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Quarter 1 Compiled Modules

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Module Code: Pasay-M8–Q1–W1-D1 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 1/ Day 1 OBJECTIVE: Factors completely polynomials with common monomial factor. YOUR LESSON FOR TODAY: • Factoring polynomials with common monomial factor We are going to factor polynomials. To factor means writing the polynomial as a product of other polynomials. Let us begin by recalling greatest common factor (GCF) TRY TO DISCOVER! Examples: What is the greatest common factor (GCF) of: 1. 12 and 18? _________ Solution: Prime factors of 12 = 2 ∙ 2 ∙ 3 Prime factors of 18 = 2 ∙ 3 ∙ 3 Notice that 12 and 18 have a factor of 2 and a factor of 3 in common, and so the greatest common factor of 12 and 18 is 2 ∙ 3 = 6. 2. 10������3 and 4������? __________ Solution: Factors of 10������3 = 2 ∙ 5 ∙ ������ ∙ ������ ∙ ������ Factors of 4������ = 2 ∙ 2 ∙ ������ Notice that 10������3 and 4������ have one factor of 2 and one factor of ������ in common, and so the greatest common factor of 10������3 and 4������ is 2 ∙ ������ = 2������. Now let us use the GCF in factoring polynomials with common monomial factor. Examples: 1. Factor completely 2������3 − 6������2 Solution: Step 1: Find the GCF We can find the GCF of any two numbers by examining their prime factors. 2������3 = 2 ∙ ������ ∙ ������ ∙ ������ 6������2 = 2 ∙ 3 ∙ ������ ∙ ������ Notice that 2������3 and 6������2 have one factor of 2 and two factor of ������ in common, and so the greatest common factor of 2������3 and 6������2 is 2 ∙ ������ ∙ ������ = 2������2 Step 2: Divide each term of the polynomial by the GCF. The quotient is the other factor. 2������3 − 6������2 2������2 2������2 ������ − 3 is the other factor Step 3: Express the polynomial as a product of the GCF and the other factor. 2������3 − 6������2 = 2������2(������ − 3 ) Therefore, 2������3 − 6������2 = 2������2(������ − 3) 2|Page

2. Factor completely 10������3 + 4������ Solution: Step 1: Find the GCF 10������3 = 2 ∙ 5 ∙ ������ ∙ ������ ∙ ������ 4������ = 2 ∙ 2 ∙ ������ GCF = 2������ Step 2: Divide each term of the polynomial by the GCF. The quotient is the other factor. 10������3 + 4������ 5������2 + 2 is the other factor 2������ 2������ Step 3: Express the polynomial as a product of the GCF and the other factor. 10������3 + 4������ = 2������(5������2 + 2) Therefore, 10������3 + 4������ = 2������ (5������2 + 2) Can we be faster? Once we know the GCF, the factored form is simply the product of the GCF and the sum of the terms in the original polynomial divided by the GCF. 3. Give the factor of 5������2 + 10������. 5������2 + 10������ = 5������ 5������2 10������ = 5������(������ + 2) Solution: ( 5������ + 5������ ) GCF = 5������ Answer: 5������(������ + 2) 4. Factor: 16������4 − 4������3 + 8������5 Solution: GCF = 4������3 16������4 − 4������3 + 8������5 = 4������3 (146������������34 − 4������3 + 48������������35) = 4������3(4������ − 1 + 2������2) 4������3 Answer: 4������3(4������ − 1 + 2������2) 5. Factor out the GCF of -6n5 – 6n3 Solution: GCF = −6������3 −6������5 − 6������3 = −6������3 (−−66������������43) = −6������3[������2 − (−1)] = −6������3(������ + 1) Answer: −6������3(������2 + 1) 6. World problem A large triangle with an area of 14������4 + 6������2 square meters is divided into smaller rectangles with areas 14������4 and 6������2 square meters. The width of the of the rectangle (in meters) is equal to the greatest common factor of 14������4 and 6������2. What is the length and width of the large rectangle? Length Solution: GCF: = 2������2, the width of the rectangle is 2������2. Width 14������4 6������2 We can find the length of the large rectangle by finding the lengths of the two smaller rectangles. We will do that by using the area formula ������ = (������)(������) (������������������������ = ������������������������������ℎ ������ ������������������������ℎ) and the fact that ������ = 2������2. 3|Page

14������4 + 6������2 = 2������2 (124������������24 + 26������������22) 14������4 + 6������2 = 2������2(7������2 + 3) The length of the large triangle is the quotient or the other factor which is 7������2 + 3. PRACTICE EXERCISES 1: A. What is the GCF of: 1. 9������2 and 6������ ______________ 2. 12������5 and 8������3 ______________ 3. 63 and 45b ______________ 4. 6������3 and 3������2 ______________ 5. 10������3, 9������2, and ������ ______________ B. Factor the following expressions and write the letter in the box that corresponds to each number to form a math term. 1. 3������3 − 4������ + 5������4 E. 4xy(y+5x2y) 2. 12������������ + 36������ L. 12������(������ + 3) 3. 15������2 − 12������ O. 3������(5������ − 4) 4. 4������������2 + 20������3������2 P. 4������������2(1 + 5������) 5. −5������������ + 3������2������2 + ������3������ S. ������(3������2 − 4 + 5������3) 12345 What Math term is formed? Do you what is that? That will be on the next chapter of our subject. PRACTICE EXERCISES 2: A. Find the GCF of the following. 1. 7������2 and 21������ ______________ 2. 9������ and −45 ______________ 3. 18������6 and 12������3 ______________ 4. 15������2, 25������2, and 55������ ______________ 5. 9������7������5 and −3������2������6 ______________ 4|Page

B. Factor completely by filing in the box. The first one is done for you. Polynomial Common Monomial The other factor Factored Form 2������ + 10 Factor or Greatest ������ + 5 2(������ + 5) Common Factor 2 −4������ − 12 3������3 − 12������2 − 9������ ������5 + ������3 15������2 − 55������ 9������7������5 + 3������2������6 18������6 − 12������3 7������2 + 21������ 6������3 − 3������2 Remember: • Factoring a polynomial means writing it as a product of other polynomials. • The GCF (greatest common factor) of two or more monomials is the product of all their common prime factors. • Steps in factoring polynomials with common monomial factor: 1. Find the GCF of all the terms in the polynomial 2. Divide each term of the polynomial by the GCF. The quotient is the other factor. 3. Express the polynomial as a product of the GCF and the other factor. PRACTICE EXERCISES 3: Find the factor of each expression. Write the letter of the correct answer in each box that contain the exercise numbers. You will discover a mathematical term. 1. 6������5 + 12������3������ − 3������������2 T. ������(������ − 8) 2. 45������4 − 15������2 I. 3(2������ + 3������) 3. 7������2������3 − 28������������ + 14������������2 M. 2(������2 − 3������ + 4) 4. 36������5 − 12������3 + 24������4 A. 3������(2������4 + 4������2������ − ������2) 5. 4������ + 4������ O. 12������3(������2 − 1 + 2������) 6. 6������ + 9������ L. 15������2(3������2 − 1) 7. ������2 − 8������ H. 2(5������ + 1) 8. 10������ + 2 G. 7������������(������������2 − 4 + 2������) 9. 2������2 − 6������ + 8 R. 4(������ + ������) 5|Page

12 3 4 5 6 7 8 9 What is the mathematical term and its meaning? ________________________________________________________________________________________________________ _____________________________________________________________ Generalization: How do we factor polynomials with common monomial factor? ________________________________________________________________________________________________________ __________________________________________________________________ Integrated the development of the following skills: 2. Critical thinking A. Solving problems 1. Communication Skills B. Comparing and analyzing A. Explaining the process of factoring polynomials with GCF B. Following instruction C. Reasoning D. Understanding messages EVALUATION: 1. Find the GCF: a. 25������5 and 40������2 b. 17������2������2 and 24������������2 c. 28������5������2, −14������4������,3 and 49������3������4 2. Complete the products: a. 36������3 + 18������2 = 6( ______ + _______) b. 36������3 + 18������2 = 18( _______ + ______) c. 36������3 + 18������2 = 18������2( _______ + ______) 6|Page

d. Which of the products in a – c is a complete factorization of 36������3 + 18������2? ____________________ Explain your answer. ____________________________________________________________________________ 3. Factor the following polynomials completely. f. 3������2 − 9 a. 6 − 18������ b. 2������2 + 5������ g. 21������2������ − 35������������2 c. 3������4 + 3������2 h. 16������2������3 − 20������3������5 d. 2������ − 4������3 i. ������������2 + ������������������ e. 3������������2 − 6������2������ j. 3������2 − 6������ − 30 4. The area of a rectangle is 14������2ℎ. One dimension is 2������. What is the other dimension? _____________ 5. Kenjo said that the factored form of 3������ + 6������2 + 15������3 is 3������(2������ + 5������2). Do you agree with him? Explain why or why not. __________________________________________________________________________________________________ ______________________________________________________________ 6. The lengths of the parallel sides of a trapezoid are represented by ������ and ������ and its height by ℎ. The area of the trapezoid can be written as 1 ������ℎ + 1 ������ℎ. Express this area as the product of two factors. 22 _____________________________________________________________ Prepared by: Anecita S. Santos Pasay City South High School References for further Enhancement: https://www.chilimath.com https://cdn.kutasoftware.com https://www.lavc.educ khan academy 7|Page

