8

Mathematics
Quarter 1 – Module 1:
Factoring Polynomials
Week 1
Learning Code M8al-Ia-1
 Mathematics – Grade 8
Alternative Delivery Mode
Quarter 1 – Module 1 – Factoring Polynomials
First Edition 2020

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Published by the Department of Education

Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writers: John Richard L.Quiambao Mary Ann L. Abundo
Zenaida W. HaliliJean Aiko C. Domingo Judy Ann G. Gallo
Editor: Luningning R.Tayamora Judy Ann G. Gallo
Rene V. Salgado Katherine A. Matarlo
Reviewers/Validators: Remylinda T. Soriano, EPS, Math
Angelita Z. Modesto, PSDS
George B. Borromeo, PSDS
Illustrator: All Writers
Layout Artist: All Writers
Management Team: Malcolm S. Garma, Regional Director
Genia V. Santos, CLMD Chief
Dennis M. Mendoza, Regional EPS in Charge of LRMS and
Regional ADM Coordinator
Maria Magdalena M. Lim, CESO V, Schools Division Superintendent
Aida H. Rondilla, Chief-CID
Lucky S. Carpio, Division EPS in Charge of LRMS and
Division ADM Coordinator

1
 8
Mathematics
Quarter 1 – Module 1:
Factoring Polynomials
Week 1
Learning Code M8al-Ia-1

2
 GRADE 8
Learning Module for Junior High School Mathematics
MODULE FACTORING POLYNOMIALS
1

WHAT I NEED TO KNOW

PPREPREVIER!
LEARNING COMPETENCY
At the end of this module, the learner will be able to factors
completely different types of polynomials (polynomials with common
monomial factors, difference of two squares, sum and difference of two
cubes, perfect square trinomials and general trinomials.

Welcome to Mathematics 8 this is a new journey for you to discover

and explore new concepts in mathematics. Series of different activities will
be encountered in this module but first let’s check your knowledge by
answering the pre-test.

WHAT I KNOW
PPREPREVIER
Choose the letter of your answer. If your answer is not among the choices write E.
!
1. What is the GCF of 2𝑥² − 4𝑥 − 6𝑥² = 2𝑥(𝑥 − 2 − 3𝑥²)?
a) 2𝑥 b)(𝑥 − 2 − 3𝑥 2 ) c)−4𝑥 d)2𝑥(𝑥 − 2 − 3𝑥 2 )
2. Which of the following is equivalent to 18𝑥 + 12𝑦?
a) 4(5𝑥 + 3𝑦)b)2(9𝑥 + 8𝑦) c)3(6𝑥 + 5𝑦) d)6(3𝑥 + 2𝑦)
3. Which of the following is equivalent to the equation 𝑥 + 5 = (𝑥 + 3)²?
a)(𝑥 + 1)(𝑥 + 4) = 0 b)(𝑥 − 1)(𝑥 + 4) = 0 c)(𝑥 + 1)(𝑥 + 4) = 5 d)(𝑥 + 1)(𝑥 + 4) = 3
4. Which of the following is a quadratic expression with a > 1?
a)𝑥 2 + 3𝑥 − 5 b) – 𝑥 2 − 2𝑥 − 1 c)3𝑥 2 − 𝑥 − 2 d) 5𝑥 2 + 25
5. Which of the following is a quadratic expression where a =1?
a) 4𝑥² − 7𝑥 − 8 b) 𝑥² − 5𝑥 − 280 c) 4𝑥² − 9 d) 9𝑥² + 𝑥 − 10
6. Factor 𝑥 3 − 8.
a)(𝑥 + 2)(𝑥 2 − 2𝑥 + 4) b)(𝑥 − 2)(𝑥 2 + 2𝑥 + 4)c)(𝑥 − 5)(3𝑥 + 1) d)(3𝑥 − 5)(𝑥 + 1)
7. Factor 𝑥 3 + 27 completely.
a)(𝑥 + 3)(𝑥 2 − 3𝑥 + 9) b)(𝑥 + 3)(𝑥 2 − 3𝑥 + 18)
c)(𝑥 + 3)(𝑥 2 − 3𝑥 − 9)d)(𝑥 + 5)(𝑥 − 3)
8. Which of the following is a quadratic expression with a=1?
a) 𝑎2 − 3𝑥 − 2 𝑏)2𝑥² + 4𝑥 − 5 𝑐) − 𝑥² + 4𝑥 + 8 𝑑)2𝑥² − 2𝑥 + 7
9. Which of the following will complete the statement below?
If the product is positive, then m and n have _____________.
a) Opposite signs
b) Both negative signs
c) Both positive signs
d) Both negative or both positive signS
10. Given the expression below, what is the sum and product of m and n?
𝑥² − 7𝑥 + 12
a) Sum: +12 b) sum:-12 c)sum:-7 d) sum: 7
Product: -7 product: +12 product:+12 product: -12

