Grade 8 Enhanced Mathematics Module - Special Products and Factoring

SCIENCE HIGH SCHOOL General Luna Road, Baguio City Philippines 2600 Telefax No.: (074) 442-3071 Website: www.ubaguio.edu E-mail Address: ub\@ubaguio.edu GRADE 8: GR8SHMATH ENHANCED MATHEMATICS SY 2023-2024 Module Title Special Products and Factoring No. of Hours 30 hours Quarter First Quarter...

SCIENCE HIGH SCHOOL General Luna Road, Baguio City Philippines 2600 Telefax No.: (074) 442-3071 Website: www.ubaguio.edu E-mail Address: ub\@ubaguio.edu GRADE 8: GR8SHMATH ENHANCED MATHEMATICS SY 2023-2024 Module Title Special Products and Factoring No. of Hours 30 hours Quarter First Quarter Duration August - September Prepared by Cesaria B. Padilla Schedule M -- F (6 hrs/week) I. Introduction Have you, at a particular time, asked yourself how manufacturers able to create boxes for their specific products using minimal materials? Or how a particular wall was painted using the least amount of paint? Or how some students were able to multiply expressions in the least amount of time? This module presents many applications of finding special products and factors which will be discussed in this module which in turn also helps you to recognize patterns and techniques in finding the said concepts. At the end of the module, you are expected to answer the question, "How can unknown quantities in geometric problems be solved?" II\. Lesson Coverage Week Lesson/ Topic Title Learning Targets (Most Essential Competencies) Expected Output Week 1 Factoring Greatest Common Monomial and Factoring by Grouping ⮚ I can multiply polynomials using the distributive property of multiplication. ⮚ I can extract the greatest common monomial factor of a certain polynomial. ⮚ I can determine the factors of polynomials with four terms or more by grouping. Pre-Assessment 1 In -- out of the box Formative Assessments (Try This 1 -- Try This 12) Summative Assessment 1 Week 2 Special Products and Factors ⮚ I can perform special products. ⮚ I can determine and factor perfect square trinomials. ⮚ I can determine the factors of the difference of two squares. ⮚ I can factor polynomial involving the sum and cbp2024 \| G8SHMATH 8 difference of two cubes. Week 3 FOIL Method and Factoring General Trinomials ⮚ I can multiply polynomials using FOIL Method. ⮚ I can determine the factors of any trinomial using the trial-and-error method. Formative Assessments (Try This 13 -- Try This 16) Summative Assessment 2 In -- out of the box Post-Assessment 1 Week 4 Factoring Completely ⮚ I can determine the factoring technique to be applied in a specific polynomial. ⮚ I can factor completely different types of polynomials by applying different methods. Week 5 Applications of Special Products and Factors ⮚ I can represent a real-life situation using key concepts of factors of polynomials. ⮚ I can solve real-life problems involving factors of polynomials. ⮚ I can interpret the meaning of the mathematical models I made. ⮚ I can use a variety of strategies in solving problems involving factors of polynomials Content Standard: The learner demonstrates an understanding of key concepts of factors of polynomials, andrational algebraic expressions. Mini task 1: Packaging Boxes Performance Standard: The learner is able to formulate real-life problems involving factors of polynomials and rational algebraic expressions and solve these problems accurately using a variety of strategies. Module Map Here is a simple map of the lessons that will be covered in this module. cbp2024 \| G8SHMATH 9 III\. STUDY GUIDES To do well in this module, you need to remember and do the following: 1\. Perform carefully all explore activities. 2. Be very observant of the process and answers you get in the explore activities since these will help you understand the lesson. 3. Watch the supplemental videos regarding the lesson which were uploaded in the google classroom. 4. Perform first all Try This activities before comparing your answers to the key answers provided. 5. Perform all Try This activities since these will help you master the skills of factoring for you to be successful in the next module, which is about Rational expressions. 6. Answer all summative assessments in your module before transferring your answers to the Quiz Assignments in your google classroom. 7. Submit your summative task on time for immediate feedback, which will also give you the confidence to do the next task. 8. Always follow the instructions given to you on how and where to submit your outputs. 9. Always read updates in the Group Chat or announcements in the google classroom. 