Module Code: Pasay-M8–Q1–W1-D2 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 1/ Day 2 OBJECTIVE: factors completely polynomials which are difference of two squares. YOUR LESSON FOR TODAY: • Factoring the Difference of Two Squares: ������������ − ������������ = (������ + ������)(������ − ������) TRY TO DISCOVER! Determine if the given expression is a perfect square or not. 1. 64 6. 36������7������3������ 2. 100 7. 100������6 3. 121������2������2 8. 1 ������2 4. 36������2������3 4 5. 81������4������2������ 9. ������4 16 10. 144������2 49 64, 100, 121������2������2, 100������6, 1 ������2, ������4, and 144������2 are perfect squares. Why? Perfect square is the product when a 4 16 49 number is multiplied by itself. What did you notice on the exponent of the variables? This time, we are going to identify polynomials which are difference of two squares. Difference of Two Squares. How does it look like? Try to understand the diagram below. Squared Terms First 2 2 Term Second Term Difference or subtraction or minus Do you understand the illustration? Now let us identify polynomials which are difference of two squares. Write your answers in the box below. 1. ������2 − 4������2 6. 225 + ������4 2. ������2 − 12 25 3. ������4 − 9������2 4. 1 ������2 − 121 7. 9������2 − 25 8. 4������2 − 9 4 9. 4������2 + 49 5. ������2 + ������2 10. 9������2 − 16������ 8|Page

Difference of Two Squares Not Difference of Two Squares Describe the terms of the difference of two squares. What can you say about the exponents of the variables? ____________________________________________________________________________________ Since you know now the polynomials which are the difference of two squares, we are going to find their factors. Example: 1. Factor 9������2 − 16 Solution: Step 1: Find the square roots of the two terms. The square root of 9������2 is 3������. The square root of 16 is 4. Step 2: Write the factorization as the sum and difference of the square roots. Therefore, 9������2 − 16 = (3������ + 4)(3������ − 4) 2. Give the factor of 4������2 − 9������4. Solution: Step 1: Find the square roots of the two terms. The square root of 4������2 is 2������. The square root of 9������4 is 3������2. Step 2: Write the factorization as the sum and difference of the square roots. Therefore, 9������2 − 16 = (3������ + 4)(3������ − 4) 3. How do we factor 9������4������2 − 16������2������4? Solution: Step 1: The square root of 9������4������2 is 3������2������. The square root of 16������2������4 is 4������������2. Step 2: Write the factorization as the sum and difference of the square roots. Therefore, 9������4������2 − 16������2������4 = (3������2������ + 4������������2)(3������2������ − 4������������2). 4. Factor 1 − ������2. Solution: Step 1: The square root of 1 is 1. The square root of ������2 is ������. Step 2: Write the factorization as the sum and difference of the square roots. Therefore, 1 − ������2 = (1 + ������)(1 − ������) 9|Page

5. Factor the binomial (������ − 3)2 − 4. Is this difference of two squares? Why? It is a difference of two squares. The first term is (������ − 3)2 and the second term is 4. Solution: Step 1: The square root of (������ − 3)2 is ������ − 3. The square root of 4 is 2. Step 2: Write the factorization as the sum and difference of the square roots. (������ − 3)2 − 4 = [(������ − 3) + 2][(������ − 3) − 2] We can combine the like terms by adding or subtracting the constants. = (������ − 3 + 2)(������ − 3 − 2) = (������ − 1)(������ − 5) Therefore, (������ − 3)2 − 4 = (������ − 1)(������ − 5) PRACTICE EXERCISES 1: A. Complete the table by writing the factors and square root. Perfect Square Factors Square Root 64 100 10 ∙ 10 or (10)2 10 121������2������2 10������3 1 100������6 (10������3)(10������3) or (10������3)2 2 ������ 1 ������ 2 (1 ������) (1 ������) or (1 2 4 22 2 ������) ������4 16 144������2 49 B. Factor completely. 1. ������2 − 100 Solution: What is the square root of ������2? ____________ What is the square root of 100? ____________ Therefore, ������2 − 100 = ( _______________)( ______________) 2. 25������2 − 1 Solution: What is the square root of 25������2? ____________ What is the square root of 1? ____________ Therefore, 25������2 − 1 = ( _______________)( ______________) 3. 4������2 − 9������2 Solution: What is the square root of 4������2? ____________ What is the square root of 9������2? ____________ Therefore, 4������2 − 9������2 = ( _______________)( ______________) 10 | P a g e

4. 16������4 − ������4 Solution: What is the square root of 16������4? ____________ What is the square root of ������4? ____________ Therefore, 16������4 − ������4 = ( _______________)( ______________) 5. (2������ − 1)2 − 49 Solution: What is the square root of (2������ − 1)2? ____________ What is the square root of 49? ____________ Therefore, (2������ − 1)2 − 49 = ( _______________)( ______________) 6. 4������2������4 − 36������4������2 Solution: What is the square root of 4������2������4? ____________ What is the square root of 36������4������2? ____________ Therefore 4������2������4 − 36������4������2 = ( _______________)( ______________) PRACTICE EXERCISES 2: 6) ������2 − 64 Factor completely. 1) ������2 − 49 2) ������2 − 144 7) ������2 − 25 3) ������2 − 9 8) ������2 − 4 4) ������2 − 121 9) ������2 − 36 5) ������2 − 289 10) ������2 − 169 Remember: Formula for Difference of Two Squares: ������������ − ������������ = (������ + ������)(������ − ������) Steps on how to factor the difference of two squares. 1. Find the square roots of the two terms. 2. Write the factorization as the sum and difference of the square roots. Generalization: Explain in your own words how to factor difference of two squares? ________________________________________________________________________________________________________ _________________________________________________________________ 11 | P a g e

PRACTICE EXERCISES 3: A. Determine if the given binomial is a difference of two squares or not. Write DTS for difference of two squares and NOT for not difference of two squares. 1. ������2 − ������2 ________ 6. 49������2 − 100������4 ________ 2. 2������2 − ������2 ________ 7. 16 − 3������2 ________ 3. ������2 + 25 ________ 8. 2ℎ4 − 36 ________ 4. ������2 − 15 ________ 9. (������ + ������)2 − 4 ________ 5. 16������2 − 25������4 ________ 10. (������ − ������)2 + 16 ________ Integrated the Development of the Following Learning Skills: 1. Communication Skills 2. Critical thinking A. Explaining the process of factoring polynomials A. Solving problems B. Following instruction B. Comparing and analyzing C. Reasoning D. Understanding messages EVALUATION: 16) ������2 − 64 17) ������2 − 25 Factor each completely. 18) ������2 − 4 1) ������2 − 49 19) ������2 − 36 2) ������2 − 144 3) ������2 − 9 20) ������2 − 169 4) ������2 − 121 21) 16������2 − 1 5) ������2 − 289 22) ������2 − 16 23) 1 − ������2 6) 4������2 − 25 24) ������2 − 9 7) 9������2 − 4 25) 16������2 − 9 8) 9������2 − 25 26) 121������2 − 36������2 9) 16������2 − 25 27) 64������2 − 25������2 10) 25������2 − 9 28) 49������2 − 4������2 11) ������2 − ������2 29) 25������2 − ������2 12) 9������2 − 4������2 30) 36������2 − ������2 13) 144������2 − 25������2 14) 121������2 − 9������2 15) 81������2 − 121������2 Prepared by: Anecita S. Santos Pasay City South High School References for Further Enhancement https://www.chilimath.com https://cdn.kutasoftware.com https://www.lavc.educ Khan academy 12 | P a g e

Module Code: Pasay-M8–Q1–W1-D3 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 1/ Day 3 OBJECTIVE: Factors completely perfect square trinomials. YOUR LESSON FOR TODAY: • Identifying and factoring perfect square trinomials TRY TO DISCOVER! Give the product of the following: 1.) (������ + ������)2 2.) (������ − ������)2 3.) (������ + 5)2 4.) (3������ − 4)2 Solution: 1) (������ + ������)2 = (������ + ������)(������ + ������) you can use FOIL Method = ������2 + ������������ + ������������ + ������2 = ������2 + 2������������ + ������2 Therefore, (������ + ������)2 = ������2 + 2������������ + ������2. 2) (������ − ������)2 = (������ − ������)(������ − ������) you can use FOIL Method Perfect Square = ������2 − ������������ − ������������ + ������2 Trinomials = ������2 − 2������������ + ������2 Therefore, (������ − ������)2 = ������2 − 2������������ + ������2. Note that in the patterns, ������ and ������ can be any algebraic expression. For example, we want to expand (������ + 5)2. In this case, ������ = ������ and ������ = 5 ARE YOU READY TO PRACTICE? 1) (������ + 5)2 = ������2 + 2(������)(5) + (5)2 = ������2 + 10������ + 25 Therefore, (������ + 5)2 = ������2 + 10������ + 25. The answer, which is ������2 + 10������ + 25 is a Perfect Square Trinomial. Because the first term, x2 is a perfect square, the middle term, 10������ is twice the product of the square roots of the first term (������) and last term (5) while the last term, 25 is a perfect square. 2) (3������ − 4)2 = (3������)2 − 2(3������)(4) + (4)2 = 9������2 − 24������ + 16 Now you already know perfect square trinomials. This time we are going to get their factors. It’s just the reverse of what we did earlier. 13 | P a g e

Perfect Square Trinomials can be factored using the following equations: ������������ +������������������ + ������������ = ( ������ + ������)������ ������������ -������������������ + ������������ = ( ������ − ������)������ Example: 1.Factor ������2 + 16������ + 64 Solution: a. Since ������2 = (������)2 and 64 = (8)2, then both the first and last terms are perfect squares. and 16������ = (2)(������)(8), then the given expression is a perfect square polynomial. b. The square root of the first term is ������ and the square root of the last term is 8, then the polynomial is factored as (������ + 8)2 or (������ + 8)( ������ + 8). 2. Factor ������2 − 6������ + 9 Solution: a. Since ������2 = (������)2 and 9 = (3)2, then both the first and last terms are perfect squares and 6������ = (2)(������)(3), then the given expression is a perfect square polynomial. b. The square root of the first term is ������ and the square root of the last term is 3, then the polynomial is factored as (������ − 3)2. Take note of the sign: Positive (+) if the middle term is positive Negative (−) if the middle term is negative We can also factor this trinomial using the sum-product pattern. Example: 3. Factor: ������2 + 8������ + 16 What are the factors of 16 that sum to 8 ? Since 4 ∙ 4 = 16 and 4 + 4 = 8, we can factor the polynomial as follows ������2 + 8������ + 16 = (������ + 4)(������ + 4) = (������ + 4)2 4. Factor ������2 − 6������ + 9 What are the factors of 9 that sum to −6? Since (−3)(−3) = 9 and (−3) + (−3) We can factor the polynomial as: ������2 − 6������ + 9 = (������ − 3)(������ − 3) = (������ − 3)2 14 | P a g e