1
 GRADE 8
Learning Module for Junior High School Mathematics
11. Factor the following expression: 𝑥² − 3𝑥 − 18
a) (𝑥 + 6)(𝑥 − 3) b) (𝑥 − 6)(𝑥 + 3) c)(𝑥 − 6)(𝑥 − 3) d) (𝑥 + 6)(𝑥 − 4)
12. What is the factored form of 𝑥² − 49?
a)(𝑥 − 7)(𝑥 − 7) b) (𝑥 − 49)(𝑥 + 49) c)(𝑥 − 7)(𝑥 + 7) d.(𝑥 − 49)(𝑥 − 49)
13. Factor the following expression: 𝑥 2 + 25?
a)(𝑥 − 5)(𝑥 − 5) b) (𝑥 + 25)(𝑥 + 25) c)(𝑥 + 5)(𝑥 − 5) d)(𝑥 − 25)(𝑥 − 25)
14. What is the factored form of 𝑥² − 6𝑥 + 8?
a)(𝑥 − 2)(𝑥 + 4) b) (𝑥 − 8)(𝑥 + 1) c)(𝑥 − 2)(𝑥 − 4) d.(𝑥 − 8)(𝑥 − 1)
15. Factor the following expression: 𝑥² + 8𝑥 + 12
a)(𝑥 − 6)(𝑥 − 2) b) (𝑥 + 6)(𝑥 + 2) c)(𝑥 + 4)(𝑥 − 3) d)(𝑥 − 4)(𝑥 − 3)

*** If you got an honest 15 points (perfect score), you may skip this module.

WHAT’S IN
PPREPREV
A.WriteP in the box if the given number is prime and C if it is composite. In
IER!
the blank, writethe prime factors of the number. The first has been done as
an example for you.
_______1. 40 = 2 · 2 · 2 · 5
_______2. 19 = __________
_______3. 56 = __________
_______4. 29 = __________
_______5. 35 = __________
_______6. 81 = __________
B. Find the GCF of the following pairs of expressions.
_______1. 16𝑥 2 and 4y𝑥 2
_______2. 27 𝑥 4 𝑦 5 and 9𝑥 3 𝑦 2
_______3. 100𝑥 5 𝑦 6 and 50𝑥 3 𝑦 3

You will be using the concept you have reviewed for this lesson… but first
read the selection provided and answer the questions that follow.

WHAT’S NEW
CLOSET REMODELING

My mother plans to remodel our closet. She

measured the dimensions of the closet and
drew the layout of her desired design and
measurements. She hired a carpenter to do the
task. Unfortunately, the layout was lost and
mother only remembers the area of portion A
and C portion. She sketched again the diagram
and include the area of portiona A and C.
Based on the diagram potions A and B are
rectangles while C are congruent squares.

Now, the carpenter has to figure out the

dimensions of each portion.

2
 GRADE 8
Learning Module for Junior High School Mathematics
WHAT IS IT
Read and answer the following questions.
1. What did mother want to do with the closet?
_____________________________________________________
2. What did mother do so that the carpenter will be able to do his task?
_____________________________________________________
3. What happened with the layout that mother prepared?
_____________________________________________________
4. What did mother do, when she learned that the layout was lost?
_____________________________________________________
5. According to mother she only remembers the area of each portion,
what do you think should the carpenter do in order to find the
dimension of each portion?
_____________________________________________________
6. Based on the diagram, what are the dimensions of
portion A? ___________________________________________
portion C? ___________________________________________
portion B? ___________________________________________
7. If the value of x is 5 inches, find the dimensions of each portion.
____________________________________________________
8. What are the dimensions of the entire closet?

FACTORING POLYNOMIALS
Factoring is the process of getting the polynomial factors of a given
number or expression. You learned how to factor out prime and composite
numbers earlier. Now, you will learn how to factor out variables. You will
also learn how to factor out polynomials by getting their greatest common
factor or by using special products.

COMMON MONOMIAL FACTOR

To factor polynomial with common monomial factor, expressed the
given polynomial as a product of the common monomial factor and the
quotient obtained when the given polynomial is divided by the common
monomial factor

EXAMPLE 1: Factor 3x+6.

Step 1: Express each term as factors. 3(x)+3(2)
Step 2: Find the greatest common factor (GCF). GCF = 3
3𝑥 6
Step 3: Divide each term by the GCF. + = 𝑥+2
3 3
Step 4: Write the GCF and the result from step 3 3(x+2)
together.

Solve It Another Way!

 GRADE 8
Learning Module for Junior High School Mathematics
EXAMPLE 2: Factor (4x+12y).
Step 1: Find the greatest common factor (GCF). 4 is the GCF
Step 2: Write the GCF as a factor of each term. 4(x)+4(3y)
Step 3: Write out, or factor out, the GCF using 4(x+3y)
the distributive property.

