomials with common monomial factor (Free download PDF )

1. Math Explorer: With Scaffolding Instruction toward Independent
Learning

2. Module 1 : Factoring Polynomials
Factoring is the reverse process of multiplication. It is
the process of finding the factors of an expression.
A polynomial is factored completely if each of its factors
can no longer be expressed as product of two other polynomials.
Here is the map of the lessons that will be covered in
this module.
Factoring PolynomialsModule
1
At the end of this module, the learners will:
1. factors completely different types of polynomials
2. find factors of product of polynomials
3. Solves problems involving factors of polynomials.

3. Ms. Lorie Jane L. Letada
Module 1
The GCF of these three monomials is (3)(x)(x) = 3 𝒙 𝟐
Common monomial factoring is the process of writing a
polynomial as a product of two polynomials, one of which is a
monomial that factors each term of the polynomial.
To ensure that the polynomial is the prime polynomial, use
the Greatest Common Factor (GCF) of the terms of the given
polynomials.
1. Find the GCF of 4𝒎 𝟐
and 𝟏𝟎𝒎 𝟒
.
Solution:
Express each as a product of prime factors.
4𝒎 𝟐
= 2 (2) m m
𝟏𝟎𝒎 𝟒
= (5) 2 m m (m) (m)
Common Factors = 2 m m
The GCF of these two monomials is (2)(m)(m) = 𝟒𝒎 𝟐
2. Find the GCF of 𝟔𝒙 𝟒
, 9𝒙 𝟐
y, and 𝟏𝟓𝒙 𝟓
𝒚.
Solution:
Express each as a product of prime factors.
𝟔𝒙 𝟒
= (2 ) 3 x x (x) (x)
9𝒙 𝟐
y = (3) 3 x x (y)
𝟏𝟓𝒙 𝟓
𝒚 = (5) 3 x x (x) (x) (x) (y)
Common Factors = 3 x x
Lesson
1
Polynomial with Common
Monomial Factor
Observe and analyze below on how to get the GCF
and answer number 3 and 4 .

4. Module 1 : Factoring Polynomials
3. Find the GCF of 8𝒚 𝟔
and 𝟏𝟎𝒚 𝟒
z.
Solution:
Express each as a product of prime factors.
8𝒚 𝟔
=
𝟏𝟎𝒚 𝟒
z =
Common Factors =
4. Find the GCF of 𝟐𝟎𝒃 𝟒
𝒄 , 5𝒃 𝟐
c, and 𝟏𝟓𝒃 𝟑
𝒄.
Solution:
Express each as a product of prime factors.
𝟐𝟎𝒃 𝟒
𝒄 =
5𝒃 𝟐
c =
𝟏𝟓𝒃 𝟑
𝒄 =
Common Factors =
Math Guro
GCF – The greatest common factor (GCF) of a set of numbers is the
largest factor that all the numbers or variables share.
Polynomial – an expression consisting of variables and coefficients, that
involves only the operations of addition, subtraction, multiplication, and
non-negative integer exponents of variables. Example: 2x +3y – 4xy
Monomial – an expression in algebra that contains one term, like 3xy
Factors – a number or algebraic expression that divides another
number or expression evenly with no remainder. Ex. The factors of 6 is
2 and 3 because (2) (3)= 9.

5. Ms. Lorie Jane L. Letada
The property of
exponents stated
𝑎 𝑚
𝑎 𝑛 = 𝑎 𝑚−𝑛
1. )
𝑥5
𝑥3 = 𝑥5−3
= 𝑥2
2. )
𝑝7
𝑝4 =
3. )
105
103 = 105−3
= 102
= 100
4. )
1525
1523 =
5. )
45 𝑥10
42 𝑥2 = 45−2
𝑥10−2
= 43
𝑥8
= (4 ∗ 4 ∗ 4 ) 𝑥8
= 64 𝑥8
6.)
36 𝑏17
32 𝑏12 =
=
=
=
A fraction is reduced to its lowest terms, when its
numerator and denominator have no common factors.
Observe the steps:
1. List the prime factors of the numerator
and denominator.
2. Find the factors common to both the numerator and
denominator.
3. Divide the numerator and denominator by all common
factors called CANCELLING.
Example 1:
To factor
polynomials w/
common monomial
factor, you should
know how to get:
1.GCF
2. Quotient of two
monomials
3.Reducing
fractions to
lowest term