PRACTICE EXERCISES 1: A. Tell whether the given trinomial is a perfect square or not. Give the reason. 1) ������2 + 14������ + 49 Yes! Because the first term, ������2 and the last term, 49 are perfect squares and the middle term is twice the product of the square roots of the first and last terms, 2(������)(7). 2) ������2 + ������ + 1 _______________________________________________________________ 3) ������2 − 8������ + 16 _______________________________________________________________ 4) ������2 + 6������ + 9 _______________________________________________________________ 5) ������2 − 10������ + 25 _______________________________________________________________ B. Factor the following. 1) ������2 − 8������ + 16 = 2) ������2 + 14������ + 49 = 3) ������2 + 18������ + 81 = 4) ������2 − 20������ + 100 = 5) 4������2 + 40������ + 25 = PRACTICE EXERCISES 2: Factor the following. 1) ������2 + 18������ + 81 2) ������2 − 20������ + 100 3) ������2 − 14������ + 49 4) ������2 + 22������ + 121 5) ������2 − 18������ + 81 6) 25������2 − 120������ + 144 7) 49������2 + 84������ + 36 8) ������2 − 16������ + 64 9) 81������2 − 180������ + 100 10) ������2 + 18������ + 81 REMEMBER: Perfect Square Trinomial is the product of squaring a binomial. A trinomial is a perfect square if the first term and last terms are perfect squares and the middle term is always twice the product of the square root of the first and last terms. 15 | P a g e

PRACTICE EXERCISES 3: Match the factors of the following polynomials and write the corresponding letter in the box in order to decode the name of a mathematician. 1) 16������2 + 40������ + 25 A) (5������ + 11)2 2) 121������2 − 198������ + 81 E) (5������ − 8)2 3) 100������2 − 180������ + 81 T) (������ + 12)2 4) 25������2 − 40������ + 16 S)(7������ + 2)2 5) 36������2 − 60������ + 25 E) (11������ − 9)2 6) 25������2 − 80������ + 64 D) (6������ − 5)2 7) 4������2 − 28������ + 49 N) (10������ − 9)2 8) 121������2 − 110������ + 25 R) (������ + 11)2 9) 25������2 + 110������ + 121 E) (10������ + 3)2 10) ������2 + 22������ + 121 S) (2������ − 7)2 11) ������2 + 24������ + 144 R) (4������ + 5)2 12) 100������2 + 60������ + 9 C) (11������ − 5)2 13) 49������2 + 28������ + 4 E) (5������ − 4)2 1 2 3 4 5 6 7 8 9 10 11 12 13 Integrated the development of the following skills: 2. Critical thinking A. Solving problems 1. Communication Skills B. Comparing and analyzing A. Explaining the process of factoring polynomials with GCF B. Following instruction C. Reasoning D. Understanding messages EVALUATION: A. Which of the following expressions are perfect square trinomials? Write PST if it is and NOT if not. 1) 4������2 − 16������ + 64 2) ������2 − 16������ + 64 3) ������2 + 16������ + 16 4) 9������2 − 18������ + 9 5) 9������2 + 18������ + 9 16 | P a g e

6) 16������2 − 36������ + 81 7) 16������2 − 72������ + 81 8) 36������2 + 12������ + 1 9) 25������2 − 10������ + 4 10) 9������2 + 6������ + 1 B. Factor each expression. 1) 4������2 − 12������ + 9 2) ������2 − 6������ + 9 3) ������2 + 10������ + 25 4) 9������2 − 12������ + 4 5) 49������2 + 14������ + 1 6) 36������2 + 132������ + 121 7) ������2 + 22������ + 121 8) ������2 − 16������ + 64 9) ������2 − 16������ + 64 10) ������2 + 8������ + 81 Prepared by: Anecita S. Santos Pasay City South High School References FOR Further Enhancement: https://www.chilimath.com https://cdn.kutasoftware.com 17 | P a g e

Module Code: Pasay-M8–Q1–W1-D4 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 1/ Day 4 OBJECTIVE: Factors completely the sum of two cubes. YOUR LESSON FOR TODAY: • Factoring the sum of two cubes TRY TO DISCOVER! The sum of two cubes can be factored into a product of a binomial times a trinomial. That is ������������ + ������������ = (������ + ������)(������������ − ������������ + ������������) same opposite always positive SO AP Don’t forget the mnemonic “SOAP”, S – same sign, O – opposite sign and AP – always positive. 1. Factor the polynomial ������������ + ������������. Solution: ������������ + ������������ = (������)3 + (3)3 Rewrite as base to a cube = (������ + 3)[(������)2 − (������)(3) + (3)2] = (������ + 3)(������2 − 3������ + 9) Therefore, ������3 + 27 = (������ + 3)(������2 − 3������ + 9). 2. Factor ������������������������ + ������������������������. Solution: 27������3 + 64������3 = (3������)3 + (4������)3 = (3������ + 4������)[(3������)2 − (3������)(4������) + (4������)2] = (3������ + 4������)(9������2 − 12������������ + 16������2) Therefore, 27������3 + 64������3 = (3������ + 4������)(9������2 − 12������������ + 16������2) 18 | P a g e

3. Factor ������������������������������ + ������������. Solution: ������������������������������ + ������������ = (5������)3 + (3)3 = (5������ + 3)[(5������)2 − (5������)(3) + (3)2] = (5������ + 3)(25������2 − 15������ + 9) Therefore, ������������������������������ + ������������ = (������������ + ������)(������������������������ − ������������������ + ������) ARE YOU READY TO PRACTICE? 4. Factor 3������������ + 24������4������ Can we consider this polynomial as sum of two cubes? It is not. If you see something like this, take out the common factors. Solution: 3������������ + 24������4������ = 3������������ (1 + 8������3) Common Sum of monomial two factor cubes = 3������������[(1)3 + (2������)3] = 3������������(1 + 2������)[(1)2 − (1)(2������) + (2������)2] = 3������������(1 + 2������)(1 − 2������ + 4������2) Therefore, 3������������ + 24������4������ = 3������������(1 + 2������)(1 − 2������ + 4������2) 5. Factor 4������3������3 + 108 What is the common factor? Solution: 4������3������3 + 108 = 4(������3������3 + 27) = 4[(������������)3 + (3)3] = 4(������������ + 3)[(������������)2 − (������������)(3) + (3)2] = 4(������������ + 3)(������2������2 − 3������������ + 9) Therefore, 4������3������3 + 108 = 4(������������ + 3)(������2������2 − 3������������ + 9) Are you ready now to answer exercises? If not, try to go back to the explanation above. Just remember the SOAP for the signs. PRACTICE EXERCISES 1: A. Give the cube root of the following. 1) 1 2) 64������3 2) 8������3 4) 343������3 3) 27������3 6) 729������3 4) 125������3 8) 1000������3 5) 216������3 10) 512������3 B. Factor each completely by filling the blanks. 1) ������3 + 125 = ( _______)3 + (_______ )3 = ( ______ + _____) [( ______)2 − (______)(______) + (_______)2 ] = _______________________________________ 19 | P a g e

2) ������3 + 64 = ( _______)3 + (_______ )3 = ( ______ + _____) [( ______)2 − (______)(______) + (_______)2 ] = _______________________________________ 3) ������3 + 8 = ( _______)3 + (_______ )3 = ( ______ + _____) [( ______)2 − (______)(______) + (_______)2 ] = _______________________________________ 4) ������3 + 27 = ( _______)3 + (_______ )3 = ( ______ + _____) [( ______)2 − (______)(______) + (_______)2 ] = _______________________________________ 5) ������3 + 1 = ( _______)3 + (_______ )3 = ( ______ + _____) [( ______)2 − (______)(______) + (_______)2 ] = _______________________________________ 6) 1 + ������3 = ( _______)3 + (_______ )3 = ( ______ + _____) [( ______)2 − (______)(______) + (_______)2 ] = _______________________________________ 7) 8������3 + 27 = ( _______)3 + (_______ )3 = ( ______ + _____) [( ______)2 − (______)(______) + (_______)2 ] = _______________________________________ 8) 8������3 + 1 = ( _______)3 + (_______ )3 = ( ______ + _____) [( ______)2 − (______)(______) + (_______)2 ] = _______________________________________ 9) 27������3 + 1 = ( _______)3 + (_______ )3 = ( ______ + _____) [( ______)2 − (______)(______) + (_______)2 ] = _______________________________________ 10) 125 + ������3 = ( _______)3 + (_______ )3 = ( ______ + _____) [( ______)2 − (______)(______) + (_______)2 ] = _______________________________________ 20 | P a g e

PRACTICE EXERCISES 2: Factor each completely. Just write the final answer. 1) 8 + ������3 2) 216 + ������3 3) 8������3 + ������3 4) 27 + ������3 5) 27������3 + ������3 6) ������3 + 8������3 7) 64 + ������3 8) 64������3 + ������3 9) 64������3 + 27������3 10) 125������3 + 8������3 Remember: Sum of Two Cubes: ������������ + ������������ = (������ + ������)(������������ − ������������ + ������������) 1. The factors of the sum of two cubes is a binomial and a trinomial. 2. For the binomial factor, the first term is the cube root of the first term of the given polynomial, while the second term is the cube root of the last term of the given polynomial. 3. For the trinomial factor, the first term is the square of the first term of the binomial factor. The middle term is the product of the terms of the binomial factor with opposite sign to it. The last term is the square of the binomial factor. PRACTICE EXERCISES 3: 6) 8������3 + 27������3 7) 125������3 + ������3 Factor completely. 8) 27������3 + 125 9) 3������3 + 24 1) 27������3 + 8������3 2) 64������3 + 8������3 3) ������3 + 125 4) 8������3 + 27������3 5) 2������3 + 54������3 21 | P a g e