Tips
• Dividing by the greatest common factor is also known as factoring
out the GCF.
• While this method cannot be applied to all polynomials, it is often the
first step in any factoring problem.
Key Points
• Factoring by a common monomial is also known as factoring by the
greatest common factor (GCF).
• When a polynomial has been written as a product consisting of prime
factors, only then it is said to be factored completely.

FACTORING DIFFERENCE OF TWO SQUARE

The factors of the difference of two squares are the sum of the square
roots of the first and second terms times the difference of their square
roots.
*The factors of 𝑎2 − 𝑏2 𝑎𝑟𝑒 ( 𝑎 + 𝑏 ) 𝑎𝑛𝑑 ( 𝑎 −𝑏 ).

EXAMPLE 1: Factor x2−4.

Step 1: Identify a2 and b2 in the expression. a2=x2 and b2=4
Step 2: Find the values of a and b by getting the a2=x2|b2=4
square root of each. a=x|b=2
Step 3: Substitute the values of a and b into the a −b2=(a−b)(a+b)
2

formula for difference of two squares. (x)2−(2)2=(x−2)(x+2)

Thus, the factors of x2−4 are x−2 and x+2.

EXAMPLE 2: Factor 9x2−4 .

Step 1: Identify a2 and b2 in the expression. a2=9x2 and b2=4
Step 2: Find the values of a and b by getting the a2=9x2|b2=4
square root of each. a=3x|b=2
Step 3: Substitute the values of a and b into the a −b2=(a−b)(a+b)
2

formula for difference of two squares. (3x)2−(2)2=(3x−2)(3x+2)

Thus, the factors of 9x2−4 are 3x−2 and 3x+2.

Tips
• Always check if the expression you are factoring is a difference of two
squares.
• Sometimes, you have to factor out the GCF first before you factor the
difference of two squares.
Key Points
• An expression is a difference of two squares if the first and second
terms are perfect squares, subtracted from each other.
 GRADE 8
Learning Module for Junior High School Mathematics
• Only expressions in the form of a difference of two squares can be
factored using the formula, a2−b2=(a−b)(a+b).

FACTORING BY GROUPING
Polynomials may not have terms with a common monomial factor,
but when the terms are grouped, a common monomial factor may appear
in each group.

EXAMPLE 1: Factor x3+6x2−36x+216.

Step 1: Group the terms in the (x3+6x2)+(−36x+216)
parentheses as a sum.
Step 2: Factor out the greatest common x2(x+6)−36(x+6)
factor, or GCF, in each group.
Step 3: Factor out the common binomial. (x+6)(x2−36)
Step 4: Continue factoring, if possible.
The binomial x2−36 can still be factored using the difference of two
squares as (x−6)(x+6).
Thus, the complete factored form is (x+6)(x−6)(x+6) or simply, (x+6)2 (x−6).

Tips
• Be careful when grouping terms preceded by a negative sign. Make
sure that the grouped terms can be simplified back as the original
expression.
• Factoring by grouping can also be used for expressions with more
than 4 terms.
Key Points
• Consider factoring by grouping when you have at least 4 terms in
the expression.
• Make sure that grouped terms have a common factor.
• Factoring out a binomial is one of the key steps in this method.

FACTORING SUM OR DIFFERENCE OF TWO CUBES

• The sum of the cubes of two terms is equal to the sum of the two
terms multiplied by the sum of the squares of these terms minus
the product of these two terms.
a³ + b³ = ( a + b ) ( a² - ab + b² )

• The difference of the cubes of two terms is equal to the difference

of the two terms multiplied by the sum of the squares of these two
terms plus the product of these two terms.
a³ - b³ = ( a - b ) ( a² + ab + b² )

EXAMPLE 1: Factor x3+27.

Currently the problem is not written in the form that we want. Each
term must be written as cube, that is, an expression raised to a power of 3.
The term with variable x is okay but the 27 should be taken care of.
Obviously, we know that 27 = (3)(3)(3) = 33.
 GRADE 8
Learning Module for Junior High School Mathematics
Rewrite the original problem as sum of two cubes, and then simplify.
Since this is the "sum" case, the binomial factor and trinomial factor will
have positive and negative middle signs, respectively

EXAMPLE 2: Factor y3-8

This is a case of difference of two cubes since the number 8 can be
written as a cube of a number, where 8 = (2)(2)(2) = 23.
Apply the rule for difference of two cubes, and simplify. Since this is
the "difference" case, the binomial factor and trinomial factor will have
negative and positive middle signs, respectively.