6. Module 1 : Factoring Polynomials
Now, you are ready to find the factor of polynomials with
common monomial.
Observe and analyze the steps:
1. Find the greatest common factor (GCF) of the terms in
the polynomial. This is the first factor.
2. Divide each term by the GCF to get the other factor.
Factor each expression.
1. 𝟏𝟎𝒚 𝟑
+ 5𝒚 𝟐
2. 𝟏𝟓𝒎 𝟓
+ 5𝒎 𝟐
3. 𝟐𝟓𝒃 𝟑
𝒄 𝟐
− 5𝒃 𝟐
𝒄
Step 1
𝟏𝟎𝒚 𝟒 = (5) (2) (y) (y) (y) (y)
5𝐲 𝟑= (5 ) (y) (y) (y)
GCF = 5 y y y
GCF = 𝟓𝒚 𝟑
Step 2
𝟏𝟎𝒚 𝟑 + 5𝒚 𝟐 = 𝟓𝒚 𝟐 (
𝟏𝟎𝒚 𝟒
𝟓𝒚 𝟐
+
𝟓𝒚 𝟑
𝟓𝒚 𝟐
)
= 𝟓𝒚 𝟐 (𝟐𝒚 𝟒−𝟐 + 𝒚 𝟑−𝟐)
= 𝟓𝒚 𝟐 (𝟐𝒚 𝟐 + y )
Step 1
𝟏𝟓𝒎 𝟓
=
5𝐦 𝟐 =
GCF =
GCF =
Step 2
𝟏𝟓𝒎 𝟓
+ 5𝒎 𝟐
=
=
=
Step 1
𝟐𝟓𝒃 𝟑 𝒄 𝟐 = (5)(5)(b)(b)(b)(b)(c)(c)
𝟓𝒃 𝟕 𝒄 = (3) (5)(b)(b) (c)
GCF = (5)(b)(b) (c)
GCF = 5𝒃 𝟐
𝒄
Step 2
𝟐𝟓𝒃 𝟑
𝒄 𝟐
− 5𝒃 𝟐
𝒄 = 5𝒃 𝟐
𝒄 ( 𝟐𝟓𝒃
𝟑
𝒄 𝟐
𝟓𝒃
𝟐
𝒄
−
𝟓𝒃
𝟐
𝒄
𝟓𝒃
𝟐
𝒄
)
= 5𝒃 𝟐
𝒄 ( 5𝒃 𝟑−𝟐
𝒄 𝟐−𝟏
– 1)
=5𝒃 𝟐
𝒄 ( 5b – 1 )
Any
number or
variables
divided by
itself is
always
equal to
1.
All
variables
always
have
imaginary
exponent
one.

7. Ms. Lorie Jane L. Letada
𝟒. 𝟏𝟐𝒉 𝟒
𝒊 𝟐
− 15𝒉 𝟐
𝒊
Factor each expression. Show your solution with the
step-by-step process.
𝟏. 𝟖𝒈 𝟔
+ 16𝒈 𝟐
4. 𝟔𝒂 𝟐
𝒃 + 𝟏𝟖𝒂𝒃
𝟐. 𝟒𝟗𝒚 𝟗
+ 21𝒚 𝟐
5. 𝟐𝟒𝒑 𝟏𝟗
𝒒 𝟏𝟐
+ 𝟒𝒑 𝟏𝟓
𝒒 𝟕
𝟑. 𝟐𝒙 𝟐
𝒚 𝟐
− 14y 6. 𝟑𝟎𝒓 𝟔
+ 𝟏𝟎𝒓 𝟐
References
Diaz, Z., Mojica M. (2013) . Next Century Mathematics 8; Quezon City ;
Phoenix Publishing House , Inc
Mathematics 8 Learner’s Module K-12; DepEd K-12 Modified Curriculum
Guide and Teacher’s Guide for Mathematics 8
http://www.math.com/school/subject1/lessons/S1U4L2DP.html
https://quickmath.com/math-tutorials/common-monomial-factors-
factoring-special-products-and.html
Let’s do exercises!
Step 1
𝟏𝟐𝒉 𝟒
𝒊 𝟐
=
15𝒉 𝟐
𝒊 =
GCF =
GCF =
Step 2
𝟏𝟐𝒉 𝟒
𝒊 𝟐
− 15𝒉 𝟐
𝒊 =
=
=

8. Module 1 : Factoring Polynomials