Integrated the development of the following skills: 2. Critical thinking 1. Communication Skills A. Solving problems B. Comparing and analyzing A. Explaining the process of factoring B. Following instruction Prepared by: C. Reasoning Anecita S. Santos D. Understanding messages Pasay City South High School EVALUATION: Find the factors of each expression. 1) ������3 + 8 2) ������3 + 125 3) ������3 + 216 4) ������3 + 64 5) ������3 + 27 6) ������3 + 1 7) ������3 + 343 8) 8������3 + 27 9) 27������3 + 125 10) 64������3 + 216 11) 8������3 + 27������3 12) 64������3 + 8������3 13) 125������3 + 27������3 14) 27������3 + 1 15) ������3 + 125 16) 216������3 + 27������3 17) 8������6 + 125������3 18) 3������3 + 24������3 19) 2������3 + 54������3 20) 3������3 + 192������3 References for Further Enhancement: https://www.chilimath.com https://cdn.kutasoftware.com https://www.lavc.educ khan academy 22 | P a g e

Module Code: Pasay-M8–Q1–W2-D1 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 2/ Day 1 OBJECTIVE: Factors completely the sum and difference of two cubes. YOUR LESSON FOR TODAY: • How to factor the Sum and Difference of Two Cubes TRY TO DISCOVER! Example 1. Factor completely x3 + 8. Solution: x3 + 8 = (x + 2)[ (x)2 – (x)(2) + (2)2 ] = ( x + 2) ( x2 – 2x + 4 ) [Grab your reader’s attention with a great quote from the document or use this space to emphasize a key point. To place this text box anywhere on the page, just drag it.] Questions: How do we get the first binomial factor? (x + 2) 3√������3 3√8 How about the trinomial factor? [ (x)2 – (x)(2) + (2)2 ] square of the last term of the binomial factor negative product of the 1st and the last term of the binomial factor square of the first term of the binomial factor Example 2. Factor completely x3y6 - 27. Solution: x3y6 - 27 = (xy2 - 3)[ (xy2)2 – (xy2)(3) + (3)2 ] = (xy2 - 3) ( x2y4 +3xy2 + 9) Questions: How do we get the first binomial factor? (xy2 - 3 ) 3√������3������6 3√27 How about the trinomial factor? [(xy2)2 + (xy2)(3) + (3)2] square of the last term of the binomial factor negative product of the 1st and the last term of the binomial factor square of the first term of the binomial factor Example 3. Factor completely 2x9 + 16 y6. 23 | P a g e

Solution: 2x9 + 16 y6 = 2(x9 - 8 y6) = 2(x3 – 2y2)[(x3)2 + (x3)(2y)2 +(2y)2 ] = 2(x3 – 2y2)(x6 +2x3y2+4y4) Questions: How do we get the first factor? The first factor 2 is obtain by getting the common monomial factor in each term. After obtaining the first factor, What to do next? Get the binomial factor as shown below. (xy2 - 3 ) 3√������3������6 3√27 How about the trinomial factor? [(xy2)2 + (xy2)(3) + (3)2] square of the last term of the binomial factor negative product of the 1st and the last term of the binomial factor square of the first term of the binomial factor ARE YOU READY TO PRACTICE? PRACTICE EXERCISES 1: Find the mising factor of x6 + y3 . x6 + y3 = (x2 + y )[(x2)2 – (x2)(y) + (y)2] =( __ +__ ) (x4 - ___ + y2) PRACTICE EXERCISES 2: Find the mising factor of 125x9 - 1 . 125x9 – 1 = ( ___ - 1) [( __ + (5x3)(1) + (1)2] = ( ___ - 1) ( __ + 5x3 + 1) Remember: Sum and Difference of Two Cubes Follow these patterns: x3 + y3 = (x + y) (x2 – xy + y2) x3 - y3 = (x - y) (x2 + xy + y2) Steps to follow when factoring Sum and Difference of Two Cubes 1. Get the cube root of the first and second terms of the given cubes. This will serve as the first binomial factor. 2. Get the trinomial factor by squaring the first term of the first factor, then getting the negative product of the 1st and the last term of the binomial factor then squaring the last term of the binomial factor. 3. Write them in factored form. 24 | P a g e

PRACTICE EXERCISES 3: 2. Character A. Accuracy Factor completely: 3x3y3 + 24y6 B. Patience 3x3y3 + 24y = ( __ )( ___ + ___ ) C. Perseverance 3x3y3 + 24y = ( __ )( ___ + ___ )( ___ - ___ + ___ ) Integrated the Development of the Following Learning Skills: 1. Communication Skills A. Understanding of words/vocabulary: common ratio, geometric sequence. B. Following instructions/directions. C. Reasoning D. Responding to ideas EVALUATION: A. Factor the following completely. 1. x3 – 1 = 2. 64 + y12 = 3. x3y6 + z15 = 4. 27x9 + y3 = 5. 1 + y9z12 = B. Extra Challenge. Factor the following completely. 1. 16x15 + 2y9z6 = 2. (x + z)3 + ( x – z )3 = 3. [(x + y)3 – 8] = 4. (x – 1)3 + 64 = 5.(x + y)3 + (x – y)3 = Prepared by: Maria Mawii A. Camacho Kalayaan National High School References for Further Enhancement: Math Builders First Year pp. 402-405 Mathematics Learners Module 8 p.35 https://www.purplemath.com/modules/specfact2.htm https://www.chilimath.com/lessons/intermediate-algebra/factoring-sum-and-difference-of-two-cubes/ 25 | P a g e

Module Code: Pasay-M8–Q1–W2-D2 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 2/ Day 2 OBJECTIVE: Factors completely general trinomials. YOUR LESSON FOR TODAY: • How to factor trinomials of the form x2 + bx + c. • How to factor trinomials of the form a x2 + bx + c. TRY TO DISCOVER! • How to factor trinomials of the form x2 + bx + c? Follow this pattern Example 1. Factor completely x2+ 10x + 16. What are the values of a,b and c in this example? Answer: a = 1 b = 10 and c = 16 Since our (m+n) = b = 10 and mn = c = 16 Step 1. Find two numbers m and n m ⚫ n = c = 16 (m + n) = b = 10 1,16 1 + 16 = 16 4,4 4+4=8 2,8 2 + 8 = 10 Step 2. Use the m and n result in step 1 as the last factors of the given trinomial and x as the first factors. ( x + 2)( x + 8) Step 3. Write them in factored form for the final answer. x2 + 10x + 16 = ( x + 2)( x + 8) Step 4. You can use the FOIL method or any other methods in checking the factors. ( x + 2)( x + 8) = [(x)(x) + 8(x) + 2 (x) + (2)(8)] = x2 + 10x + 16 Example 2. Factor completely x2 - 3x - 4. What are the values of a,b and c in this example? Answer: a = 1 b = -3 and c = -4 Since our (m+n) = b = -3 and mn = c = - 4 Step 1. Find two numbers m and n M⚫n=c=-4 (m + n) = b = -3 1,- 4 1 + (-4) = -3 -1,4 -1 + 4 = 3 2,-2 2 + (-2) = 0 26 | P a g e

Step 2. Use the m and n result in step 1 as the last factors of the given trinomial and x as the first factors. ( x + 1)( x -4) Step 3. Write them in factored form for the final answer. x2 - 3x – 4 = ( x + 1)( x - 4) Step 4. You can use the FOIL method or any other methods in checking the factors. ( x + 1)( x - 4)= [(x)(x) + (-4)(x) + (1) (x) + (1)(-4)] = x2 - 3x – 4 • How to factor trinomials of the form a x2 + bx + c? Example 3. Factor completely 6x2 +x - 2. What are the values of a,b and c in the trinomial expression 6x2 +x – 2? Answer: a = 6 b = 1 and c = -2 Here are some steps to follow in factoring 6x2 +x - 2. Step 1. Multiply the leading coefficient(a) and the constant(c). 6x2 +x – 2 (6)(-2) = -12 Step 2. List factors of a ⚫ c = -12 which gives (a + c) = b = 1 as well. a ⚫ c = -12 a+c=1 1, -12 1 +( -12) = -11 -1, 12 3, -4 -1 + 12 = 11 -3, 4 3 + (-4) = -1 -3 + 4 = 1 Step 3. Rewrite the trinomial6x2 +x – 2 as four-term expression by changingx into -3x and 4x to make 6x2 +4x – 3x– 2. Step 4. Factor 6x2 +4x – 3x– 2 by grouping. 6x2 +4x – 3x– 2 = (6x2 – 3x) + (4x – 2) Step 5. Factor out the common binomial and write the remaining factor as sum or difference of binomial. 6x2 +4x – 3x– 2 = (6x2 – 3x) + (4x – 2) = 3x (2x – 1) +2 (2x – 1) = (3x + 2)(2x – 1) Example 4. Factor completely 2x2 +7x + 3. What are the values of a,b and c in the trinomial expression 2x2 +7x + 3? Answer: a = 2 b = 7 and c = 3 Here are some steps to follow in factoring 2x2 +7x + 3. Step 1. Multiply the leading coefficient(a) and the constant(c). 2x2 +7x + 3 (2)(3) = 6 Step 2. List factors of a ⚫ c = 6 which gives (a + c) = b = 7 as well. a⚫c=6 a+c=7 1, 6 1+6=7 2, 3 2+3=5 Step 3. Rewrite the trinomial 2x2 +7x + 3 as four-term expression by changing 7x into x and 6x to make 2x2 +x + 6x + 3. 27 | P a g e