FACTORING BY PERFECT SQUARE TRINOMIALS

Whenever you multiply a binomial by itself twice, the resulting
trinomial is called a perfect square trinomial
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a − b)2
Notice that all you have to do is to use the base of the first term and
the last term
In the model just described, the first term is a2 and the base is a. the
last term is b2 and the base is b

EXAMPLE 1: Factor x2 + 2x + 1
Step 1: Identify the first term and the last term a = x2 b=1
Step 2: Find the square root of the first and last a=x b=1
term
Step 3: Put the bases inside parentheses (a ± b)2 . (x + 1)2
*NOTE: The sign must be the same as the sign of
the middle term

. EXAMPLE 2: Factor p2 – 18p + 81

Step 1: Identify the first term and the last term a = p2 b = 81
Step 2: Find the square root of the first and last a=p b=9
term
Step 3: Put the bases inside parentheses (a ± b)2 . (p - 9)2
*NOTE: The sign must be the same as the sign of
the middle term
 GRADE 8
Learning Module for Junior High School Mathematics

FACTORING BY GENERAL TRINOMIAL

In factoring Quadratic Trinomials where a = 1
1. List down all the factors of the last term;
2. Identify which factor pair sums up the middle term; then
3. Write each factor in the pairs as the last term of the binomial factors.

EXAMPLE 1: Factor x2 + 5x + 6
STEP 1: Identify the factors of the last term
6 = 3·2
-3 · -2
6·1
-6· -1
STEP 2: Check the sums of the pairs of potential factors, and identify which
factor pair sums up the middle term:
6 = 3+2 =5
-3 + -2 = -5
6+1 =7
-6 + -1 = -7
STEP 3: Write each factor in the pairs as the last term of the binomial
factors
Since I need my factors to sum to plus-five, then I'll be using the
factors 2 and 3
x2 + 5x + 6 = (x + 2)(x + 3)

EXAMPLE 2: Factor x2−8x+15.

STEP 1: Identify the factors of the last term
15 = -1 · -15
-3 · -5
STEP 2: Check the sums of the pairs of potential factors, and identify which
factor pair sums up the middle term:
15 = -1 + -15 = -16
-3 + -5 = -8
STEP 3: Write each factor in the pairs as the last term of the binomial
factors
x2 – 8x + 15 = (x - 3)(x - 5)

Tip
Always check if you have factored the quadratic expression correctly
by multiplying back the binomial factors, obtaining the original expression.

Key Points
• Some quadratics with a=1 can be written as the product of two
binomial factors, (x+m)(x+n).
• The sum of m and n is the coefficient of the middle term, while
the product of m and n is the last term in a quadratic expression.
• If the product is positive, then m and n are either both negative or
both positive. If the product is negative, then m and n have
 GRADE 8
Learning Module for Junior High School Mathematics

FACTORING BY GENERAL TRINOMIAL

Factoring Quadratic Trinomial with a > 1 using the 6 steps:
STEP 1: Identify a, b and c
STEP 2: Multiply a and c
STEP 3: Write down all possible factors of ac
STEP 4: Identify which factor sums up to b
STEP 5: Rewrite the original equation into four terms splitting the
middle term
STEP 6: Use factoring by grouping

EXAMPLE 1: Factor 3x2 + 14x + 8

STEP 1: identify a, b and c a = 3 b = 14 c = 8
STEP 2: multiply a and c ac = (3) (8) = 24
STEP 3: write down all possible factors:
factors of ac 2 (12) -3(-8)
-2 (-12) 4(6)
3(8) -4(-6)

STEP 4: identify which factor sums b = 14

up to b 2 + (12) = 14

STEP 5: rewrite the original 3x2 + 14x + 8

equation into four terms splitting 3x2 + 12x + 2x + 8
the middle term
STEP 6: use factoring by grouping 3x2 + 12x + 2x + 8
(3x2 + 12x) + (2x + 8)
3x ( x + 4) + 2 ( x + 4 )
(3x+2) (x+4)

Tips
• There are other techniques for factoring quadratics. Feel free to look
for one that you are most comfortable with.
• Practice factoring quadratics. The more problems you solve, the easier
it will get.

Key Points
When factoring quadratic expressions of the form ax2+bx+c, where a>1:
• The factors of a are the first terms of the binomial factors and the
factors of c are the second terms.
• If c is positive, then its factors are either both positive or both
negative. Shorten the list of factors by using only positive factors
of c if b is positive. Use only negative factors if b is negative.
 GRADE 8
Learning Module for Junior High School Mathematics

WHAT’S MORE

Let’s begin your individual activities. Are you ready?

Activity 2 – COMPLETE ME!

Complete the table below.

Greatest
Common Quotient of
Factored
Polynomial Monomial Polynomial and
Form
Factor CMF
(CMF)
5x + 10 5 x+2 5(x + 2)
3xy2 3xy2(2x + y)
3a4 – 9a3b + 6a2b2 a2 – 3ab + 2b2
3a2b2 2 – 5abc + 4a2b2c2
12WI3N5 – 16WIN + 20WINNER

ACTIVITY 3: WHAT’S UP FOR THE NEW NORMAL!

