Math 8_q1_mod1

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 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e. songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. 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 Zenaida W. HaliliJean Editor: Luningning R.Tayamora Rene V. Salgado

Mary Ann L. Abundo Aiko C. Domingo

Judy Ann G. Gallo

Judy Ann G. Gallo 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

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

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GRADE 8 Learning Module for Junior High School Mathematics MODULE

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FACTORING POLYNOMIALS

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

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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.

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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. Step 2: Find the greatest common factor (GCF). Step 3: Divide each term by the GCF. Step 4: Write the GCF and the result from step 3 together. Solve It Another Way!

3(x)+3(2) GCF = 3 3𝑥 6 + = 𝑥+2 3 3 3(x+2)

GRADE 8 Learning Module for Junior High School Mathematics

EXAMPLE 2: Factor (4x+12y). Step 1: Find the greatest common factor (GCF). Step 2: Write the GCF as a factor of each term. Step 3: Write out, or factor out, the GCF using the distributive property.

4 is the GCF 4(x)+4(3y) 4(x+3y)

Tips • •

Key • •

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. 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. Step 2: Find the values of a and b by getting the square root of each. Step 3: Substitute the values of a and b into the formula for difference of two squares. Thus, the factors of x2−4 are x−2 and x+2.

a2=x2 and b2=4 a2=x2|b2=4 a=x|b=2 2 a −b2=(a−b)(a+b) (x)2−(2)2=(x−2)(x+2)

EXAMPLE 2: Factor 9x2−4 . Step 1: Identify a2 and b2 in the expression. Step 2: Find the values of a and b by getting the square root of each. Step 3: Substitute the values of a and b into the formula for difference of two squares. Thus, the factors of 9x2−4 are 3x−2 and 3x+2.

a2=9x2 and b2=4 a2=9x2|b2=4 a=3x|b=2 2 a −b2=(a−b)(a+b) (3x)2−(2)2=(3x−2)(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).

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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 •

•

Key • • •

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

a = x2 a=x

b=1 b=1

(x + 1)2

a = p2 a=p

b = 81 b=9

(p - 9)2

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 STEP 2: multiply a and c STEP 3: write down all possible factors of ac

a = 3 b = 14 c = 8 ac = (3) (8) = 24 factors: 2 (12) -3(-8) -2 (-12) 4(6) 3(8) -4(-6)

STEP 4: identify which factor sums up to b

b = 14 2 + (12) = 14

STEP 5: rewrite the original equation into four terms splitting the middle term STEP 6: use factoring by grouping

3x2 + 14x + 8 3x2 + 12x + 2x + 8 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 Monomial Factor (CMF) 5 3xy2

Polynomial 5x + 10 3a4 – 9a3b + 6a2b2

Quotient of Polynomial and CMF x+2

Factored Form 5(x + 2) 3xy2(2x + y)

a2 – 3ab + 2b2 2 – 5abc + 4a2b2c2

3a2b2 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 B C D E

8x2 – 80x 9y2 – 36y 4x2 + 32xy 12x2y – 8xy2 36x4 – 42x2

G I L N O

4y2

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

9y

15x3y2 – 30xy 18x4y + 9xy7 6x4 – 10x3 + 2x 8y5 – 24y4 – 16y2 5x3y – 15xy2 +25xy

5xy

6x2

4x3

R S T V

3x2

4x5 – 8x4 – 4x3 5x3y – 20x2y2 + 100xy 15x3y2 – 20x2y3 + 12x4y 6x5 –15x4 – 21x3 + 27x2

6x2

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 2. 𝑥 2 − 7𝑥 + 12 3. 𝑥 2 − 4𝑥 + 4 4.𝑥 2 − 16 5.𝑥 2 + 5𝑥 6.𝑥 2 − 81

7.𝑥 2 − 18𝑥 + 81 8. 6𝑥 2 − 3𝑥 9. 8𝑥 3 − 1 10. 𝑥 3 + 27𝑦 3 11. 4𝑥 2 − 20𝑥 + 25

GRADE 8 Learning Module for Junior High School Mathematics 1

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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.

2.

x2 – y2

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 inside the box, if NO, escape this part.

If your answer is YES, write each factor inside the box, if NO, escape this part.

3.

4.

16x4 – 49y2

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

If your answer is YES, write each factor inside the box, if NO, escape this part.