Step 4. Factor 2x2 +x + 6x + 3 by grouping.2x2 +x + 6x + 3= (2x2 +x) + (6x + 3) Step 5. Factor out the common binomial and write the remaining factor as sum or difference of binomial. 2x2 +x + 6x + 3= (2x2 +x) + (6x + 3) = x (2x + 1) +3 (2x + 1) = (2x + 1)(x + 3) ARE YOU READY TO PRACTICE? PRACTICE EXERCISES 1: Factor completely x2 + 9x + 8. Step 1. Find two numbers m and n m⚫n=c= (m + n) = b = Step 2. Use the m and n result in step 1 as the last factors of the given trinomial and x as the first factors. ( x +__)( x + __) Step 3. Write them in factored form for the final answer. x2 + 9x + 8 = ( x +__)( x + __) PRACTICE EXERCISES 2: Factor completely 5x2 +7x + 2. Here are some steps to follow in factoring 5x2 +7x + 2. Step 1. Multiply the leading coefficient(a) and the constant(c). 5x2 +7x + 2 ( )( ) = Step 2. List factors of a ⚫ c = which gives (a + c) = b = as well. a•c= a+c= Step 3. Rewrite the trinomial 5x2 +7x + 2, as four-term expression by changing 7x into and to make ______________. Step 4. Factor ___________ by grouping. Step 5. Factor out the common binomial and write the remaining factor as sum or difference of binomial. ____________________________________ 28 | P a g e

Remember: Steps to follow in factoring trinomials of the form x2 + bx + c. 1. Find two numbers m and n whose product is the constant c and whose sum is b. 2. Use the m and n result in step 1 as the last factors of the given trinomial and x as the first factors. 3. Write the factored form for the final answer. Steps to follow in factoring trinomials of the form ax2 + bx + c. 1. Multiply the leading coefficient(a) and the constant(c). 2. Find 2 factors of ac whose sum is b. 3. Rewrite the trinomial as a four-term expression by replacing the middle term b with the sum of the factors. 4. Group the terms with a common factor. 5. Factor out the common binomial factor and write the remaining factor as sum or difference of a binomial. Integrated the Development of the Following Learning Skills: 2. Character A. Accuracy 1. Communication Skills B. Patience A. Understanding of words/vocabulary: common ratio, geometric sequence. C. Perseverance B. Following instructions/directions. C. Reasoning D. Responding to ideas EVALUATION: Factor the following completely. 1. x2+ 6x + 8 2. x2 - x – 12 3. x2 - 9x + 20 4. x2 + 9x + 18 5. x2 - 48x – 100 6. 6x2 + 7x + 2 7. 4x2- 23x + 15 8. 12x2 + 20x + 3 9. 6x2 + 13x + 6 10. 7x2 + 41x - 6 Prepared by: Maria Mawii A. Camacho Kalayaan National High School References for Further Enhancement: Math Builders First Year pp. 406-412 Mathematics Learners Module 8 pp. 39-41 and 43-44 https://www.mesacc.edu/~scotz47781/mat120/notes/factoring/trinomials/a_is_not_1/trinomials_a_is_n ot_1.html 29 | P a g e

Module Code: Pasay-M8–Q1–W2-D3 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 2/ Day 3 OBJECTIVE: Solving Problems Involving Factors of Polynomials. YOUR LESSON FOR TODAY: Analyze and solve problems involving factors of polynomials TRY TO DISCOVER! Example 1. Find two consecutive integers whose product is 110. Here are simple steps to follow. a. representation Let x = the first integer and x + 1 = the second integer b. equation x (x + 1) = 110 c. solution x (x + 1) = 110 Use DPMA x2 + x = 110 Bring all terms to one side so that the other side is zero x2 + x - 110 = 0 Factor the left side (x + 11)(x - 10) = 0 Use the zero product property to set each factor to zero x + 11 = 0 x - 10 = 0 Solve each equation x = -11 x = 10 If x = -11 then x + 1 = -11 + 1 = -10 and If x = 10 then x + 1 = 10 + 1 = 11 d. answer (-11 and -10) and (10 and 11) e. checking x (x + 1) = 110 x (x + 1) = 110 -11(-11 + 1) = 110 10(10 + 1) = 110 -11(-10) = 110 10(11) = 110 110 = 110 110 = 110 Example 2. A rectangular swimming pool is twice as long as it is wide. A small concrete walkway surrounds the pool. The walkway is constant 2 feet wide and has an area of 196 square feet. Find the dimensions of the pool. Here are simple steps to follow. f. representation Let w = the width of the pool and 2w = the length of the pool Area of the pool A = w(2w) Area of the pool and the walkway A = (w + 4)(2w + 4) 30 | P a g e

g. equation Total Area = Area of the pool and the walkway - Area of the pool h. solution 196 = [(w + 4)(2w + 4)] – [w(2w)] i. answer 196 = [(w + 4)(2w + 4)] – [w(2w)] j. checking 196 = 2w2 + 12w + 16 – 2w2 196 = 12w + 16 196 -16 = 12w 180 = 12������ 12 12 W = 15 The width of the pool is 15ft and the length is 30 ft 196 = [(w + 4)(2w + 4)] – [w(2w)] 196 = [(15 + 4)][(2)(15) + 4] – [15(2)(15)] 196 = 19(34) – 450 196 = 196 Example 3. The length of a one leg of a right triangle is 7 meters more than the length of the other leg. The length of the hypotenuse is 13 meters. Find tche length of the two legs. Note: Use the Pythagorean Theorem to solve the problem. Pythagorean Theorem states that the square of the longest side (hypotenuse) is equal to the sum of the squares of the two legs. c2 = a2 + b2 a b Here are simple steps to follow. a. representation Let x = the shorter leg x + 7 = the longer leg 13 = the hypotenuse x+7 13 b. equation c2 = a2 + b2 x c. solution 132 = x2 + (x + 7)2 132 = x2 + (x + 7)2 Simplify 132 and (x + 7)2 169 = x2 + (x2 + 14x + 49) Combine similar terms 169 = 2x2 + 14x + 49 Tranpose 169 to the right side 0 = 2x2 + 14x – 120 Factor by GCMF 0 = 2 ( x2 + 7x – 60) Factor ( x2 + 7x – 60) 0 = 2 ( x + 12) ( x – 5) Use the zero product property to set each factor to zero 2 = 0 x + 12 = 0 x–5=0 x = -12 x=5 d. answer The reasonable solution is x = 5 , which means that the shorter leg e. checking is 5 meters and the longer leg is 12 meters. 132 = x2 + (x + 7)2 169 = 52 + (5 + 7)2 169 = 25 + 144 169 = 169 31 | P a g e

ARE YOU READY TO PRACTICE? PRACTICE EXERCISES 1: Find the area of the shaded region. 3x + 1 2x + 1 2x 5x + 3 a. representation Area of the smaller rectangle A = _______________ Area of the bigger rectangle A = _______________ Area of the shaded region A = _______________ b. equation c. solution d. answer e. checking PRACTICE EXERCISES 2: Suppose that the length of one leg of a right triangle is 7 centimeters shorter than the other leg. The hypotenuse is 2 centimeters longer than the longer leg. Find the lengths of all three sides of the right triangle. Note: Use the Pythagorean Theorem to solve the problem. c2 = a2 + b2 a. representation b. equation c. solution d. answer e. checking 32 | P a g e

Remember: Steps to follow in solving problems involving factors of polynomials. 1. Read and analyze the problem carefully. 2. Identify what is asked in the problem. 3. Choose a variable that will represent the quantity. 4. Translate into an equation. 5. Solve the equation using knowledge on factoring. 6. Before giving the final answer check it to the original problem if it make sense. Integrated the Development of the Following Learning Skills: 2. Character 1. Communication Skills A. Understanding of words/vocabulary: common ratio, geometric sequence. A. Accuracy B. Following instructions/directions. C. Reasoning B. Patience D. Responding to ideas C. Perseverance EVALUATION: 1. The product of two consecutive integers is 224. Find the value of each integer. 2. The area of the square is numerically equal to twice its perimeter. What is the length of the side? 3. Suppose that the length of one leg of a right triangle is 2 inches less than the length of the other leg. If the hypotenuse is 10 inches, find the lengths of each leg. 4. The combined area of two squares is 45 centimeters. Each side of one square is twice as long as a side of the other square. Find the lengths of the sides of each square. 5. The sum of the areas of a square and a rectangle is 64 square centimeters. The length of the rectangle is 4 centimeters more than a side of the square, and the width of the rectangle is 2 centimeters more than a side of the square. Find the dimensions of the rectangle and the square. Prepared by: Maria Mawii A. Camacho Kalayaan National High School References for Further Enhancement: MSA Intermediate Algebra p. 88 https://www.mesacc.edu/~scotz47781/mat120/notes/factoring/trinomials/a_is_not_1/trinomials_a_is_n ot_1.html https://sites.google.com/site/algebra2polynomialfunctions/home/9-solving-equations-using-factoring 33 | P a g e

Module Code: Pasay-M8–Q1–W2-D4 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 2/ Day 4 OBJECTIVE: Illustrates rational algebraic expressions. YOUR LESSON FOR TODAY: In this module, students should be able to illustrate rational algebraic expressions. TRY TO DISCOVER! Activity 1. Loop the word Direction: Look for the following terms and ring the words. 1. ALGEBRA 6. SET 2. EXPRESSION 7. ZERO 3. FRACTION 8. UNDEFINED 4. NUMERATOR 9. DOMAIN 5. DENOMINATOR 10. RATIONAL Activity 2. Translating Verbal phrases to Mathematical phrases. Direction: Translate the following verbal phrases to mathematical phrases. 1. ������ divided by 18 2. The quotient of 4������2 and 5������3. 3. The product of 17 and ������ divided by 3������ 4. Six times the sum of 13 and a number, divided by 16 5. The reciprocal of the product of ������ and ������. Take Note! A Rational Algebraic Expression is a quotient of two polynomials, in which the denominator is zero. It is a fraction that contains at least one variable. In symbols, ������ where P and Q are polynomials and ������ ≠ 0. ������ 34 | P a g e