Find the greatest common monomial factor and write the matching letter
on the blank above the answer
A 8x2 – 80x G 15x3y2 – 30xy R 4x5 – 8x4 – 4x3
B 9y2 – 36y I 18x4y + 9xy7 S 5x3y – 20x2y2 + 100xy
C 4x2 + 32xy L 6x4 – 10x3 + 2x T 15x3y2 – 20x2y3 + 12x4y
D 12x2y – 8xy2 N 8y5 – 24y4 – 16y2 V 6x5 –15x4 – 21x3 + 27x2
E 36x4 – 42x2 O 5x3y – 15xy2 +25xy

4y2 9y 5xy 6x2 4x3 3x2 6x2

5xy 4y2 4x 9xy 8x 2x 4xy 9xy 5xy x2y 8x 8y2 4x 9xy 8y2 15xy

Activity 4:HOPE IN THE DARK

Amid the Covid -19 pandemic, people still find time to smile. The negativity brought
by the pandemic has not overcome the positive outlook of most people in the world.
Factor the following polynomials. The box below contains the answers. Write the
word that corresponds to your choice on the respective blanks below to reveal an
inspiring message from the movie, ‘Twilight”.

1.𝑥 2 + 7𝑥 + 6 7.𝑥 2 − 18𝑥 + 81

2. 𝑥 2 − 7𝑥 + 12 8. 6𝑥 2 − 3𝑥
3. 𝑥 2 − 4𝑥 + 4 9. 8𝑥 3 − 1
4.𝑥 2 − 16 10. 𝑥 3 + 27𝑦 3
5.𝑥 2 + 5𝑥 11. 4𝑥 2 − 20𝑥 + 25
6.𝑥 2 − 81
 GRADE 8
Learning Module for Junior High School Mathematics

1 2 3 4

5 3 6 7 8 9 10

3 11
-Stephenie Meyer, Twilight

DARK (𝑥 + 9)(𝑥 − 9)
NIGHT (𝑥 + 4)(𝑥 − 4)
LIKE (𝑥 − 4)(𝑥 − 3)
I (𝑥 + 6)(𝑥 + 1)
THE (𝑥 − 2)(𝑥 − 2)
WITHOUT 𝑥(𝑥 + 5)
WE (𝑥 − 9)(𝑥 − 9)
STARS (2𝑥 − 5)(2𝑥 − 5)
WOULD 3𝑥(2𝑥 − 1)
SEE (𝑥 + 3𝑦)(𝑥 2 − 3𝑥𝑦 + 9𝑦 2 )
NEVER (2𝑥 − 1)(4𝑥 2 + 2𝑥 + 1)

ACTIVITY 5: AM I FACTORABLE?

1. x2 – y2 2. 9m2 – 25n2
Is the expression a difference of two Is the expression a difference of two
squares? Shade your answer. squares? Shade your answer.
YES NO YES NO
If your answer is YES, write each factor If your answer is YES, write each factor
inside the box, if NO, escape this part. inside the box, if NO, escape this part.

3. 16x4 – 49y2 4. 36a2b2 – 81c4

Is the expression a difference of two Is the expression a difference of two
squares? Shade your answer. squares? Shade your answer.
YES NO YES NO
If your answer is YES, write each factor If your answer is YES, write each factor
inside the box, if NO, escape this part. inside the box, if NO, escape this part.
 GRADE 8
Learning Module for Junior High School Mathematics

5. 25x2 – 10y2 6. 100x2y6 – z4

Is the expression a difference of two Is the expression a difference of two
squares? Shade your answer. squares? Shade your answer.
YES NO YES NO
If your answer is YES, write each factor If your answer is YES, write each factor
inside the box, if NO, escape this part. inside the box, if NO, escape this part.

7. 121x2 + 9y2 8. x2 – (y – z)4

Is the expression a difference of two Is the expression a difference of two
squares? Shade your answer. squares? Shade your answer.
YES NO YES NO
If your answer is YES, write each factor If your answer is YES, write each factor
inside the box, if NO, escape this part. inside the box, if NO, escape this part.

ACTIVITY 6: COLOR ME!

Color the square blue if the given expression is factored correctly, if not, skip
the square.