YES

NO

If your answer is YES, write each factor inside the box, if NO, escape this part.

GRADE 8 Learning Module for Junior High School Mathematics 5.

6.

25x2 – 10y2

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 inside the box, if NO, escape this part.

If your answer is YES, write each factor inside the box, if NO, escape this part.

7.

8.

121x2 + 9y2

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

If your answer is YES, write each factor inside the box, if NO, escape this part.

NO

If your answer is YES, write each factor 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. a2 – b2 = (a + b)(a – b)

y2 – 4 = (y + 4)(y + 4)

b4 – 16 = (b2 – 4)(b2 – 4)

9a2 – 4b2 = (3a – 2b) (3a – 2b)

m2– 9 = (m – 3)(m + 3)

4x2 – 25 = (2x + 5)(2x – 5)

36x2 − y2 = (6x – y)(6x + y)

36x2 – 1 = (6x – 1)(6x – 1)

p2– 144 = (p + 12)(p – 12)

9b2– 25 = (3b + 5)(3b – 5)

a2b2 – 16 = (ab – 4)(ab + 4)

4b4 – 49d2 = (2b2 + 7d) (2b2 + 7d)

25m2– 9 = (5m + 3) (5m – 3)

100z6 – 81 = (10z3 – 9) (10z3 – 9)

49x2 − 4y2 = (7x – 2y) (7x + 2y)

81b2 – 49 = (9b – 7)(9b + 7)

x4 – y6 = (x2 + y3) (x2 + y3)

m6 – 100 = (m3 – 10) (m3 – 10)

4r4 – 81t2 = (2r2 + 9t) (2r2 + 9t)

144x2 − 25y2 = (12x – 5y) (12x + 5y)

121y2 − 36x2= (11y – 6x) (11y + 6x)

16w4 – 25z6 = (4w2 + 5z3) (4w2 + 5z3)

x2y2 – 9z2 = (xy – 3z) (xy – 3z)

25n4 – 144 (5n2 + 12) (5n2 + 12)

9u2 − 4v2 = (3u + 2v) (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 2 1 MAN 𝑥 + 12𝑥 + _______ (𝑥 − 5)2 2 ENOUGH 𝑥 2 − 10𝑥 + ______ (3𝑥 − 4𝑦)2 3 WITH _____ − 22𝑥 + 121 (𝑥 − 11)2 4 HIS 4𝑥 2 + ____ + 49 (6𝑥 − 𝑦)2 5 NEAREST 9𝑥 2 − ______ + 16𝑦 2 (5 + 2𝑥)2 6 COMES _____ + 8𝑥𝑦 + 𝑦 2 (10𝑥 + 9)2 7 PERFECTION (7 + 6𝑥)2 25𝑥 2 − 100𝑥 + _____ 8 INSIGHT 36𝑥 2 − 12𝑥𝑦 + ____ (2𝑥 + 7)2 9 THE 16𝑥 2 + ______ + 9𝑦 2 (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

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-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. 1) 𝑥 − 64 3

Column A ____________

Column B (2𝑥 + 1)(4𝑥 − 2𝑥 + 1) 2

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

6) 3𝑥 3 + 24𝑦 3

____________

1

1

1

(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 J: x2 +15 36x+

Trinomial K: x2 − −9 70x

Trinomial L: x2 +11 10x+

Colors

Numbers

Trinomial I: x2 −10 21x+

1. x−4

2. x−15

4. x+4

5. x−7

6. x−6

7. x+9

8. x+1

9. x−1

10. x−14 Red: x−5 Light Green: x+2 Dark Blue: x−8 Pink x−3

11. x+8 Orange: x−10 Dark Green: x+5 Light Purple: x+10 Gray: x+6

3. x+12

12. x−9 Yellow: x+7 Light Blue: x−2 Dark Purple: x+18 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 3 13. Factor 𝑥 − 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) 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 -

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https://www.dummies.com/education/math/algebra/how-to-factorthe-difference-of-two-perfect-cubes/ https://www.dummies.com/education/math/algebra/how-to-factor-the-sum-oftwo-perfect-cubes/ https://www.basic-mathematics.com/factoring-perfect-squaretrinomials.html https://www.cliffsnotes.com/study-guides/algebra/algebra-ii/factoringpolynomials/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? ________________________

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