Each of the following are examples of rational expression: ������ 5 ������2 + ������2 ������2 − 4������ + 5 ������2 − ������2 6 , ������ − 10 , 4 , ������ + 1 , ������ + ������ These expressions cannot be considered as rational expression: 4������2 ������ + 1 2������ − 5������3 ������32 , √������ , 1 3������3 Based on the examples, can you illustrate a Rational Algebraic Expression? ________________________________________________________________________________________________________ __________________________________________. Example 1. Which of the following are rational expressions.? Explain your answer. a. ������−8 b. 15+������ c. ������3−8 3 5������+3 5−√������ Solution: 1. Yes. It is a rational expression, because both the numerator and denominator are polynomials. 2. Yes. Rational, since both numerator and denominator are polynomials. 3. No. Since the denominator does not define a polynomial. Example 2. Determine the replacement set or domain of each. Solution: a. 3������+6 b. ������2+2������−1 c. ������+1 ������ 8 ������−5 1. The denominator, x, should not be zero. Thus, the domain is the set of real numbers except 0. 2. The denominator is a constant, so, the domain is the set of real numbers. 3. The domain is the set of all real numbers except -5. Example 3. What are the non-permissible values of x in the following rational expressions? a. 18������2+2������ b. 3+������ c. 7������2 4������2−4������−15 Solution: ������−6 4������+5 1. The denominator cannot be zero, that is, ������ − 6 ≠ 0. It follows that ������ ≠ 6. The non-permissible value is 6. 2. Since 4������ + 5 ≠ 0, so ������ ≠ − 5. The non-permissible value is - 5. 44 3. The denominator 2������2 + 5������ − 3 ������ℎ������������������������ ������������������ ������������ ������������������������. To determine the non-permissible values of x, factor the trinomial. 2������2 + 5������ − 3 ≠ 0 (2������ − 1)(������ + 3) ≠ 0 Recall the zero-product property, if ������������ = 0, ������ℎ������������ ������ = 0 ������������ ������ = 0 or both ������������ ≠ 0, then 2������ − 1 ≠ 0, ������������ ������ ≠ 1 and ������ + 3 ≠ 0, ������������ ������ ≠ −3 2 The non-permissible values are 1 ������������������ − 3. 2 PRACTICE EXERCISES 1: Activity 3. Your Turn! A. Put a check (√) beside the expression if it is a rational expression and cross (X) if it is not. 1. (������+4)(������−4) 3. ������−1 ������−2 √������ 2. 4������3 4. ������2−������2 3 ������7 9 35 | P a g e

B. Determine the replacement set or domain of each. 5. 3������+6 6. ������2+2������−1 7. 1 ������2+3������+2 ������ ������+6 C. What are the non-permissible values of x in the following rational expressions? 8. 18������2+2������ 9. ������+2 10. 5������2−1 ������2−2������−3 ������+2 3������−5 PRACTICE EXERCISES 2: Activity 4. Classify Me: Classify the different expressions into rational algebraic expressions or not algebraic expressions. Match each destination to the Bus company that gives land transport services. Write the destinations into the appropriate column. Rational Algebraic Expressions Not Rational Algebraic Expressions (Philtranco – Pasay Terminal) (Victory Liner – Pasay Terminal) 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. ������ + 1 ������2 − 4 5 ������2 − 4������ + 5 5������2 √������ − 2 ������ + ������ 2������ + 10 ������ − 1 Baguio Iloilo 4 Cagayan De Oro Camarines Norte ������2 − 100 3(������ − 1) 7������5 ������2 + 3������������ + 2 ������2 − ������2 5������ − 2������2 √������ Isabela 5 + ������ Legaspi Naga 1 Pangasinan 4������2 0 6������2 Olongapo Zambales Remember: A Rational Algebraic Expression is a quotient of two polynomials, in which the denominator is zero. It is a fraction that contains at least one variable. In symbols, ������ where P and Q are polynomials and ������ ≠ 0. ������ 36 | P a g e

PRACTICE EXERCISES 3: Activity 5. Write it. 1. In your own words, how will you illustrate Rational Algebraic Expressions? 2. What is the difference between a rational number and a rational expression? 3. Discuss the importance of identifying the non-permissible value/s of a variable in a rational expression. 4. Answer the following questions briefly: a. Is every polynomial a rational expression? b. Is every rational expression a polynomial? _________________________________________________________________________ 2. Character A. Accuracy Integrated the Development of the Following Learning Skills: B. Patience 1. Communication Skills C. Perseverance A. Understanding of words/vocabulary: rational algebraic expression B. Following instructions/directions. C. Reasoning D. Responding to ideas EVALUATION: A. Put a check (√) beside the expression if it is a rational expression and cross (X) if it is not. 1. (������+2)(������−2) 3. ������+3������−1 ������+2 √������+6 2. 4������4 4. 64−������2 2 ������−6 6������3 B. Determine the replacement set or domain of each. 5. 3������−18 6. ������2−2������������+������2 7. 6 ������2−4������+3 10 ������−3 C. What are the non-permissible values of x in the following rational expressions? 8. 2������2+8������ 9. 4������−2 10. 5������2−25 ������2+2������−3 ������−7 4������+5 Prepared by: Rosalyn C. Bantay Pasay City East High School References for Further Enhancement: Grade 8 Learner’s Module, pp. 68-70 https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:rational/x2ec2f6f830c9fb89:cancel- common-factor/a/intro-to-rational-expressions 37 | P a g e

Module Code: Pasay-M8–Q1–W3-D1 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 3/ Day 1 OBJECTIVE: Simplifies rational algebraic expressions using common monomial factor and difference of two squares. YOUR LESSON FOR TODAY: • Simplifying rational algebraic expressions using common monomial factor and difference of two squares. TRY TO DISCOVER! Do you still remember, how do we reduce fraction in simplest form? Now, Given the following examples: 1. 3������ + 9 = 3(������ + 3) 3. 5������3������2 + 15������2������3 − 20������������4 = 5������������(������2������ + 3������������2 − 4������3) 2. 4������ − 24������ = 4(������ − 6������) ➢ What factoring method is used in examples 1, 2 and 3? 4. ������2 − 4 = (������ + 2)(������ − 2) 4. 4������2 − 9������2 = (2������ − 3������)(2������ + 3������) ➢ What factoring pattern is generated in examples 4 and 5? Look the at the two sets of rational expressions? 1. 2������2������3 2. 15������4+20������2 12������2 4������������ Is there any factoring method that could be used for the numerators and denominators? What are the prime factors of numerator? denominator? Can you identify the common factors between the numerator and denominator? Can we express the common factor as 1? If we are to multiply 1 to the other factor? Can we say that what factor is left is the simplest form of the given polynomial? Take a look to the next set of rational algebraic expressions: 3. ������2−16 4. ������������+������������2 1−������2 2������−8 Is there any factoring method that could be used for the numerators and denominators? What are the factors of the polynomial located on the numerator? Denominator? Can you identify what is the common factor between the numerator and denominator? What is the simplest form of the given rational expression? Solution: #1: 2������2������3 = 2∙������∙������∙������∙������∙������ = ������∙������∙������ ∙ 2∙������∙������ = ������∙������∙������ Answer: ������������2 #2 #3 4������������ 2∙2∙������∙������ 2∙������ 2∙������∙������ 2∙������ 2������ #4 3������4+21������2 = 3������2(������2+7) = ������2+7 ∙ 3������2 = ������2+7 Answer: ������2+7 12������2 2∙2∙3∙������2 2∙2 3������2 ������2−16 2∙2 4 = (������+4)(������−4) = ������+4 ∙ ������−4 2������−8 = ������+4 Answer: ������+4 ������������+������������2 2(������−4) 2 ������−4 2 2 1−������2 = ������������(1+������) = ������������ ∙ 1+������ = ������������ Answer: ������������ (1−������)(1+������) 1−������ 1+������ 1−������ 1−������ Can you give your own examples? How about if we have (x^2-y^2)/(y-x), how do we simplify the given rational algebraic expression? 38 | P a g e

Take note that ������ − ������ is also equal to −1(������ − ������) Thus, ������2 − ������2 (������ + ������)(������ − ������) ������ + ������ ������ − ������ ������ + ������ ������ − ������ = −1(������ − ������) = −1 ∙ ������ − ������ = −1 = −1(������ + ������) Now do you understand simplifying rational expressions using common monomial factor and difference of two squares In simplifying rational algebraic expression is in simplest form when the numerator and the denominator of the expression have no common factors other than 1 and -1. Check if the polynomials have common monomial factor or if the pattern is difference between two squares. ARE YOU READY TO PRACTICE? Identify the factoring method to be used and simplify the following rational algebraic expressions: 1. 4������3������3 6. ������2−������2 2. ������+������ 3. 3������������ 7. 4. 8������+16������ ������2−������2 5. 8. ������−������ Solutions: 16 ������2−5������ 9. 2������2−18 3������−9 3������−15 ������2−49 6������−24 21−3������ ������2−������2 ������−������ 3������−12 18������2−12������ 10. 3������2+6������ #1 Factoring Method: Greatest Common Monomial Factoring Solution: 4������3������3 = 2∙2∙������∙������∙������∙������∙������∙������ = 2∙2∙������∙������∙������∙������ ∙ ������∙������ = 2∙2∙������∙������∙������∙������ = 4������2 3������������ 3∙������������ 3 ������∙������ 3 3 #2 Factoring Method: Greatest Common Monomial Factoring Solution: 8������+16������ = 8(������+2������) = ������+2������ ∙ 8 = ������+2������ 16 8∙2 28 2 #3 Factoring Method: Greatest Common Monomial Factoring Solution: ������2−5������ = ������(������−5) = ������ ∙ ������−5 = ������ 3������−15 3(������−5) 3 ������−5 3 #4 Factoring Method: Greatest Common Monomial Factoring Solution: 6������−24 = 6(������−4) = 2 ∙ 3(������−4) = 2 ������������ 2 3������−12 3(������−4) 1 3(������−4) 1 #5 Factoring Method: Greatest Common Monomial Factoring Solution: 18������2−12������ = 2∙3∙������(3������−2) = 2(3������−2) ∙ 3������ = 2(3������−2) ������������ 6������−4 3������2+6������ 3������(������+2) (������+2) 3������ (������+2) ������+2 #6 Factoring Method: Difference of Two Squares Solution: ������2−������2 = (������−������)(������+������) = ������−������ ∙ ������+������ = ������−������ ������������ ������ − ������ ������+������ ������+������ 1 ������+������ 1 #7 Factoring Method: Difference of Two Squares Solution: ������2−������2 = (������−������)(������+������) = ������+������ ∙ ������−������ = ������+������ ������������ ������ + ������ ������−������ ������−������ 1 ������−������ 1 #8 Factoring Method: Difference of Two Squares Solution: 2������2−18 = 2(������−3)(������+3) = 2(������+3) ∙ ������−3 = 2(������+3) ������������ 2������+6 3������−9 3(������−3) 3 ������−3 3 3 #9 Factoring Method: Difference of Two Squares Solution: ������2−������2 = (������+������)(������−������) = ������+������ ∙ ������−������ = ������+������ ������������ − 1(������ + ������) ������−������ −1(������−������) −1 ������−������ −1 #10 Factoring Method: Difference of Two Squares Solution: ������2−49 = (������−7)(������+7) = ������+7 ∙ ������−7 = ������+7 21−3������ −1(3������−21) −1(3) ������−7 −3 How was your score? I hope you were able to answer them correctly, if not go over the examples again, and take a retest on the item you answered incorrectly. 39 | P a g e