10. Keep in touch with your teacher for queries/concerns via messenger. 11. Remember to do your best in every activity given to you. IV\. PRE-ASSESSMENT 1 Let us find out how much you already know about the topics covered in this module. In each item, choose the letter of the option that you think correctly answers the question. DO NOT leave any item unanswered. Check for the correct answers found at the end of the module. Take note of the items that you were not able to correctly answer and try answering them again as you go through this module. 1. What is the greatest common monomial factor of 16x 3yz 2 and 24x 2yz 4? a\. 2xyz c. 8xyz b\. 6xz d. 8x 2yz 2 2\. Find the product of (xy + 5) (xy -- 5). a. x 2y 2 + 10xy + 25 c. x 2y 2 + 25 b\. x 2y 2 -- 10xy + 25 d. x 2y 2 -- 25 3\. What is the square of the binomial (6x 2y -- 7)? a\. 36x 4y 2 + 49 c. 36x 4y 2 + 84x 2y + 49 b\. 36x 4y 2 -- 49 d. 36x 4y 2 -- 84x 2y + 49 4\. Find the product of (m + 5n) (m2 -- 5mn + 25n 2 ). a. m3 + 125n 3 c. m3 + 10m2n -- 50mn 2 + 125n 3 b\. m3 -- 125n 3 d. m3 -- 10m2n -- 50mn 2 + 125n 3 5\. Write x 2 + 5x -- 6 as product of its factors. a. (x + 1) (x -- 6) c. (x -- 1) (x + 6) b\. (x + 2) (x -- 3) d. (x -- 1) (x + 3) 6\. What is the factored form of 15a 2 -- 17ab -- 4b 2? a\. (3a + 2b) (5a -- 2b) c. (5a + b) (3a -- 4b) b\. (3a -- 2b) (5a -- 2b) d. (3a -- b) (3a + 4b) cbp2024 \| G8SHMATH 10 7\. When factored completely, 5x 3 -- 45x is equal to \_\_\_\_\_. a. 5x (x + 3)2 c. x (5x + 9) (x -- 5) b\. 5 (x 2 + 3) (x -- 3) d. 5x (x + 3) (x -- 3) 8\. The area of a square is equal to (9a 2 + 30a + 25) square units. Find the binomial that represents the length of one side of the square. a. (a + 25) units c. (3a + 10) units b\. (3a + 5) units d. (9a + 5) units 9\. One of the factors of 2a 2 + 5a -- 12 is a + 4. Which is the other factor? a\. 2a -- 3 c. 2a -- 8 b\. 2a + 3 d. 2a + 8 10\. Express 12ab -- 12a -- 8b 2 + 8b as product of its factors. a. 4 (3a -- 2b) (b -- 1) c. 4 (a -- 2b) (3b -- 1) b\. -- 4 (3a + 2b) (b + 1) d. 4 (3a + 2b) (b -- 1) 11\. The area of a rectangle is represented as 27p 3 -- 8q 3. Find the dimensions of the rectangle. a. (3p -- 2q) and (9p 2 -- 6pq + 4q 2 ) c. (3p -- 2q) and (9p 2 -- 12pq + 4q 2 ) b\. (3p -- 2q) and (9p 2 + 6pq + 4q 2 ) d. (3p -- 2q) and (9p 2 + 12pq + 4q 2 ) 12\. Find the missing term so that 49m4 + \_\_\_\_\_ + 100n 4 forms a perfect square trinomial. a. 70 m2n 2 c. 70 mn b\. 140 m2n 2 d. 140 mn 13\. Factor completely: y 3 + y 2 -- y -- 1. a. (y 2 + 1) (y -- 1) c. (y + 1) (y + 1) (y -- 1) b\. (y + 1) (y 2 -- 1) d. (y -- 1) (y -- 1) (y + 1) 14\. A square of side y is cut out from a larger square of side x. What is the area of the remaining figure? a\. x 2 -- y 2 b\. (x -- y)2 c\. y 2 -- x 2 d\. (y -- x)2 15\. Factor completely: 4x 2 + 100 a\. (2x + 10) (2x + 10) c. 4 (x 2 + 25) b\. 4 (x + 5) (x + 5) d. 2 (2x 2 + 50) 16\. The length of a box is two inches less than thrice the height. The width is one inch more than twice the height. The box has a volume of 715 cubic inches. Which of the following equations can be used to find the height of the box? a\. H (3H -- 1) (2H + 2) = 715 c. H (3H -- 2) (2H + 1) = 715 b\. H (2H -- 2) (3H + 1) = 715 d. H (2 -- 3H) (1 + 2H) = 715 cbp2024 \| G8SHMATH 11 17\. A portion from a square yard is allotted for the building of a fountain. How large is the area remaining in the garden? a\. 9x 2 + 3x -- 11 b\. 9x 2 -- 15x -- 11 c\. 9x 2 -- 9x -- 11 d\. 9x 2 -- 11 18\. The side of a square is x cm long. The length of a rectangle is 4 cm longer than a side of this square, and the width is 4 cm shorter. Which statement is true? a\. The area of the square is greater than the area of the rectangle. b. The area of the rectangle is greater than the area of the square. c. The area of the square is equal to the area of the rectangle. d. The relationship cannot be determined from the given information. 19. A factory makes rectangular cartons for its products. If the volume of each carton is given by V = y 3 + 14y 2 + 40y, which are the possible dimensions of the carton? a\. y (y + 14) (y -- 4) c. y (y + 5) (y + 8) b\. y (y + 40) (y + 1) d. y (y + 4) (y + 10) 20\. Based on your answer in the previous number, what are the dimensions of the carton if y = 6? a\. 6 by 20 by 2 c. 6 by 11 by 14 b\. 6 by 46 by 7 d. 6 by 10 by 16 V. LESSON PROPER How long would it take you to solve the multiplication expressions on the right without using pen and paper? Give yourself three minutes to solve the expressions. You may get a calculator to check your answers. How many answers did you get correctly? Did you solve them easily? Now, look at the expressions. What do you notice? Can these be solved using other techniques? 