9a2 – 4b2 =
a2 – b2 = y2 – 4 = b4 – 16 = m2– 9 =
(3a – 2b)
(a + b)(a – b) (y + 4)(y + 4) (b2 – 4)(b2 – 4) (m – 3)(m + 3)
(3a – 2b)

4x2 – 25 = 36x2 − y2 = 36x2 – 1 = p2– 144 = 9b2– 25 =

(2x + 5)(2x – 5) (6x – y)(6x + y) (6x – 1)(6x – 1) (p + 12)(p – 12) (3b + 5)(3b – 5)

4b4 – 49d2 = 25m2– 9 = 100z6 – 81 = 49x2 − 4y2 =

a2b2 – 16 =
(2b2 + 7d) (5m + 3) (10z3 – 9) (7x – 2y)
(ab – 4)(ab + 4)
(2b2 + 7d) (5m – 3) (10z3 – 9) (7x + 2y)

x4 – y6 = m6 – 100 = 4r4 – 81t2 = 144x2 − 25y2 =

81b2 – 49 =
(x2 + y3) (m3 – 10) (2r2 + 9t) (12x – 5y)
(9b – 7)(9b + 7)
(x2 + y3) (m3 – 10) (2r2 + 9t) (12x + 5y)

121y2 − 36x2= 16w4 – 25z6 = x2y2 – 9z2 = 25n4 – 144 9u2 − 4v2 =
(11y – 6x) (4w2 + 5z3) (xy – 3z) (5n2 + 12) (3u + 2v)
(11y + 6x) (4w2 + 5z3) (xy – 3z) (5n2 + 12) (3u – 2v)
 GRADE 8
Learning Module for Junior High School Mathematics

ACTIVITY 7 :PERFECT MATCH

Fill in each blank in column A to make each expression a perfect
square trinomial. Then, match each completed perfect square trinomial
with its factors in column B.
A B
1 2
𝑥 + 12𝑥 + _______ MAN (𝑥 − 5)2
2 𝑥 2 − 10𝑥 + ______ ENOUGH (3𝑥 − 4𝑦)2
3 _____ − 22𝑥 + 121 WITH (𝑥 − 11)2
4 4𝑥 2 + ____ + 49 HIS (6𝑥 − 𝑦)2
5 9𝑥 2 − ______ + 16𝑦 2 NEAREST (5 + 2𝑥)2
6 _____ + 8𝑥𝑦 + 𝑦 2 COMES (10𝑥 + 9)2
7 25𝑥 2 − 100𝑥 + _____ PERFECTION (7 + 6𝑥)2
8 36𝑥 2 − 12𝑥𝑦 + ____ INSIGHT (2𝑥 + 7)2
9 16𝑥 2 + ______ + 9𝑦 2 THE (𝑥 + 6)2
10 ______ + 180𝑥 + 81 TO (4𝑥 + 𝑦)2
11 25 + _____ + 4𝑥 2 LIMITATIONS (4𝑥 + 3𝑦)2
12 49 + 84𝑥 + _____ ADMIT (5𝑥 − 10)2

➢ REVEAL THE QUOTE

1 2 3 4 5 6 7

8 9 10 11 6 12
-JOHANN WOLFGANG VON GOETHE, German poet and playwright

Activity: 8 Factoring Sum and Difference of Two Cubes

The Coronavirus pandemic (COVID-19) is now affecting every part of
the world, disrupting people’s lives and creating fear, anxiety, sorrow and
hardship. However, as the world endures quarantines and closures, God
offers peace and healing through His Word. In Psalm 23:4, we can find
strength and hope at this troubling time.

To find out what it says, factor the following binomials completely in

Column A. Then, write the phrase of words beside each item that
corresponds to your answer in Column B.

Column A Column B
1) 𝑥 − 64
3
____________ 2
(2𝑥 + 1)(4𝑥 − 2𝑥 + 1)
I walk through
2) 8𝑥 3 + 1 ____________ 3(𝑥 + 2𝑦)(𝑥 2 − 2𝑥𝑦 + 4𝑦 2 )
For You are
 GRADE 8
Learning Module for Junior High School Mathematics
3) 27𝑥 12 + 125𝑦 12 ____________ (𝑥𝑦 2 − 6)(𝑥 2 𝑦 4 + 6𝑥𝑦 2 + 36)
no evil,
4) 𝑥 9 + 𝑦 21 ____________ (10𝑥 2 + 𝑦 3 )(100𝑥 4 − 10𝑥 2 𝑦 3 + 𝑦 6 )
Theycomfort me
5) 𝑥 3 𝑦 6 − 216 ____________ 2(1 − 6𝑥)(1 + 6𝑥 + 36𝑥 2 )
Your rod
1 1 1
6) 3𝑥 3 + 24𝑦 3 ____________ (𝑥 − 3)(𝑥 2 + 3 𝑥 + 9)

and your staff

7) 343 − 𝑥 6 ____________ (3𝑥 4 + 5𝑦 4 )(9𝑥 8 − 15𝑥 4 𝑦 4 + 25𝑦 8 )
the darkest valley
8) 2 − 72𝑥 3 ____________ (𝑥 − 4)(𝑥 2 + 4𝑥 + 16)
even though
1
9) 𝑥 3 − 27 ____________ (𝑥 3 + 𝑦 7 )(𝑥 6 − 𝑥 3 𝑦 7 + 𝑦 14 )

I will fear
10) 1000𝑥 6 + 𝑦 9 ____________ (7 − 𝑥 2 )(49 + 7𝑥 2 + 𝑥 4 )
with me;

ACTIVITY 9: COLOR ME!

