DLL June Week 1 Factoring Polynomials

GRADE 8 DAILY LESSON LOG

School Rufo de la Cruz Integrated School Teacher DIANA LIZA G. BAÑO Teaching Dates and Time June 11-15, 2017 10:00-11:00 Grade 8 – Love 11:00-12:00 Grade 8 – Faith Faith 1:00-2:00 Grade Grade 8 - Hope Hope

Monday

Tuesday

Grade Level 8 Learning Area Mathematics Quarter First Quarter

Wednesday

Thursday

Friday

I. OBJECTIVES

A. Content Standards B. Performance Standards C. Learning Competencies/Objectives Write the LC code for each

The learner demonstrates understanding understanding of key concepts of factors of polynomials, rational algebraic expressions, linear equations and inequalities in two variables, systems of linear equations and inequalities in two variables and linear functions. The learner is able to formulate real-life problems involving factors of polynomials, rational algebraic expressions, linear equations and inequalities in two variables, systems of linear equations and inequalities in two variables and linear functions, and solve these problems accurately using a variety of strategies.

Lesson 1

Lesson 1

Factors completely different

Lesson 1


Lesson 1


Lesson 1



types of polynomials





(polynomials with common





monomial factor, difference of

monomial factor, difference




two squares, sum and

of two squares, sum and




difference of two cubes,





perfect square trinomials, and

perfect square trinomials,




general trinomials).

and general trinomials).




(M8AL-Ia-b-1)

(M8AL-Ia-b-1)

(M8AL-Ia-b-1)

(M8AL-Ia-b-1)

(M8AL-Ia-b-1)

Day 1 Objective:

Day 2 Objective:

Day 3 Objective:

Day 4 Objective:

Day 5 Objective:

Factors completely polynomials with common monomial factor.

Factors completely difference

Factors completely sum or

Factors completely perfect


of two squares.

difference of two cubes .

square trinomials.

types of polynomials (polynomials with common monomial factor, difference of two squares, sum and difference of two cubes, perfect square trinomials).

FACTORING DIFFERENT TYPES TYPES OF POLYNOMIALS II. CONTENT

FACTORING POLYNOMIALS WITH COMMON MONOMIAL FACTOR

FACTORING DIFFERENCE OF TWO SQUARES.

FACTORING SUM AND DIFFERENCE OF TWO CUBES

FACTORING PERFECT SQUARE TRINOMIAL

FACTORING POLYNOMIALS WITH COMMON MONOMIAL FACTOR, DIFFERENCE OF

TWO SQUARES, SUM AND DIFFERENCE OF TWO CUBES, PERFECT SQUARE TRINOMIAL

III. LEARNING RESOURCES

A. References 1. Teacher’s Guide pages 2. Learner’s Material pages

31,33 29,31

34-35 33-34

37 35

35-40 37-38

3. Textbook pages 4. Additional Materials from Learning Resource (LR) portal B. Other Learning Resources

e-math pp. 437-439

e-math pp. 443-445

e-math pp. 448-449, Elem. Alg. pp.209

e-math pp. 451-455, Elem Alg. pp.206-207

Today, we will learn how to factor polynomials with common monomial factor.

Today, we will learn how to factor polynomials which are difference of two squares.

Today, we will learn how to factor polynomials which are sum of two cubes.

Today, we will learn how to factor polynomials which are perfect square trinomials.

It is important to factor

It is important to factor

It is important to factor sum

polynomials with common

difference of two squares to

of two cubes to prepare

monomial factor to prepare

prepare ourselves in solving

ourselves in solving problems

ourselves in solving problems

problems involving factoring.

involving factoring.

It is important to factor perfect square trinomials to prepare ourselves in solving problems involving factoring.

At the end of the lesson, you


IV. PROCEDURES

Learning Episode 1: A. MOTIVATION 1. Presentation

2. Importance

involving factoring.

3. Formative



Today, we will have our independent learning activity on factoring completely different types of polynomials specifically polynomials with common monomial factor, difference of two squares, sum and difference of two cubes, and perfect square trinomials It is important that you will be able tofactor completely different types of polynomials to prepare yourselves in solving mathematical problems involving factoring. In the Independent learning

common monomial factor.

B. PROBE AND RESPOND 1. Review /Drill

2. Pre-requisite Skills

are difference of two squares.

are sum of two cubes.