98 x 102 34 x 34 49 x 49 The multiplication expressions shown above can be performed easily by using patterns. 3x - 1 3x - 1 4 -- 3x 3 Explore cbp2024 \| G8SHMATH 12 The process of finding products of polynomials using short methods is called special products. Rewriting a polynomial as a product of polynomial factors is called factoring polynomials. Study the solutions shown below. Now, try answering the following. 39 x 39 62 x 58 42 x 42 Were you able to make use of the knowledge that you gained? Are these applicable to other multiplication problems? The problems you answered show one of the many situations where we can apply knowledge of special products. In this module, you will learn the different techniques of multiplying polynomials using special products. In this module, you will do varied activities, which will help you answer the question, "How can unknown quantities in geometric problems be solved? The reverse of finding related products is a process called factoring. In this module, you will also learn the different techniques of factoring. In -- Out Of the BOx Let us begin by answering the question, "How can unknown quantities in geometric problems be solved? Write your response in the provided space "IN THE BOX". After the lesson on special products and factoring, you will be asked to put what you learned in the space "OUT OF THE BOX". I can now say that\... 98 x 102 (100 -- 2) (100 + 2) 10 000 -- 4 9996 34 x 34 (30 + 4) (30 + 4) 900 + 2 (30) (4) + 16 1 156 47 x 47 (50 -- 3) (50 -- 3) 2500 -- 2 (50) (3) + 9 2 209 IN THE BOX I think\... OUT OF THE BOX cbp2024 \| G8SHMATH 13 In Grade 7, you learned how to multiply polynomials. Your knowledge and skills in multiplying polynomials will be very useful in special products and factoring polynomials. Common Monomial Factoring Multiply the following polynomials by applying the distributive property of multiplication. 1. 8x (y + 9) 2\. 6p 2 (p − 12) 3\. 4an (a + 5b) 4\. 2w (3w3 − 5w2 + 1) 5\. 3pq 3 (p 2 − 5pq -- 2q) Were you able to arrive with the following answers? 1\. 8x (y + 9) = 8xy + 72x 2\. 6p 2 (p − 12) = 6p 3 − 72p 2 3\. 4an (a + 5b) = 4a 2n + 20abn 4\. 2w (3w3 − 5w2 + 1) = 6w4 − 10w3 + 2w 5\. 3pq 3 (p 2 − 5pq -- 2q) = 3p 3q 3− 15p 2q 4 − 6pq 4 What if we are required to do the opposite of multiplication? In this case, it is to factor the common monomial. INTRODUCTION (Explore) Note: Apply the product rule of exponents in multiplying variables cbp2024 \| G8SHMATH 14 Example 1. 8x 2 y + 72x 8x 2 y + 72x 8x is the greatest common monomial factor 8x (\_\_\_\_\_ + \_\_\_\_\_) → Divide each of the terms by the greatest common monomial factor. » 8x 2 y divided by 8x = xy » 72x divided by 8x = 9 8x (xy + 9) → Factored Form When it comes to variables, why do we have to pick the one with a lower exponent? Study the next example. Example 2. 6p 3 − 72p 2 Express the terms to their prime factors 6p 3 = 3 2 p p p Align the common factors 72 p 2 = 3 3 2 2 2 p p 3 2 p p = 6p 2 Multiply the common factors to get the greatest common monomial factor: 6 p 2 What did you notice? You can also get the greatest common monomial factor by considering this. 6p 3 − 72p 2 6p 2 (\_\_\_\_ -- \_\_\_\_) → 6p 2 is the greatest common monomial factor. 6p 2 (p -- 12) → Factored Form We need to arrive at 8x (xy + 9) where the common monomial factor is 8x Numeral What is the greatest common factor of the constants in each term? Variables What variable is common to all the terms? \*Pick the one with lower exponent Divide each term by the greatest common monomial factor to get the final answer. 6p 2 divided by 6p 2 = p -- 72p 2 divided by 6p 2 = -- 12 Numeral Between 6 and 72, the GCF is 6 Variable p is a variable common to both terms but between p 2 and p 3 , p 2 is the greatest variable common factor. INTERACTION (Firm-up & Deepen) cbp2024 \| G8SHMATH 15 Example 3. 6w4 − 10w3 + 2w Express the terms to their prime factors 6w4 = 3 2 w w w w Align the common factors − 10w3 = 5 2 w w w 2w = 2 w 2 w = 2w Multiply the common factors to get the greatest common monomial factor: 2w 2w (\_\_\_\_ -- \_\_\_\_ + \_\_\_\_\_) → 2w is the greatest common monomial factor. 2w ( 3w3 -- 5w2 + 1) → Factored Form Example 4. 