Directions: Factor each trinomial. Identify the binomial factors from below
and record the number with the color. Color the picture according to your
answers.
Trinomial A: x2 −14 48x+ Trinomial B: x2 + −3 40x

Trinomial C: x2 −13 30x− Trinomial D: x2 +16 63x+

Trinomial E: x2 −11 18x+ Trinomial F: x2 − −6 40x

Trinomial G: x2 +14 72x− Trinomial H: x2 + −5 6x

 GRADE 8
Learning Module for Junior High School Mathematics

Trinomial I: x2 −10 21x+ Trinomial J: x2 +15 36x+

Trinomial K: x2 − −9 70x Trinomial L: x2 +11 10x+

1. x−4 2. x−15 3. x+12

Numbers

4. x+4 5. x−7 6. x−6

7. x+9 8. x+1 9. x−1
10. x−14 11. x+8 12. x−9
Red: x−5 Orange: x−10 Yellow: x+7
Colors

Light Green: x+2 Dark Green: x+5 Light Blue: x−2

Dark Blue: x−8 Light Purple: x+10 Dark Purple: x+18
Pink x−3 Gray: x+6 Black: x+3
 GRADE 8
Learning Module for Junior High School Mathematics

WHAT I HAVE LEARNED

• For all polynomials, first factor out the greatest common factor (GCF).
• For a binomial, check to see if it is any of the following:
1. difference of squares: x 2 – y 2 = ( x + y) ( x – y)
2. difference of cubes: x 3 – y 3 = ( x – y) ( x 2 + xy + y 2)
3. sum of cubes: x 3 + y 3 = ( x + y) ( x 2 – xy + y 2)
• For a trinomial, check to see whether it is either of the following
forms:
1. x 2 + bx + c:
➢ List down all the factors of the last term;
➢ Identify which factor pair sums up the middle term; then
➢ Write each factor in the pairs as the last term of the binomial
factors.
2. ax 2 + bx + c:
STEP 1: Identify a, b and c
STEP 2: Multiply a and c
STEP 3: Write down all possible factors of ac
STEP 4: Identify which factor sums up to b
STEP 5: Rewrite the original equation into four terms splitting
the middle term
STEP 6: Use factoring by grouping

3.
• For polynomials with four or more terms, regroup, factor each group,
and then find a pattern as in steps 1 through 3.

WHAT I CAN DO

Computers
Digital images are composed of thousands of tiny pixels rendered assquares,
as shown below. Suppose the area of a pixel is 4x 2 _ 20x _ 25. What is
thelength of one side of the pixel?
 GRADE 8
Learning Module for Junior High School Mathematics

ASSESSMENT
Choose the letter of your answer. If your answer is not among the choices
write E.
1. Which of the following will complete the statement below?
If the product is positive, then m and n have _____________.
e) Both negative or both positive signs
f) Opposite signs
g) Both negative signs
h) Both positive signs
2. Given the expression below, what is the sum and product of m and n: 𝑥² − 7𝑥 + 12?
b) Sum: +12 b) sum:-12 c)sum:-7 d) sum: 7
Product: -7 product: +12 product:+12 product: -12
3. Factor the following expression: 𝑥² − 3𝑥 − 18
b) (𝑥 − 6)(𝑥 + 3) b) (+6)(𝑥 − 3) c)(𝑥 − 6)(𝑥 − 3) d) (𝑥 + 6)(𝑥 − 4)
4. What is the factored form of 𝑥² − 49?
a)(𝑥 − 7)(𝑥 + 7) b) (𝑥 − 49)(𝑥 + 49) c)(𝑥 − 7)(𝑥 − 7) d.(𝑥 − 49)(𝑥 − 49)
5. Factor the following expression: 𝑥 2 + 25?
a)(𝑥 − 5)(𝑥 − 5) b) (𝑥 + 25)(𝑥 + 25) c)(𝑥 + 5)(𝑥 − 5) d)(𝑥 − 25)(𝑥 − 25)
6. What is the factored form of 𝑥² − 6𝑥 + 8?
a)(𝑥 − 2)(𝑥 + 4) b) (𝑥 − 8)(𝑥 + 1) c)(𝑥 − 2)(𝑥 − 4) d.(𝑥 − 8)(𝑥 − 1)
7. Factor the following expression: 𝑥² + 8𝑥 + 12
a)(𝑥 − 6)(𝑥 − 2) b) (𝑥 + 6)(𝑥 + 2) c)(𝑥 + 4)(𝑥 − 3) d)(𝑥 − 4)(𝑥 − 3)
8. What is the GCF of 2𝑥² − 4𝑥 − 6𝑥² = 2𝑥(𝑥 − 2 − 3𝑥²)?
a) 2𝑥 b)(𝑥 − 2 − 3𝑥 2 ) c)−4𝑥 d)2𝑥(𝑥 − 2 − 3𝑥 2 )
9. Which of the following is equivalent to 18𝑥 + 12𝑦?
a) 6(3𝑥 + 2𝑦) b)2(9𝑥 + 8𝑦) c)3(6𝑥 + 5𝑦) d)4(5𝑥 + 3𝑦)
10. Which of the following is equivalent to the equation 𝑥 + 5 = (𝑥 + 3)²?
a)(𝑥 + 1)(𝑥 + 4) = 0 b)(𝑥 − 1)(𝑥 + 4 = 0 c)(𝑥 + 1)(𝑥 + 4) = 5 d)(𝑥 + 1)(𝑥 + 4) = 3
11. Which of the following is a quadratic expression with a > 1?
a)4𝑥 2 + 3𝑥 − 5 b) – 𝑥 2 − 2𝑥 − 1 c)3𝑥 2 − 𝑥 − 2 d) 5𝑥 2 + 25
12. Which of the following is a quadratic expression where a > 1?
a) 𝑥² − 7𝑥 − 8 b) 5𝑥² − 5𝑥 − 280 c) 4𝑥² − 9 d) 9𝑥² + 𝑥 − 10
13. Factor 𝑥 − 8.
3