Factor completely the following polynomials. 1. 12a2 – 16a4 2. 8d + 2d2 – 4d3

Teacher defines factoring and greatest common monomial factor (GCF) and state the steps in factoring the greatest common monomial factor Teacher says: Factoring is the process of finding the factors of a number or polynomial. It is the reverse process of getting the products. GETTING THE COMMON MONOMIAL FACTOR Greatest Common Factor (GCF) is the greatest factor/number that can divide each given number. Greatest Common Monomial Factor (GCMF) is the greatest factor contained in every term of an algebraic expression. GCMF is the common factor having the greatest numerical factor and with variables having the least degree.

Factor completely the following polynomials. 1. x2 – y2 2. h2 – 4

which are perfect square trinomials. Factor completely the following polynomials. 1. x3 – 8 2. x3+27

Expected Answer: 1. 4a2(3 – 4a2) 3. 2d(4+ 2d – 2d2)

Expected Answer: 1. (x + y)(x – y) 3. (h + 2)(h – 2)

Expected Answer: 1. (x – 2)(x2 + 2x + 4) 2. (x + 3)(x 2 – 3x + 9)

Teacher describe a polynomial which is a difference of two squares.

Teacher describe and state the steps in factoring a polynomial which is a sum or difference of two cubes (STC/DTC)

Teacher defines perfect square trinomial and state the steps in factoring perfect square trinomial (PST).

Teacher says: Sum/Difference of two cubes is the product of a special case of a binomial factor and a trinomial factor.

Teacher says:

Teacher says: Difference of two squares (DTS) is the product of multiplying the sum and difference of two terms. So, the factored form of a polynomial that is a difference of two squares (DTS) is the sum and difference of the square roots of the first and last terms. Pattern: (F stand for first term and L stands for the last term) F

2

2

 L









 F  L  F  L 

F

2



2

L

F

2



2

L



Here are the steps to factor polynomials that is a difference of two cubes (DTC). a. Find the cube roots of the first and last terms. b. Write their difference as the first factor. c. For the second factor, get the trinomial factor by: i. Squaring the first term of the first factor; ii. Adding the product of the first and second terms of the first factor.

Perfect square trinomial is the result of squaring a binomial. A perfect square trinomial has first and last terms which are perfect squares and a middle term which is twice the product of the square root of first and last terms. The following are the PST: x2 + 4x + 4 x2 + 6x + 9

x2 + 2xy + y 2 Reasons why they are PST: 1. Two of the terms must be perfect squares, x 2 and y2. 2. There must be no sign before x2 or y2. 3. If you multiply x and y and

items worth 20 points in all.

iii. Squaring the last term of the first factor d. Write them in factored form.

Here are the steps to factor polynomials with common monomial factor. a. Find the greatest common factor of the numerical coefficients. b. Find the variable with the least exponent that appears in each term of the polynomial. c. The product of the greatest common factor in (a) and (b) is the GCF of the polynomial. d. To completely factor the given polynomial, divide the polynomial by its GCF, the resulting quotient is the other factor.

Here are the steps to factor polynomials that is a sum of two cubes (STC). a. Find the cube roots of the first and last terms. b. Write their sum as the fi rst factor. c. For the second factor, get the trinomial factor by: i. Squaring the first term of the first factor; ii. Subtracting the product of the first and second terms of the first factor. iii. Squaring the last term of the first factor d. Write them in factored form. Pattern: (F stand for first term and L stands for the last term) F 

3

double the result, you get the second term, 2xy, or its additive inverse, -2xy. After recognizing a PST, we can now proceed to how PST is factored. These are the rules: 1. Get the square root of the first and last terms. 2. List down the square root as sum/difference of two terms as the case may be. Note: PST come in two forms: one in which the middle term is positive and the other in which the middle term is negative.

You can use the following relationships to factor perfect square trinomials: F

F

Learning Episode 2: A. Modelling

The teacher gives illustrative examples on factoring polynomials with common

The teacher gives illustrative examples on how to factor difference of two squares.

2

2 FL  L

 F  L  or  F  L  F  L  2



3

2

2

 FL  L



F

3

 L

 F  L  F

2

2

 FL  L



The teacher gives illustrative examples on how to factor sum of two cubes.