3p 3q 2 − 15p 2q 3 + 6pq 2 Express the terms to their prime factors 3p 3q 2 = 3 p p p q q Align the common factors − 15p 2q 3 = −3 5 p p q q q 6pq 2 = 3 2 p q q 3 p q q = 3pq 2 Multiply the common factors to get the greatest common monomial factor: 3 pq 2 You can also get the greatest common monomial factor by considering this. 3p 3q 2 − 15p 2q 3 + 6pq 2 3pq 2 (\_\_\_\_ -- \_\_\_\_ + \_\_\_\_) 3pq 2 (p 2 − 5pq + 2) → Factored Form Numeral Among 3, 15 and 6, the GCF is 3. Variable p is a variable common to all terms but among p 3 , p 2 and p, p is the greatest variable common factor q is a variable common to all terms but among q 2 , q 3 and q 2 , q 2 is the greatest variable common factor Divide each term by the common monomial factor to get the final answer.3p 3q 2 divided by 3pq 2 = p 2 − 15p 2q 3 divided by 3pq 2 = − 5pq 6pq 2 divided by 3pq 2 = 2 Divide each term by the greatest common monomial factor to get the final answer. 6w4 divided by 2w = 3w3 − 10w3 divided by 2w = -- 5w2 2w divided by 2w = 1 cbp2024 \| G8SHMATH 16 Complete this table. Polynomial Common Monomial Factor Remaining factor Factored Form Ex. 18x 3y 4 -- 12x 2y 5 6x 2y 4 3x -- 2y 6x 2y 4 (3x -- 2y) 1\. 5x + 10 2\. -- 8x -- 20 3\. 3x 5 -- 12x 4 -- 9x 3 4\. y 4 + 3y 2 5\. 2m5n 4 + mn 4 6\. 17x 2 + 34x + 51 7\. 18m2n 4 -- 12m2n 3 + 24m2n 2 The exercises that you have just performed involved only one technique of factoring polynomials. What if a polynomial expression does not have a Common Monomial Factor or Greatest Common Monomial Factor (GCMF), for example x 2 + 10x + 25, or x 2 -- 4? Will you immediately conclude that the expression is prime or non-factorable? How can these be written in product form? You will learn the other factoring techniques in the next lessons. The counterpart of each factoring technique is also presented. Note: To check if your answer is correct, multiply the common monomial factor with the remaining factor and it must give you back the polynomial. Try This 1 cbp2024 \| G8SHMATH 17 Factoring by Grouping Some expressions have a common binomial factor. Let us recall the multiplication of binomials. Review Multiply (a + b) (c + d). Solution: Use the FOIL method. Photo Credits: What is the FOIL Method?. \[Online Image\]. (2020). Calcworkshop. https://calcworkshop.com/polynomials/foil-method/ Now, let us reverse the process by looking for the factors of the given product. Example 1. Factor ac + ad + bc + bd. Solution 1: Group the terms in such a way that each group of binomials will have GCMF. (ac + ad) + (bc + bd) The common factor is a. The common factor is b. a (c + d) + b (c + d) The common factor is (c + d) Factor also the common binomial. Thus, ac + ad + bc + bd = (c + d) (a + b). The factors obtained are the same as the ones given in the review. Interchanging the factors is simply applying the commutative property of multiplication. Meaning, 2(3) is the same as 3(2). Solution 2: Form another set of groups of binomials. Make sure that each group will have GCMF. From ac + ad + bc + bd, let's try (ac + bc) + (ad + bd) (ac + bc) + (ad + bd) The common factor is c. The common factor is d. c (a + b) + d (a + b) The common factor is (a + b) Factor also the common binomial. Thus, ac + ad + bc + bd = (a + b) (c + d). The factors obtained are the same as the ones in solution 1. Notice that you will have a binomial common to both terms after factoring the GCMF. Always make sure to have this. Notice that you will have a binomial common to both terms after factoring the GCMF. Always make sure to have this. INTRODUCTION (Explore) INTERACTION (Firm-up & Deepen) cbp2024 \| G8SHMATH 18 Example 2. Factor xy + 3x -- 6y -- 18. Solution 1: (xy + 3x) + (-- 6y -- 18) The common factor is x. The common factor is -6. x (y + 3) + -- 6 (y + 3) Simplify this. x (y + 3) -- 6 (y + 3) The common factor is (y + 3) Factor also the common binomial. Thus, xy + 3x + 6y + 18 = (y + 3) (x -- 6). Solution 2: Let's try another set of grouping. (xy -- 6y) + (3x -- 18) The common factor is y. The common factor is 3. y (x -- 6) + 3 (x -- 6) The common factor is (x -- 6) Factor also the common binomial. Thus, xy + 3x + 6y + 18 = (x -- 6) (y + 3) which is the same (y + 3) (x -- 6) of solution 1. Example 3. Factor (x + y) (x -- 2) + (x + y) (x -- 3). Solution: (x + y) (x -- 2) + (x + y) (x -- 3) first term second term The common factor of the two terms is (x + y) Factor also the common binomial. (x + y) (x -- 2) + (x + y) (x -- 3) = (x + y) \[(x -- 2) + (x -- 3)\] = (x + y) \[x -- 2 + x -- 3\] → combine like terms = (x + y) (2x -- 5) The operation is always addition (+). The operation is always addition (+). Note: You can form different groupings of binomials but remember that these binomials must have GCMF and you must arrive with common binomials after factoring the GCMF. The operation at the middle