a)(𝑥 + 2)(𝑥 2 − 2𝑥 + 4) b)(𝑥 − 2)(𝑥 2 + 2𝑥 + 4) c)(𝑥 − 5)(3𝑥 + 1) d)(3𝑥 − 5)(𝑥 + 1)

14. Factor 𝑥 3 + 27 completely.
a)(𝑥 + 3)(𝑥 2 − 3𝑥 + 9) b)(𝑥 + 3)(𝑥 2 − 3𝑥 + 18)
c)(𝑥 + 3)(𝑥 2 − 3𝑥 − 9) d)(𝑥 + 5)(𝑥 − 3)
15. Which of the following is a quadratic expression with a=1?
b) 𝑎² − 3𝑥 − 2 𝑏)2𝑥² + 4𝑥 − 5 𝑐) − 𝑥² + 4𝑥 + 8 𝑑)2𝑥² − 2𝑥 + 7
 GRADE 8
Learning Module for Junior High School Mathematics

ADDITIONAL ACTIVITIES
 GRADE 8
Learning Module for Junior High School Mathematics

E-Search
➔ https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-factor
➔ https://bit.ly/2VKO35v
➔ https://bit.ly/2VKbVG9
➔ https://bit.ly/2xht4O9
➔ https://bit.ly/3bNrqTw
➔ https://bit.ly/2VLBwPa
➔ https://bit.ly/2VNwH8a
➔ https://bit.ly/2yTF3So
➔ https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-factor

REFERENCES
-
• https://www.dummies.com/education/math/algebra/how-to-factor-
the-difference-of-two-perfect-cubes/
• https://www.dummies.com/education/math/algebra/how-to-factor-the-sum-of-
two-perfect-cubes/
• https://www.basic-mathematics.com/factoring-perfect-square-
trinomials.html
• https://www.cliffsnotes.com/study-guides/algebra/algebra-ii/factoring-
polynomials/summary-of-factoring-techniques
• https://www.britannica.com/biography/Johann-Wolfgang-von-Goethe
• Worktext in Mathematics (E-Math) by Orlando A, Orence and Marilyn O.
Mendoza
 GRADE 8
Learning Module for Junior High School Mathematics

PISA-Based Worksheet

CLOSET REMODELING
My mother plans to remodel our closet. She measured the dimensions of
the closet and drew the layout of her desired design and measurements.
She hired a carpenter to do the task. Unfortunately, the layout was lost
and mother only remembers the area of portion A and C portion. She
sketched again the diagram and include the area of portiona A and C.
Based on the diagram potions A and B are rectangles while C are
congruent squares.

Now, the carpenter has to figure out the dimensions of each portion.

Read and answer the following questions.

1. What did mother want to do with the closet? _________________________
2. What did mother do so that the carpenter will be able to do his task?
_____________________________________________________________________
3. What happened with the layout that mother prepared? _______________
4. What did mother do, when she learned that the layout was lost? _____
5. According to mother she only remembers the area of each portion,
what do you think should the carpenter do in order to find the
dimension of each portion? __________________________________________
6. Based on the diagram, what are the dimensions of
portion A? __________________________________________________________
portion C? __________________________________________________________
portion B? __________________________________________________________
7. If the value of x is 5 inches, find the dimensions of each portion. _____
8. What are the dimensions of the entire closet? ________________________

16