PRACTICE EXERCISES 1: Identify the factoring method to be used and simplify the following rational algebraic expressions: 1. 7������3������ 6. ������2−������2 7. (������+������)(������−������) 8������������ 8. 9. ������2−������2 2. 2������+16������ 10. ������−������ 14 3������2−27 3������−9 3. ������2−3������ ������2−������2 5������−15 ������−������ 4. 3������−9 ������2−4 6−3������ 4������−12 5. 4������2+6������ 6������2+9������ PRACTICE EXERCISES 2: Trivia: It is a much-used method of cooking by Tagalogs in preparing vegetables. Fruit vegetables such as upo, sitaw, kalabasa, puso ng saging, sayote and carrots, among others are often sautéed. Vegetables are cooked in a small portion of oil with garlic and a small amount of meat, shrimp or small fish then seasoned with salt or fish sauce to taste. https://fnri.dost.gov.ph/index.php/publications/writers-pool-corner/57-food-and-nutrition/210-bahay- kubo-sings-a-variety-of-veggies-for-the-picking Direction: Simplify the given Rational Algebraic Expression in column A with its simplest form on Column B to reveal the answer on the Trivia. Write the corresponding letter above the item number on the answer Box. Column A Column B 1. 4������−12 A3 4������−20 3������+5������ 2. ������2−������2 G ������−3 ������−������ ������−5 3. ������2−25 I −1(������ − ������) 2������−10 4. 6������−8������ N ������+5 9������2−16������2 2 5. 9������+15������ S2 9������2−25������2 3������+4������ Answer Box: 123245 40 | P a g e

Remember: In simplifying rational algebraic expression is in simplest form when the numerator and the denominator of the expression have no common factors other than 1 and -1. To simplify a rational algebraic expression: 1. Factor both numerator and denominator simultaneously into their prime factors, 2. Identify the Greatest Common Factor that is equal to 1. 3. Apply the Fundamental Property of Fractions If a represents a real number and b and c represents non-zero numbers, ������������ = ������ ������������ ������ PRACTICE EXERCISES 3: 1. Sophia simplified the rational algebraic expression given below. ((45������������++54))−−54= 4(������+1)−4 5(������+1)−5 = 4(������+1)−4 She cancelled out ������ + 1 5(������+1)−5 = 4−4 5−5 =0 0 Discuss if Sophia did it correctly or not. 2. Shaira’s solution is shown below. 49 − ������2 = (7−������)(7+������) 2������ + 14 2(������+7) = (7−������)(7+������) She cancelled out ������ + 7 and 7 + ������ 2(������+7) = 7−������ 2 Discuss if Shaira’s solution is correct or not. 3. Matthew has a math homework he needed to finish before the given deadline. One of the given problems was so difficult he cannot solve it off. Can you help him simplify the rational algebraic expression given below? 3������2−27 (9−������2) 4. A container with 4������ − 20 boxes of powdered milk costs 4������2 − 100 pesos. How much a box of powdered milk cost? Express your answer on simplest form. 5. Based on your understanding, how is rational expression different from rational number? Cite one example to justify your answer. _______________________________________________________________________ 2. Character A. Accuracy Integrated the Development of the Following Learning Skills: B. Patience 1. Communication Skills C. Perseverance A. Understanding of words/vocabulary: Simplifying, Rational Algebraic Expression. B. Following instructions/directions. C. Reasoning D. Responding to ideas 41 | P a g e

EVALUATION: Identify the factoring method to be used and simplify the following rational algebraic expressions: 1. 8������3������ 6. ������2−������2 16������������������ (������+������)(������−������) 2. 2������+16������ 7. ������2−������2 22 ������−������ 3. ������2−2������ 8. 4������2−36 3������−6 2������+6 4. 2������−7 9. ������2−������2 6������−42 ������−������ 5. 3������2+9������ 10. ������2−16 5������2+15������ 12−3������ Prepared by: Rosalyn C. Bantay Pasay City East High School References for Further Enhancement: Grade 8 Learner’s Module, pp. 77-78 https://www.instructorweb.com/les/simplifyingrational.asp 42 | P a g e

Module Code: Pasay-M8–Q1–W3-D2 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 3/ Day 2 OBJECTIVE: Simplifies rational algebraic expressions using perfect square trinomials. YOUR LESSON FOR TODAY: In this module, students will be able to simplify rational algebraic expressions using perfect square trinomials. Unlocking difficulties Perfect square trinomial - are polynomials which are the results of squaring binomials. Square of a Sum (������ + ������)2 = ������2 + 2������������ + ������2 Square of a Difference (������ − ������)2 = ������2 − 2������������ + ������2 Examples: ▪ ������������ + ������������ + ������ ������������ ������ ������������������������������������������ ������������������������������������ ������������������������������������������������������ ������������������������������ ������������ ������������ ������ℎ������ ������������������������������������������ ������������ ������������������������������������������������ (������ + ������) ▪ ������������ − ������������ + ������ ������������ ������ ������������������������������������������ ������������������������������������ ������������������������������������������������������ ������������������������������ ������������ ������������ ������ℎ������ ������������������������������������������ ������������ ������������������������������������������������ (������ − ������) ▪ ������������������ + ������������ + ������ ������������ ������ ������������������������������������������ ������������������������������������ ������������������������������������������������������ ������������������������������ ������������ ������������ ������ℎ������ ������������������������������������������ ������������ ������������������������������������������������ (������������ + ������) TRY TO DISCOVER! Activity 1: Draw Me a Smiley Face Direction: Draw a smiley face if the given trinomial is a perfect trinomial. If not, then draw a sad face. 1. ������2 + 8������ + 16 2. ������2 − 2������ − 4 3. ������2 + 2������ + 3 4. ������2 − 6������ + 9 5. ������2 + 2������������ + ������2 Activity 2: List the Factors Direction: Factor the following polynomials. 1. 9������2 + 30������ + 25 2. 25������2 − 40������ − 16 3. 49������2 + 84������ + 36 4. 16������2 − 40������������ + 25������ 5. 25������2 + 80������������2 + 64������ Look at the following polynomials: ������2 + 2������������ + ������2; ������2 + 6������ + 9; 4������2 + 20������ + 25. Is it possible factoring out a Greatest Common Factor? Is the given polynomial, Binomial? Is the pattern Difference of Two Binomials? Or Trinomial? Is it in standard form? Is it a Perfect Square? What pattern? ������2 + 2������������ + ������2 ������������ ������2 − 2������������ + ������2? 43 | P a g e

ARE YOU READY TO PRACTICE? Solution (������ + 1)(������ + 1) Go over with the following examples and solutions: 3(������ + 1) 1. (������+1)(������+1) ������ + 1 3������+3 ������ + 1 ������ + 1 ������ + 1 Steps 3 ∙ ������ + 1 1. Factor both numerator and denominator ������ + 1 simultaneously into their prime factors using Perfect 3 Square Trinomial. 2. Identify the Common Factor that is equal to 1. Solution (������ − 3)(������ − 3) 3. Apply the Fundamental Property of Fractions. 4(������ − 3) 4. Write your final answer. ������ − 3 ������ − 3 2. ������2−6������+9 ������ − 3 ������ − 3 4 ∙ ������ − 3 4������−12 ������ − 3 Steps 4 1. Factor both numerator and denominator Solution simultaneously into their prime factors using Perfect (������ + 4)(������ + 4) Square Trinomial. (������ + 4)(������ + 4) 2. Identify the Common Factor that is equal to 1. ������+4, ������+4 3. Apply the Fundamental Property of Fractions. ������+4 ������+4 4. Write your final answer. ������ + 4 ������ + 4 3. ������2+8������+16 ������ + 4 ∙ ������ + 4 (������+4)2 1 Steps Solution 1. Factor both numerator and denominator 2(������2 − 10������ + 25) simultaneously into their prime factors using Perfect ������ − 5 Square Trinomial. 2(������ − 5)(������ − 5) = ������ − 5 2. Identify the Common Factor that is equal to 1. ������ − 5 ������ − 5 3. Apply the Fundamental Property of Fractions. 2(������ − 5) ������ − 5 1 ∙ ������ − 5 4. Write your final answer. 2(������ − 5) 4. 2������2−20������+50 ������−5 Steps 1. Factor both numerator and denominator simultaneously into their prime factors using Perfect Square Trinomial. 2. Identify the Common Factor that is equal to 1. 3. Apply the Fundamental Property of Fractions. 4. Write your final answer. 44 | P a g e