2



2

2 FL  L

 F  L  or  F  L  F  L  





 L

 F  L  F 3

2

2

The teacher gives illustrative examples on how to factor difference of two cubes.

Factor completely:

1. 4x + 8 Solution: a. Find the greatest common factor of the numerical coefficients. The GCF of 4 and 8 is 4. b. Find the variable with the least exponent that appears in each term of the polynomial. x only appears on the first terman. Therefore, we don’t have a common variable. c. The product of the greatest common factor in (a) and (b) is the GCF of the polynomial. Factors of 4x: ( 2 2)( x ) Factors of 8: (23) The GCF is (22) or 4. Hence, 4 is the GCF of 4x + 8 . d. To completely factor the given polynomial, divide the polynomial by its GCF, the resulting quotient is the other factor. Thus, the factored form of 4x + 8is 4 (2x + 2)

2. 15x2 – 55x Solution: a. Find the greatest common factor of the numerical coefficients. The GCF of 15 and 55 is 5. b. Find the variable with the least exponent that appears in each term of the polynomial.

Examples: 1. 4x2 – 36y2 Solution: The square root of 4x2 is 2x and the square root of 36y2 is 6y. To write their factors, write the product of the sum and difference of the square roots of 4x2 – 36y2 , that is ( 2x + 6y) (2x – 6y). 2. x 2 – 36 = ( x + 6)( x – 6) Solution: √ =  ; √ 36 = 6 3. 16a6 – 25b2 = (4a3 – 5b)(4a3 + 5b) Solution: √16 = 4 ; √25 

= 5

Examples:

Examples:

1. a3 + 512

1. 9x2 + 30x + 25

Solution: The cube root of a 3 is a. The cube root of 512 is 8. So, ( a + 8 ) is the first factor. To get the second factor: Square the first term of the fist factor ( a + 8), that is (a) 2 = a2 Subtract the product of the first and second terms of the first factor (a + 8), that is (8)a = -8a. Square the last term of the first factor ( a + 8 ), that is (8) 2 = 64 So, ( a2 – 8a + 64 ) is the second factor. The factored form of a3 + 512 is ( a + 8)( a 2 – 8a + 64 ).

solution: The square root of 9x 2 is 3x. The square root of 25 is 5. (3x)2 + 2(5.3x) + 5 2 So, the factors are (3x + 5)(3x + 5) or (3x + 5) 2

2. Factor x3 + 729 a. Cube roots of the f irst and last terms:   √ =  ; √ 729 = 9 b. Binomial Factor: (x + 9) c. Trinomial Factor: (x)2 – (x)(9) + (9) 2 = (x2 – 9x + 81) d. The factored form of x3 + 729 = (x + 9)(x 2 – 9a + 81) 3

3. a – 512 Solution: The cube root of a 3 is a.

2. n2 + 16n + 64 Solution: a. Since n2 = (n)2 and 64 = (8)2, then both the first and last terms are perfect squares. And 2( n)(8) = 16n, then the given expression is a perfect square polynomial. b. The square root of the first term is n and the square root of the last term is 8, then the polynomial is factored as ( n + 8)2

3. x2 – 22x + 121 Solution: The square root of x2 is x. The square root of 121 is 11. X2 – 2(11 . x) + (11) 2 So, the factors are (x – 11)(x – 11) 0r (x – 11)2. 4. 4r 2 – 12r + 9 Solution: a. Since 4 r 2 = (2r )2 and 9 = (3)2, and since – 12r = 2(2 )(3) then it follows the

xis both common to all terms and 1 is the smallest exponent for x, thus, x is the GCF of the variables. c. The product of the greatest common factor in (a) and (b) is the GCF of the polynomial. Factors of 15x 2: (3) (5) (x2) Factors of 55x: (5) (11) (x) The GCF is (5)(x) Hence, 5x is the GCF of 15x2 – 55x. d. To completely factor the given polynomial, divide the polynomial by its GCF, the resulting quotient is the other factor. Thus, the factored form of 15x 2 – 55x is 5x(3x – 11)

So, ( a – 8 ) is the first factor. To get the second factor: Square the first term of the fist factor ( a – 8), that is (a)2 = a2 Add the product of the first and second terms of the first factor (a – 8), that is (8)a =

c. 27a2 – 36a3 + 45a5

b. Binomial Factor: (7 – y) c. Trinomial Factor: (7)2 + (7)(y) + (y) 2 = (49 + 7y + y 2) d. The factored form of 343 – y 3 = (7 – y)( 49 + 7y + y 2)