should always be addition (plus sign). This will only change if the GCMF contains negative like in the case of solution 1 of Example 2. cbp2024 \| G8SHMATH 19 It is --1 which is common to --a and b to be able to produce a binomial common to both terms. It is --b which is common to --2bc and b to be able to produce a binomial common to both terms. Example 4. Factor 2ac -- 2bc -- a + b. Solution 1: 2ac -- 2bc -- a + b = (2ac -- 2bc) + (-- a + b) = 2c (a -- b) + 1 (-- a + b) = 2c (a -- b) -- 1 (a -- b) = (a -- b) (2c -- 1) Solution 2: 2ac -- 2bc -- a + b = (2ac -- a) + (-- 2bc + b) = a (2c -- 1) + b (-- 2c + 1) = a (2c -- 1) -- b (2c -- 1) = (2c -- 1) (a -- b) Factor the following. 1. 5m3 + 5m2 -- m -- 1 6. 8m2 -- 4mn -- 6m + 3n 2\. 3x 2 -- 7xy + 3x -- 7y 7. 3p 2q + 6pq -- 5p -- 10 3\. 8xy -- zw + 8xw -- yz 8. (y 2 -- 2) (y -- 4) + (y + 3) (y -- 4) 4\. 6mx -- y + 2my -- 3x 9. (5b + 6) (a -- 1) -- (a -- 1) (2b + 7) 5\. 10b 3 + 25b -- 4b 2 -- 10 10. (2x -- 5) (x + 2) -- (x + 2) (x -- 1) WhO has the COrreCt ansWer? Ana and Anita were asked to show the factors of 8x 3y + 16x 2 -- 24x 4y 2. Both of them used the distributive property of multiplication to check their answers. Anita claimed that her answer is correct, but Ana insisted that her answer is also correct. If you are their close friend and you don't want that this will cause misunderstanding between them, what would you tell them? Note: To check if your answer is correct, multiply the binomials by using FOIL Method and it must give you back the polynomial. Ana's Answer 4x 2 (2xy + 4 -- 6x 2y 2 ) Anita's Answer 8x 2 (xy + 2 -- 3x 2y 2 ) Try This 2 cbp2024 \| G8SHMATH 20 In the first lesson, you were able to bring out the greatest common monomial factor (GCMF) of a polynomial and that you can check if your answers are correct by multiplying the GCMF with the other factor using the distributive property of multiplication. You also applied your knowledge on GCMF when factoring 4 terms through the method called factoring by grouping. In this lesson, you are expected to observe patterns in multiplying polynomials. This would help you explore shortcuts in multiplying polynomials with special characteristics. The results of multiplying these kinds of polynomials are called special products. But why are they called special products? Having a deep understanding of these special products will also help you easily factor their counterparts. Factoring Difference of Two Squares Multiply the following polynomials using the FOIL Method. Do not forget to combine like terms if there are any. 1. (a + b) (a -- b) 2\. (x + 4) (x -- 4) 3\. (2y + 3) (2y -- 3) 4\. (5x + 2y) (5x -- 2y) 5\. (7 + 4z 2\) (7 -- 4z 2) Were you able to arrive with these answers? 1\. (a + b) (a -- b) = a 2 -- b 2 2\. (x + 4) (x -- 4) = x 2 -- 16 3\. (2y + 3) (2y -- 3) = 4y 2 -- 9 4\. (5x + 2y) (5x -- 2y) = 25x 2 -- 4y 2 5\. (7 + 4z 2\) (7 -- 4z 2\) = 49 -- 16z 4 Photo Credits: \[Untitled Image for Multiplying Two Binomials Using the FOIL method\]. Mesacc. http://www.mesacc.edu/\~scotz47781/mat120/notes/polynomials/f oil\_method/foil\_method.html INTRODUCTION (Explore) cbp2024 \| G8SHMATH 21 1.) What have you noticed with your answers? 2.) Can you see any pattern? What are those patterns? 3.) Do you think you can find the correct answer immediately without undergoing the FOIL Method? Let's multiply the expressions. 1. (a + b) (a -- b) → a 2 + ab -- ab -- b 2 = a 2 -- b 2 2\. (x + 4) (x -- 4) → x 2 + 4x -- 4x -- 16 = x 2 -- 16 3\. (2y + 3) (2y -- 3) → 4y 2 + 6y -- 6y -- 9 = 4y 2 -- 9 4\. (5x + 2y) (5x -- 2y) → 25x 2 + 10xy -- 10xy -- 4y 2 = 25x 2 -- 4y 2 5\. (7 + 4z 2 ) (7 -- 4z 2 ) → 49 + 28z 2 -- 28z 2 -- 16z 4 = 49 -- 16z 4 Notice that the two middle terms have opposite signs which when combined will lead to zero leaving only the first and last terms in the final answer. Find the product of the following: 1\. (5d + 7) (5d -- 7) 2\. (4m + 13n) ( 4m -- 13n) 3\. (11 -- 9c) (11 + 9 c) 4\. (x -- 4y 2 ) (x + 4y 2 ) 5\. (2d 3 + 6f) (2d 3 -- 6f) Note: The product of the sum and difference of two terms is the difference of the squares of the terms. In symbols, (x + y) (x -- y) = x 2 -- y 2. Notice that the product is always a binomial since the middle terms will always add up to zero. Try This 3 Process Questions INTERACTION (Firm-up & Deepen) cbp2024 \| G8SHMATH 22 You have noticed that when you multiply sum and difference of two terms, the result will always be a binomial whose first and last terms are both squares and a minus sign in the middle. This is called Difference of Two Squares. Which of the following is a difference of two squares? a.) 9x 2 -- 16 b.) -- 4 + 9x 6 c.) x 4 -- 8 d.) x 3 -- 25 Example 1: Factor. a. x 2 -- 64 b. 25x 2 -- 4y 4 c. 36x 6y 2 -- 9 Solutions: a. Find the square roots of the terms 2 = x 64 = 8 Substitute in the pattern. x 2 -- 64 = (x)2 -- (8)2 = (x + 8) (x -- 8) Try This 4 cbp2024 \| G8SHMATH 23 b\. Find the square roots of the terms 25 2 = 5x since (5x) (5x) = 25x 2 4 4 = 2y 2 since (2y 2 ) (2y 2 ) = 4y 4 Substitute in the pattern. 