Factors polynomial using sum and difference of two squares

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(toc) Table of Contents

Content Standard

The learner demonstrates understanding of key concepts of factors of polynomials (Polynomials with common monomial factor)

Performance Standard

The learner is able to formulate real-life problems involving factors of polynomials (with common monomial factor)

Competency

Factors completely different types of polynomials (polynomials with common monomial factor), M8AL-Ia-b-1


I. OBJECTIVES

Tells whether the given polynomials can be factored using sum and difference of two squares or not.


II. CONTENT

Factoring Sum and Difference of Two Squares


III. LEARNING RESOURCES

Teacher’s Guide (TG) in Mathematics 8, pp. 34 - 35

Learner’s Module (LM) in Math 8, pp. 32 - 33

Moving Ahead With Mathematics, pp. 196 - 197

IV. PROCEDURES

ACTIVITY: REMEMBER ME?
The teacher will guide the students to answer the following;
Recall finding the special product in this form:

$$( x + 8) ( x – 8 ) = x^2 – 64$$
$$( 2x + 4) ( 2x – 4) = 4{x^2} – 16$$
$$( 3a + 5) ( 3a – 5) = 9{a^2} – 25$$

QUESTIONS:

1. What have you observe on the product?
Expected answer: when we multiply two binomials with positive and
negative signs in between the product has two terms.
2. What do you call this product?
Ans. Binomial
3. Is there a relation between the special product and sum and difference
of two squares based on the given above?
Ans. Yes!

Teacher will process the different responses of the learners.

B. Establishing a purpose for the lesson

ACTIVITY: FIND MY PRODUCT!

Note: Let the learners identify the pattern.

a. $(x + 1) (x – 1)$

$= {x ^2} – 1$

b. $(x + y) (x – y)$

$= x ^2 – y^ 2$

1. What is the product of the binomials?

Ans. $x ^2 – 1$ and ${x^2} –{ y^2}$

2. Did you observe any pattern?

Ans. Yes!

3. If we are going to reverse the process, is it possible to find any

pattern?

Ans. Yes, it is possible.

Teacher must process the responses of the learners.


C. Presenting examples of the new lesson

ACTIVITY: Let’s EXPLORE!

Examples of Factoring polynomials using sum and difference of two squares.

1. ${x^ 2} – 1$

$= (x + 1) (x – 1)$

2. $x^2 – y^2$

$= (x + y) (x – y)$

3. $x^2 – 4$

$= (x + 2)(x – 2)$

QUESTIONS:

1. How many factors did you obtain?

Ans. Two factors

2. What are your observations based on the factors?

Expected ans. The factors are the positive square roots of each
term.

3. What is the operation on the first factor? How about the second
factor?

Ans. First factor- positive, second- negative
(Teacher must guide every responses of the learner and discuss the topic)

D. Discussing new concepts and practicing new skills #1

ACTIVITY: COMPARE US!

Take a look of the following:

$(x + 1)(x-1)  \to  x^2 – 1 \to  (x+1)(x-1)$

1. What is being shown on the first arrow?

Ans. Showed the product of two binomials.

2. How about the second arrow?

Ans. Showed the factors of the product

3. What are your observation/s?

Expected ans. The factors are the positive square roots of each term.


Teacher must guide the different responses of the learner.

E. Discussing new concepts and practicing new skills #2

ACTIVITY: Tell Me What I am?

Teacher will group the learners into five. Allow the learners find their own group members but the leader are chosen by the teacher.

Instructions: Using the pattern you have learned, tell whether the following can be factored using sum and difference of two squares.

1. $x^2+9$

2. $x^2-9$

3. $4{x^2}-25$

4. $y^2-16$

5. $36{y^2}+21$

F. Developing Mastery

ACTIVITY: CREATE YOUR OWN!

Let the learners formulate their own given binomials that can be factored using sum and difference of two terms. Let them solve on the board. (Answers may vary)

Teacher will select volunteer from the class.

G. Finding practical applications of concepts and skills in daily living

ACTIVITY: GIVE YOUR OWN!

Teacher will guide the students to give real situation that relates factoring sum and difference of two squares.

Example:

A $5m x 5m$ landscaping is to be done in one corner of a $20mx20m$ garden. Find the area of the field that was not affected by the project.
(teacher will guide the learner in leading the correct answer)

H. Making Generalizations and abstractions about the lesson

Guide Questions for Generalization:

  • Describe a polynomial that can be factored using sum and difference of two squares?

Ans. The sign in between is negative.

  • What have you observed on the first term? How about the second term?

Ans. First term- Perfect square, second term- perfect square

  • What can you conclude based on your observation? Possible answer: (First term) 2 – (Second term) 2 =(First term + Second term) (First term – Second term)

I. Evaluating learning

Instructions: Factor each of the following polynomials:

1. $a^2 – 16 = (a + 4) (a – 4)$

2. $9{x^2} – 4 = (3x + 2) (3x – 2)$

3. $64c^2 – 1 = (8c + 1) (8c – 1)$

4. $100{y^2} – 49{z^2} = (10y + 7z) (10y – 7z)$

5. $y^2 – 81 = (y + 9) (y – 9)$

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