PRACTICE EXERCISES 1: Simplify the following rational algebraic expressions using perfect square trinomials. 1. (������−1)(������−1) 6. ������2+6������+9 (������+3)2 2������−2 2. 3(������−2)(������−2) 7. ������2+10������+25 2������2−50 (������−2) 3. 2(������+1)2 8. ������2−25 ������2+10������+25 3(������+1) 4. ������2+4������+4 9. ������2−9 ������2+6������+9 6������+12 5. 2������−10 10. ������2+4������ ������2−10������+25 ������2+8������+16 PRACTICE EXERCISES 2: Trivia: Mang Delfin is an active Kagawad in their Barangay. Every night, he goes around to check the peace and order in their community. Mang Delfin explained to his family that barangay officials are needed to safeguard the community. What value is shown in this situation? (The said trivia was asked during the 28th National Quiz Bee) To find out the answer on the trivia, simplify the following rational algebraic expression using perfect square trinomial. Write the letter that corresponds to the simplest form of the given rational algebraic expression. A (������−6)(������−6) E ������2−16 N ������2−9 ������2−8������+16 ������2+6������+9 3������−18 I ������2+12������+36 O 2������2+8������+8 C 2(������+1)2 (������+6)2 ������2−4 3(������+1) D ������2+6������+9 M ������2+16������+64 T ������2+8������ 2������2−128 ������2+16������+64 5������+15 ANSWER BOX: 2(������ + 1) 2(������ + 2) ������ + 8 ������ + 8 1 ������ ������ + 8 ������ + 4 ������ − 3 ������ 3 ������ − 2 2(������ − 8) 2(������ − 8) ������ + 8 2(������ − 8) ������ − 4 ������ + 3 ������ + 8 ������ − 6 ������ − 3 ������ + 3 3 ������ + 3 5 ������ + 3 ������ + 4 ������ + 3 1 2(������ + 1) ������ − 6 ������ 1 2(������ + 2) ������ − 3 5 ������ − 4 5 ������ − 2 ������ + 3 3 3 ������ + 8 45 | P a g e

Remember: In simplifying rational algebraic expression is in simplest form when the numerator and the denominator of the expression have no common factors other than 1 and -1. To simplify a rational algebraic expression: 4. Factor both numerator and denominator simultaneously into their prime factors using Perfect Square Trinomial, 5. Identify the Greatest Common Factor that is equal to 1. 6. Apply the Fundamental Property of Fractions If a represents a real number and b and c represents non-zero numbers, ������������ = ������ ������������ ������ PRACTICE EXERCISES 3: Activity 5. Write it. 1. How will you distinguish Perfect Square Trinomials from other Trinomials? 2. Explain why ������2+������2 does not simplify to 1. (������+������)2 3. Complete the graphic organizer? _________________________________________________________________________ 2. Character Integrated the Development of the Following Learning Skills: A. Accuracy 1. Communication Skills B. Patience A. Understanding of words/vocabulary: perfect square trinomial, simplify. C. Perseverance B. Following instructions/directions. C. Reasoning D. Responding to ideas 46 | P a g e

EVALUATION: Simplify the following rational algebraic expressions using perfect square trinomials. 1. (������+2)(������+2) 6. ������2−8������+16 (������−4)2 6������+12 2. 4(������+3)(������+3) 7. ������2−10������+25 2������2−50 2(������+3) 8. ������2−49 3. (������−5)2 ������2−14������+49 2(������−5) 9. ������2−9 ������2−6������+9 4. ������2+4������+4 6������+12 5. 3������+9 10. ������2−4������ ������2+6������+9 ������2+8������+16 Prepared by: Rosalyn C. Bantay Pasay City East High School References for Further Enhancement: Grade 8 Learner’s Module, pp. 77-78 https://www.mathwarehouse.com/algebra/rational-expression/how-to-simplify-rational-expressions.php 47 | P a g e

Module Code: Pasay-M8–Q1–W3-D3 DEPARTMENT OF EDUCATION, NATIONAL CAPITAL REGION SCHOOLS DIVISION OF PASAY CITY MODULE IN MATHEMATICS 8 First Quarter / Week 3/ Day 3 OBJECTIVE: Simplifies rational algebraic expressions using the sum and difference of two cubes. YOUR LESSON FOR TODAY: In this module, students will be able to simplify rational algebraic expressions using the sum and difference of two cubes. Unlocking difficulties ������3 + ������3 = (������ + ������)(������2 − ������������ + ������2) Sum of two cubes ������3 + ������3 = (������ − ������)(������2 + ������������ + ������2) Difference of two cubes Examples: ������3 + 1 = (������ + 1)(������2 − ������ + 1) ������3 + 8 = (������ + 2)(������2 − 2������ + 4) ������3 − 1 = (������ − 1)(������2 + ������ + 1) ������3 − 8 = (������ − 2)(������2 + 2������ + 4) TRY TO DISCOVER! Activity 1: Is it a Yes or a No! Direction: Write YES if the given binomial is a sum or difference of two cubes, otherwise, write NO. 6. ������3 + 64 7. ������3 − 27 8. ������3 − 49 9. 125 + ������3 10.8������3 + 25 Activity 2: List the Factors Direction: Fill in the blank/s with the correct term or expression. 6. ������3 − 27 = (_______)(������3 + 3������ + 9) 7. 64������3 − 1 = (4������ − 1)(_______________) 8. ������3 − 125 = (������ − 5)(_______________) 9. ������3 + 1000 =(_______)(������2 − 10������ + 100) 10. 8������3 − 27������3 = (2������ − 3������) (_______________) Here are hint questions that we should take note that in factoring polynomials: Is it possible factoring out a Greatest Common Factor? Is the given polynomial, Binomial? • Is the pattern Difference of Two Binomials? • Is it Sum or Difference of Two Cubes? Or Trinomial? Is it in standard form? Is it a Perfect Square? 48 | P a g e

ARE YOU READY TO PRACTICE? We follow certain steps in simplifying Rational Algebraic Expressions using sum and difference of two cubes: Illustrative Examples: 1. ������3−27 (������−3)(������2+3������+9) Steps Solution 1. Factor both numerator and denominator (������ − 3)(������2 + 3������ + 9) simultaneously into their prime factors using sum and (������ − 3)(������2 + 3������ + 9) difference of two cubes. 2. Identify the Common Factor that is equal to 1. (������ − 3)(������2 + 3������ + 9) 3. Apply the Fundamental Property of Fractions. (������ − 3)(������2 + 3������ + 9) 4. Write your final answer. ������ − 3 (������2 + 3������ + 9) ������ − 3 ∙ (������2 + 3������ + 9) 1 2. ������3+64 Solution (������ + 4)(������2 − 4������ + 16) ������+4 ������ + 4 Steps ������ + 4 1. Factor both numerator and denominator ������ + 4 (������2 − 4������ + 16) ������ + 4 simultaneously into their prime factors using sum and 1 ∙ ������ + 4 difference of two cubes. ������2 − 4������ + 16 2. Identify the Common Factor that is equal to 1. 3. Apply the Fundamental Property of Fractions. 4. Write your final answer. 3. ������3−8 ������2+2������+4 Steps Solution (������ − 2)(������2 + 2������ + 4) 1. Factor both numerator and denominator ������2 + 2������ + 4 simultaneously into their prime factors using sum and ������2 + 2������ + 4 ������2 + 2������ + 4 difference of two cubes. ������ − 2 ������2 + 2������ + 4 1 ∙ ������2 + 2������ + 4 2. Identify the Common Factor that is equal to 1. ������ − 2 3. Apply the Fundamental Property of Fractions. 4. Write your final answer. 4. (������−4)(������2+4������+16) ������������3−64������ Steps Solution (������ − 4)(������2 + 4������ + 16) 1. Factor both numerator and denominator = ������(������3 − 64) simultaneously into their prime factors (������ − 4)(������2 + 4������ + 16) using sum and difference of two cubes. = ������(������ − 4)(������2 + 4������ + 16) (������ − 4)(������2 + 4������ + 16) 2. Identify the Common Factor that is equal to 1. (������ − 4)(������2 + 4������ + 16) 1 (������ − 4)(������2 + 4������ + 16) 3. Apply the Fundamental Property of ������ ∙ (������ − 4)(������2 + 4������ + 16) Fractions. 1 4. Write your final answer. ������ 49 | P a g e

PRACTICE EXERCISES 1: Simplify the following rational algebraic expressions using sum and difference of two cubes. 1. (������+2)(������2−2������+4) ������3+8 2. (3������+1)(9������2−3������+1) 27������3+1 3. ������3+1 3(������+1) 4. ������3−������3 ������2−������2 5. 8������3−27������3 2������−3������ PRACTICE EXERCISES 2: Trivia: It is an Amusement Park located on Pasay which was owned by Star Parks Corporation (SPC), a subsidiary of Elizalde Holdings Corporation (EHC). The owner of SPC and EHC was Spanish-Filipino businessman Fred Elizalde, the husband of the Philippines’ first prima ballerina Lisa Macuja Elizalde. Lisa’s father, Cesar Macuja, worked for Elizalde as president of SPC. Before its inception in 1991, the amusement park used to be a carnaval. It was originally part of the Toy and Gift Fair, an annual Christmas trade fair under Philippine International Corporation (Philcite), one of the companies also owned by Elizalde. Direction: To find out the answer on the trivia, simplify the following rational algebraic expression using sum and difference of two cubes. Write the letter that corresponds to the simplest form of the given rational algebraic expression. A (������−2)(������2+2������+4) I ������2+4������+16 T 4+3������ ������3+64 64+27������3 ������−2 C (������+5)(������2−5������+25) R 27������3+1 Y 216������3−343������3 ������3+125 2(3������+1) 6������−7������ S 8������3−125������3 4������2+10������������+25������2 ANSWER BOX: 2������ − 5������ 16 − 12������ + 9������2 (������2 + 2������ + 4) 9������2 − 3������ + 1 2 1 1 16 − 12������ + 9������2 36������2 + 42������������ + 49������2 ������ − 4 50 | P a g e


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