25x 2 -- 4y 4 = (5x)2 -- (2y 2 )2 = (5x + 2y 2 ) (5x -- 2y 2 ) c\. Find the square roots of the terms 36 6 2 = 6x 3y since (6x 3y) (6x 3y) = 36 6 2 9 = 3 Substitute in the pattern. 36x 6y 2 -- 9 = (6x 3y)2 -- (3)2 = (6x 3y + 3) (6x 3y -- 3) Factor the following. 1. 100x 2 -- 81 2\. x 6y 8 -- 121z 4 3\. x 2 -- 49y 10 4\. 16a 6 -- 25b 2 5\. a 2 b 4 -- 81 Example 2: Factor Completely a\. 18x 4 -- 32 b. 81x 3y 2 -- 49xy 4 c. (2x -- 1) 2 -- 16 In these items, you are expected to apply your skill on common monomial factoring. Solutions: a. 18x 4 -- 32 = 2 (9x 4 -- 16) Factor out 2. = 2 (3x 2 + 4) (3x 2 -- 4) Factor the resulting difference of two squares. b. 81x 3y 2 -- 49xy 4 = xy 2 (81x 2 -- 49y 4 ) Factor out xy 2 = xy 2 (9x + 7y 2 ) (9x -- 7y 2 ) Factor the resulting difference of two squares. c. Find the square roots of the terms (2x -- 1) 2 = 2x -- 1 16 = 4 Substitute in the pattern. (2x -- 1)2 -- 16 = (2x -- 1)2 -- (4)2 = \[(2x -- 1) + 4\] \[(2x -- 1) -- 4\] Simplify = (2x + 3) (2x -- 5) Note: Factoring completely means to make the polynomial a product of combinations of monomials, binomials or trinomials that are prime. Try This 5 cbp2024 \| G8SHMATH 24 Factor the following. 1. 5x 3 -- 5x 2\. 36x 3y 5 -- xyz 6 3\. (3m --4)2 -- 64 Factoring Perfect Square Trinomials In the previous section, you learned that multiplying the sum and difference of two terms will always lead to difference of two squares. What about multiplying the sum of two terms or difference of two terms to itself? Let us compare these two cases using the table below. Sum and Difference of Two Terms Square of a Binomial (a + 2b) (a -- 2b) (a + 2b) (a + 2b) can be written also as (a + 2b)2 (x -- 5y) (x + 5y) (x + 5y) (x + 5y) can be written also as (x + 5y)2 (3x -- 4y) (3x + 4y) (3x -- 4y) (3x -- 4y) can be written also as (3x -- 4y)2 (6x 2 + 5yz) (6x 2 -- 5yz) (6x 2 + 5yz) (6x 2 + 5yz) can be written also as (6x 2 + 5yz)2 Observe the characteristics of each special product. What have you noticed? Find the product of the following polynomials using the FOIL Method. Do not forget to combine like terms if there are any. 1. (a + 2)2 or (a + 2) (a + 2) 2\. (x + 5)2 or (x + 5) (x + 5) 3\. (2y -- 4)2 or (2y -- 4) (2y -- 4) 4\. (4x -- 3y)2 or (4x -- 3y) (4x -- 3y) 5\. (6 + z 2 )2 or (6 + z 2 ) (6 + z 2 ) Photo Credits: \[Untitled Image for Multiplying Two Binomials Using the FOIL method\]. Mesacc http://www.mesacc.edu/\~scotz47781/mat120/notes/polynomials/f oil\_method/foil\_method.html Try This 6 INTRODUCTION (Explore) cbp2024 \| G8SHMATH 25 1.) What have you noticed with your answers? 2.) Can you see any pattern? What are those patterns? 3.) Do you think you can find the correct answer immediately without undergoing the FOIL Method? Let us multiply the expressions. 1. (a + 2)2 or (a + 2) (a + 2) → a 2 + 2a + 2a + 4 = a 2 + 4a + 4 2\. (x + 5)2 or (x + 5) (x + 5) → x 2 + 5x + 5x + 25 = x 2 + 10x + 25 3\. (2y -- 4)2 or (2y -- 4) (2y -- 4) → 4y 2 -- 8y -- 8y + 16 = 4y 2 -- 16y + 16 4\. (4x -- 3y)2 or (4x -- 3y) (4x -- 3y) → 16x 2 -- 12xy -- 12xy + 9y 2 = 16x 2 -- 24xy + 9y 2 5\. (6 + z 2 )2 or (6 + z 2 ) (6 + z 2 ) → 36 + 6z 2 + 6z 2 + z 4 = 36 + 12z 2 + z 4 Notice that the two middle terms have same signs and are like terms. The sum of the middle terms will lead to a value which is the same as doubling the term. The result of squaring a binomial is called Perfect Square Trinomial. Let us analyze further the examples. Problem Solution: FOIL Method Solution: Using Patterns 1\. (a + 2)2 a 2 + 2a + 2a + 2 2 → (Combine like terms) a 2 + 4a + 4 (a)2 + 2(a) (2) + (2)2 a 2 + 4a + 4 2\. (x + 5)2 x 2 + 5x + 5x + 25 → (Combine like terms) x 2 + 10x + 25 x 2 + 2(x) (5) + (5)2 x 2 + 10x + 25 3\. (2y -- 4)2 4y 2 -- 8y -- 8y + 16 → (Combine like terms) 4y 2 -- 16y + 16 (2y)2 + 2(2y) (-- 4) + (-- 4)2 4y 2 -- 16y + 16 4\. (4x -- 3y)2 16x 2 -- 12xy -- 12xy + 9y 2 → (Combine like terms) 16x 2 -- 24xy + 9y 2 (4x)2 + 2 (4x) (-- 3y) + (-- 3y)2 16x 2 -- 24xy + 9y 2 5\. (6 + z 2 )2 36 + 6z 2 + 6z 2 + z 4 → (Combine like terms) 36 + 12z 2 + z 4 (6)2 + 2 (6) (z 2 ) + (z 2 )2 36 + 12z 2 + z 4 Photo Credits: Roberts, D. (n.d.). Factoring