8a. Square the last term of the first factor ( a – 8), that is (8)2 = 64 So, ( a2 + 8a + 64 ) is the second factor. The factored form of a3 – 512 is( a – 8)( a2 + 8a + 64 ). 4. Factor 343 – y3 a. Cube roots of the f irst and last terms:  √ 343 = 7 ;    = 

Solution: Find the GCF of each term. Factors of 27a2 : (33)(a2) Factors of 36a3: (22)(32)(a3) Factors of 45a5: (32)(5)(a5) The GCF is (32)(a2) or 9a2. So, 27a2 – 36a3 + 45a 5= 9a2( 3 – 4a + 5a3).

Learning Episode 3: A. Guided Practice

Factor completely the following polynomials. 1. 5x + 10 2. 25y3 – 55y2 3. 12b4 – 16b2 + 20b3 Expected Answer:

given expression is a perfect square trinomial. b. The square root of the first term is 2r and the square root of the last term is 3 so that its factored form is (2r – 3)2.

Factor completely the following polynomials. 1. h2 – 4 2. 36x2 – 25y4 3. 25x4 – 9y6 Expected Answer:

Factor completely the following polynomials. 1. z3 + 1 = 2. b3 – 125

Factor the following: 1. x2 + 20x + 100 2. 4 – 36m + 81m2 3. 16a2 + 12 + 9

Expected Answer: 1. (z + 1)(z2 – z + 1)

Expected Answer: 1. (x + 10) 2

2

2

B. Independent Practice

Learning Episode 4: A. Evaluation

2. 5y2(5y – 11) 3. 4b2(3b2 – 4 + 5b) Factor completely the following polynomials. 1. 8b – 2 2. 12x3 – 24x2 3. 2y3 + 16y + 32y 2

2. (6x + 5y2)( 6x – 5y2) 3. (5x2 + 3y3)( 5x2 – 3y3) Factor completely the following polynomials. 1. w2 – 64 2. 49x2 – 4 3. 121a2 – 16c4

Expected Answer: 1. 2(4b – 1) 2. 12x2(x – 2) 3. 2y (y 2 – 8 + 16y)

Expected Answer: 1. (w + 8)(w – 8) 2. (7x + 2)( 7x – 2) 3. (11a + 4c 2)( 11a – 4c2)

Factor completely the following polynomials. 1. 2a + 4 2. 3y10 – 12y7 3. 18x3 – 27x + 3x 2

Factor completely the following polynomials. 1. b2 – 49 2. 100h4 – 64 3. 4m6 – 81n8

Expected Answer: 1. 2(a + 2) 2. 3y7(y3 – 4) 3. 3x(6x2 – 9 + x)

Expected Answer: 1. (b + 7)(b – 7) 2. (10h2 + 8)( 10h2 – 8) 3. (2m2 + 9n4)( 2m2 – 9n4)

3. (4a + 3) 2 Factor completely the following polynomials. 1. 64 + a 3 2. 27 – q3

Factor the following: 1. d2 – 6d + 9 2. 25 + 10n + n 2 3. 4f 2 – 20f + 25

Expected Answer: 1. (4 + a)(16 – 4a + a 2) 2. (3 – q)(9 + 3q + q 2)

Expected Answer: 1. (d – 3)2 2. (5 + n) 2 3. (2f – 5)2

Factor completely the following polynomials. 1. 8 + c 3 2. t3 –216

Factor completely. 1. 9a2 – 6a + 1 2. b2 + 24b + 144 3. 49 + 70c + 25c 2

Expected Answer: 1. (2 + c)( 4 – 2c + c 2) 2. (t – 6)(t2 + 6t + 36)

Expected Answer: 1. (3a – 1)2 2. (b + 12) 2 3. (7 – 5c)2

Factor the following polynomials completely. (2pts each item) 1. xy +xz 2. 4x −20 3. 15x + 10x + 5 4. 25− 9r  5. 4x −   6. x  − 8 7. x  + 125 8. x  − 8 x + 1 6 9. x  +14x+49 10. 25x + 20x + 4 Expected Answer: 1. x(y + z) 2. 4(x – 5) 3. 5(3x2 + 2x + 1) 4. (5 + 3r)(5 – r) 5. (2x2 + y)(2x2 – y) 6. (x – 2)(x2 + 2x + 4) 7. (x + 5)(x 2 – 5x + 25) 8. (x – 4)2 9. (x + 7) 2 10. (5x + 2) 2