Perfect Square Trinomials. \[Online Image\]. MathBitsNotebook. https://mathbitsnotebook.com/Algebra1/Factoring/FCPerfSqTri.html Process Questions INTERACTION (Firm-up & Deepen) cbp2024 \| G8SHMATH 26 You can further understand the concept of the square of a binomial using algebra tiles. Example 1: Arrange the following tiles to form a square. Then find the area. We can also solve the area algebraically. Example 2: Arrange the following tiles to form a square. Then find the area. x 2 --x 2 x 1 -1 -x (x + 2)2 = x 2 + 2x + 2x + 4 (x + 2)2 = x 2 + 2 (2x) + 4 (x + 2)2 = x 2 + 4x + 4 Algebra Tiles To find the area, you can just count the number of tiles you have. 1 → x 2 4 → x Area: x 2 + 4x + 4 4 → 1 Areasquare = s 2 x -- 3 cbp2024 \| G8SHMATH 27 We can also solve the area algebraically. Example 3: Arrange the following tiles to form a square. Then find the area. Do you already have an idea why the square of a binomial is called Perfect Square Trinomial? Find the product of the following: 1\. (5m + 8)2 2\. (2n -- 9)2 3\. (5p + 4q)2 4\. (9a 3 -- 7b 2 )2 5\. (6ab 2 -- 3c 4 )2 You have learned that the square of a binomial or multiplying a binomial to itself will lead you to a Perfect Square Trinomial. Can you identify the characteristics of a Perfect Square Trinomial? (x -- 3)2 = x 2 -- 3x -- 3x + 9 (x + 2)2 = x 2 + 2 (-- 3x) + 9 (x + 2)2 = x 2 -- 6x + 9 Areasquare = s 2 1 → x 2 8 → x Area: x 2 + 8x + 16 16 → 1 (x +4)2 = x 2 + 4x +4x + 16 (x + 4)2 = x 2 + 2 (4x) + 16 (x + 4)2 = x 2 +8x + 16 Areasquare = s 2 x + 4 cbp2024 \| G8SHMATH 28 When factoring a Perfect Square Trinomial, you are then expected to arrive at a square of a binomial or a binomial multiplied to itself. Example 1: Determine whether these are perfect square trinomials. If YES, factor the trinomial. a. x 2 -- 10x + 25 c. x 8 + 6x 4 + 36 b\. 4x 6 + 36x 3 + 81 d. x 2 + 6x -- 9 Solutions: a. x 2 -- 10x + 25 x x --5 -- 5 2(x) (--5) = -- 10x Thus, x 2 -- 10x + 25 = (x -- 5) 2 b\. 4x 6 + 36x 3 + 81 2x 3 2x 3 9 9 2 (2x 3 )(9) = 36x 3 Thus, 4x 6 + 36x 3 + 81 = (2x 3 -- 9) 2 c\. x 8 + 6x 4 + 36 x 4 x 4 6 6 2 ( x 4\) (6) = 12x 4 d\. x 2 + 6x -- 9 This is negative Determine whether these are perfect square trinomials. If YES, factor the trinomial. 1. y 2 + 2y + 1 2\. 4p 2 + 12p + 36 3\. 9y 6 -- 60y 3z 4 + 100z 8 4\. 81p 2 -- 90pq -- 25q 2 5\. 11 -- 154z + 49z 2 Middle Term Yes The first and last terms are perfect squares. The middle term is twice the product of the values that were squared. Middle Term Yes The first and last terms are perfect squares. The middle term is twice the product of the values that were squared. Middle Term No This is NOT a Perfect Square Trinomial since the middle term is 6x 4 , instead of 12x 4 No This is NOT a Perfect Square Trinomial since one of the perfect squares is negative. \*A perfect square is always positive. If you square a negative number, the result is positive. Try This 8 cbp2024 \| G8SHMATH 29 Example 2: Factor the following completely. a. 2x 3 + 28x 2 + 98x b\. x 4 -- 24x 3y + 144x 2 y 2 c\. 10x 7y + 120x 4y 2 + 360xy 3 Solutions: a. 2x 3 + 28x 2 + 98x Factor out first the GCMF which is 2x. 2x (x 2 + 14x + 49) The trinomial is a PST. Factor it also. 2x (x + 7)2 or 2x (x + 7) (x + 7) b\. x 4 -- 24x 3y + 144x 2 y 2 Factor out first the GCMF, which is x 2. x 2 (x 2 -- 24xy + 144y 2 ) The trinomial is a PST. Factor it also. x 2 (x -- 12y)2 or x 2 (x -- 12y) (x -- 12y) c\. 10x 7y + 120x 4y 2 + 360xy 3 Factor out first the GCMF which is 10xy 10xy (x 6 -- 12x 3y + 36y 2 ) The trinomial is a PST. Factor it also. 10xy (x 3 -- 6y)2 or 10xy (x 3 -- 6y) (x 3 -- 6y) Factor the following completely. 1. 5x 12 + 50x 6y 2 + 125y 4 2\. 3m4 -- 6m3 + 3m2 3\. 4p 3 -- 24p 2 + 36p 4\. 3p 7 -- 24p 5 + 48p 3 What are the factors of Sum of Two Squares? (x + 3) (x -- 3) when multiplied will result to x 2 -- 9 which is Difference of Two Squares (DOTS) (x + 3) (x + 3) when multiplied will result to x 2 + 6x + 9 which is Perfect Square Trinomial (PST) (x -- 3) (x -- 3) when multiplied will result to x 2 + 6x + 9 which is Perfect Square Trinomial (PST) Can you find two binomials whose product is x 2 + 9? In this case, the sum of two squares is considered PRIME, meaning the factors are one (1) and itself. You already learned two special products and how to factor the results of these special products by applying the patterns you observed. In the next section, you will learn the last special product. Try This 9 cbp2024 \| G8SHMATH 30 Product of a Binomial and a Trinomial (x + y) (x 2 -- xy + y 2 ) = x 3 + y 3 (x -- y) (x 2 + xy + y 2 ) = x 3 -- y 3 Factoring Sum and Difference of Two Cubes Study the following examples. Find the product of the given expressions by applying the distributive property of multiplication that you learned in Grade 7. 