B. Assignment/Project

V. REMARK

VI.REFLECTION

A. No. of learners who earned 80% in the evaluation. B. No. of learners who require additional

Reflect on your teaching and assess yourself as a teacher. Think about your students’ progress this week. What works? What el se needs to be done to help the students learn? Identify what help your i nstructional supervisors can provide for you so when you meet them, you can ask them relevant questions.

who scored below 80%. C. Did the remedial lessons work? No. of learners who have caught up with the lesson.

D. No. of learners who continue to require remediation

E. Which of my teaching strategies worked well? Why did these work?

F. What difficulties did I encounter which my principal or supervisor can help me solve? G. What innovation or localized materials did I use/discover which I wish to share with other teachers?

ATTACHMENT (TUESDAY, JUNE 6, 2017) REMARKS: Lesson Outline STEM – Bohr and Becquerel

SPS – Ben Arda and Bea Lucero

Barley

Learning Episode 1

Learning Episode 1

Learning Episode 1

A. MOTIVATION Introduction: Today, we will learn have a short quiz on factoring polynomial with common monomial factor. Today also we will learn how to factor polynomials which are difference of two squares. Rationale: It is important to factor different types of polynomials to prepare ourselves in solving problems involving factoring. Assessment: At the end of the lesson, you will factor polynomials which are difference of two squares.

B. PROBE AND RESPOND A. Review/Drill: Do the evaluation on Lesson plan dated June 5, 2017.

B. Prerequisite Skills: Do the prerequisite skills on Lesson plan dated June 6, 2017. Learning Episode 2(MODELLING) Do the modelling on Lesson plan dated June 6, 2017. Learning Episode 3

A. GUIDED PRACTICE (Dyad/Triad) Do the guided practice on Lesson plan dated June 6, 2017.

B. INDEPENDENT PRACTICE (Individual Work) Do the independent practice on Lesson plan dated June 6, 2017. Learning Episode 4 (ASSESSMENT) A. Evaluation Do the evaluation on Lesson plan dated June 6, 2017.

B. A greement/Assignment

A. MOTIVATION Introduction: Today, we will continue to learn how to factor polynomials with common monomial factor.

A. MOTIVATION Introduction: Today, we will continue to learn how to factor polynomials with common monomial factor.

Rationale: It is important to factor different types of

Rationale: It is important to factor different types of

polynomials to prepare ourselves in solving problems involving factoring. Assessment: At the end of the lesson, you will factor polynomials with common monomial factor.

polynomials to prepare ourselves in solving problems involving factoring. Assessment: At the end of the lesson, you will factor polynomials with common monomial factor.

B. PROBE AND RESPOND A. Review/Drill: Check the assignment and discuss the answer of the assignment given yesterday, June 5, 2017.

B. Prerequisite Skills: Learning Episode 2(MODELLING) The discussion of the answers of the assignment will serve as ex the modelling part


B. Prerequisite Skills: Learning Episode 2 (MODELLING) The discussion of the answers of the assignment will serve as ex the modelling part

also.

also.

Learning Episode 3

Learning Episode 3

A. GUIDED PRACTICE (Dyad/Triad) B. INDEPENDENT PRACTICE (Individual Work) Do the independent practice on Lesson plan dated June 5, 2017. Learning Episode 4 (ASSESSMENT) A. Evaluation Do the evaluation on Lesson plan dated June 5, 2017.

B. Agreement/Assignment

A. GUIDED PRACTICE (Dyad/Triad) B. INDEPENDENT PRACTICE (Individual Work) Do the independent practice on Lesson plan dated June 5, 2017. Learning Episode 4 (ASSESSMENT) A. Evaluation Do the evaluation on Lesson plan dated June 5, 2017.


ATTACHMENT (WEDNESDAY, JUNE 7, 2017) REMARKS: Lesson Outline STEM – Bohr and Becquerel


Barley

* Do the Lesson plan dated June 6, 2017.

*Reteach: FACTORING POLYNOMIALS WITH COMMON MONOMIAL FACTOR Learning Episode 1

*Reteach: FACTORING POLYNOMIALS WITH COMMON MONOMIAL FACTOR Learning Episode 1

A. MOTIVATION Introduction: Today, we will continue to learn how to factor

A. MOTIVATION Introduction: Today, we will continue to learn how to factor

polynomials with common monomial factor.

Rationale: It is important to factor different types of polynomials

polynomials with common monomial factor.

Rationale: It is important to factor different types of polynomials

to prepare ourselves in solving problems involving factoring.

to prepare ourselves in solving problems involving factoring.

Assessment: At the end of the lesson, you will factor polynomials

Assessment: At the end of the lesson, you will factor polynomials

with common monomial factor.


B. Prerequisite Skills: Learning Episode 2 (MODELLING) The discussion of the answers of the assignment will serve as ex the modelling part also. Learning Episode 3

A. GUIDED PRACTICE (Dyad/Triad) Factor completely the following polynomials. Expected Answer: 1. 10x + 51. 5(2x + 1) 2. 25x3 – 55x42. 5x3(5 – 11x) 3. 6x3 – 3x2 + 9x3. 3x(2x2 – x + 3)

B. INDEPENDENT PRACTICE (Individual Work) Factor completely the following polynomials. Expected Answer: 1. 2y + 41. 2(y + 4) 2. 6y5 + 3y22. 3y2(2y3 – 1) 3. 2y3+ 16y + 32y 23. 2y2(6y2 – 2y+ 9)(y2 – Learning Episode 4 (ASSESSMENT) A. Evaluation Factor completely the following polynomials. Expected Answer: 1. 9 – 18w1. 9(1 – 2w) 2. 8w5 – 4w22. 4w2(2w3 – 1) 3. 5w+ 10w2 – 15w33. 5w(1 + 2w – 3w2)


with common monomial factor.


B. Prerequisite Skills: Learning Episode 2 (MODELLING) The discussion of the answers of the assignment will serve as ex the modelling part also. Learning Episode 3

A. GUIDED PRACTICE (Dyad/Triad) Factor completely the following polynomials. Expected Answer: 1. 10x + 51. 5(2x + 1) 2. 25y3 – 55y 4 2. 5y3(5 – 11y) 3. 6x3 – 3x2 + 9x 3. 3x(2x2 – x + 3)

B. INDEPENDENT PRACTICE (Individual Work) Factor completely the following polynomials. Expected Answer: 1. 11c + 221. 11(c + 2) 2. 3d2 – 24d2. 3d(d – 8) 3. 12x4 – 4x3 + 18x23. 2x2(6x2 – 2x + 9)(y2 – Learning Episode 4 (ASSESSMENT) A. Evaluation Factor completely the following polynomials. Expected Answer: 1. 9 – 18w1. 9(1 – 2w) 2. 6y5 – 3y2 2. 3y2(2y3 – 1) 3. 5x + 10x2 – 15x33. 5x(1 + 2x – 3x2)


ATTACHMENT (THURSDAY, JUNE 8, 2017) REMARKS: Lesson Outline STEM – Bohr and Becquerel


Barley

* Do the Lesson plan dated June 7, 2017.

Learning Episode 1

Learning Episode 1



B. Prerequisite Skills: Do the prerequisite skills on Lesson plan dated June 6, 2017. Learning Episode 2(MODELLING) Do the modelling on Lesson plan dated June 6, 2017. Learning Episode 3




B. Prerequisite Skills: Do the prerequisite skills on Lesson plan dated June 6, 2017. Learning Episode 2 (MODELLING) Do the modelling on Lesson plan dated June 6, 2017. Learning Episode 3


B. INDEPENDENT PRACTICE (Individual Work) Do the

B. INDEPENDENT PRACTICE (Individual Work) Do the

independent practice on Lesson plan dated June 6, 2017. Learning Episode 4 (ASSESSMENT) A. Evaluation Do the evaluation on Lesson plan dated June 6, 2017.

independent practice on Lesson plan dated June 6, 2017. Learning Episode 4 (ASSESSMENT) A. Evaluation Do the evaluation on Lesson plan dated June 6, 2017.



DLL June Week 1 Factoring Polynomials

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