1. (x + 3) (x 2 -- 3x + 9) 2\. (y + 4) (y 2 -- 4y + 16) 3\. (2z + 5) (4z 2 -- 10z + 25 4\. (x -- 3) (x 2 + 3x + 9) 5\. (y -- 4) (y 2 + 4y + 16) 6\. (2z -- 5) (4z 2 + 10z + 25 1.) Compare the expressions in numbers 1 -- 3 to the expressions in numbers 4 -- 6. What did you notice? 2.) Compare also the answers of numbers 1 -- 3 to the answers of numbers 4 -- 6. What did you notice? 3.) Can you find a shortcut in multiplying these kinds of expressions without applying the distributive property of multiplication? Let us find the product of the expressions. 1. (x + 3) (x 2 -- 3x + 9) = x 3 + 27 4. (x -- 3) (x 2 + 3x + 9) = x 3 + 27 2\. (y + 4) (y 2 -- 4y + 16) = y 3 + 64 5. (y -- 4) (y 2 + 4y + 16) = y 3 + 64 3\. (2z + 5) (4z 2 -- 10z + 25) = 8z 3 + 125 6. (2z -- 5) (4z 2 + 10z + 25) = 8z 3 + 125 These special forms of multiplication of a binomial and trinomial generate an expression (x + 3) (x 2 --3x + 9) = x 3 -- 3x 2 + 9x 3x 2 -- 9x + 27 Find the sum of two lines x 3 + 27 INTRODUCTION (Explore) Process Questions INTERACTION (Firm-up & Deepen) cbp2024 \| G8SHMATH 31 called the Sum and Difference of Two Cubes. cbp2024 \| G8SHMATH 32 Sum and Difference of Two Cubes x 3 + y 3 = (x + y) (x 2 -- xy + y 2 ) x 3 -- y 3 = (x -- y) (x 2 + xy + y 2 ) Find the product of the given expressions. 1. (x + 6) (x 2 -- 6x + 36) 2\. (2m + x) (4x 2 -- 2mx + x 2 ) 3\. (a 2b 4 + ab 2c 3 + c 6 ) (ab 2 -- c 3 ) 4\. (10 -- x) (100 + 10x + x 2 ) If you are tasked to factor sum and difference of two terms, all you have to follow the equations given on the right. Example 1: Factor the following. a. x 3 -- 8 b. x 3 + 125 c. 343y 3 -- 1 d. 1000x 6 y 3 + z 3 Solutions: a. b\. x 3 + 125 = (x)3 + (5)3 = (x + 5) \[(x)2 -- (5) (x) + (5)2 \] = (x + 5) (x 2 -- 5x + 25) c\. 343y 3 -- 1 = (7y)3 -- (1)3 = (7y -- 1) \[ (7y)2 + (7y)(1) + (1)2 \] = (7y -- 1) (49ty 2 + 7y + 1) Find the two numbers/expressions which will produce the two cubed terms (x) 3 -- (2) 3 (x-- 2) \[(x) 2 + (2)(x) + (2) 2 \] Square the last term of the binomial. Multiply the two terms of the binomial. Find the opposite sign. Square the first term of the binomial (x-- 2) (x 2 + 2x + 4) Simplify the expression above x 3 -- 8 Try This 10 cbp2024 \| G8SHMATH 33 Note: The middle term is the opposite of the product of the terms of the binomial. d\. 1000x 6 + z 3 = (10x 2 )3 + (z)3 = (10x 2 + z) \[(10x 2 )4 -- (10x 2 )(z) + (z)2 \] = (10x 2 + z) (100x 4 -- 10x 2z + z 2 ) Factor the following. 1. 8x 3 -- 1 2\. x 6 --125z 21 3\. m9n 15 + 27p 3 Example 2: Factor completely a\. 81x 4y 7z + 24xyz 4 b. 4t3 -- 500s 6 Solution: a. 81x 4y 7z + 24xyz 4 Factor out the GCMF, which is 3xyz. = 3xyz (27x 3y 6 + 8z 3 ) = 3xyz \[ (3xy 2 )3 + (2z)3 \] = 3xyz (3xy 2 + 2z) \[ (3xy 2 )2 -- (3xy 2 ) (2z) + (2z)2 \] = 3xyz (3xy 2 + 2z) (9x 2y 4 -- 6xy 2z + 4z 2 ) b\. 4t3 -- 500s 6 Factor out the GCMF, which is 4. = 4 (t3 -- 125s 6 ) = 4 \[ (t)3 -- (5s 2 )3 \] = 4 (t -- 5s 2 ) \[ (t)2 + (t) (5s 2 ) + (5s 2 )2 \] = 4 (t -- 5s 2 ) (t2 + 5s 2 t + 25s 4 ) Factor the following. 1. 343v 3p 2 + 27p 2w3 2\. 7c 7 -- 56c 4 Try This 11 Try This 12 cbp2024 \| G8SHMATH 34 Analyze these problems carefully.renOvatIOn plan Baguio City allotted one lane of Harrison Road for night market vendors. Each vendor is entitled to occupy a square space of the road. To be able to give wider space for the movement of people, the city plans to decrease the width by 50 cm and increase the length by 50 cm. Find the new area allotted for each vendor? a\) Represent the unknowns Let: x = the measure of each side of the square selling space Then, x + 50 = New length x -- 50 = New width b\) Write your expression as a polynomial. (x + 50) (x -- 50) = x 2 -- 2 500 Remember that Area = (Length) (Width) c\) If the dimensions of each old selling space were 3 meters by 3 meters, what will be the area of each new space? Note: 1 meter = 100 cm then 3 meters = 300 cm New Area = x 2 -- 2 500 = 300 2 -- 2 500 = 90 000 -- 2 500 = 87 500 square centimeters or 8.75 sq. m. tIlInG prOJeCt Peter wants to tile his rectangular floor. He has two kinds of square tiles to choose from. One kind of tile measures 12 inches on the side and the other kind measures 15 inches on the side. His rectangular floor has dimensions of 180" by 240". Peter hired your services to help him decide which tile to use. a. How many small square tiles are needed to cover the floor? b\. How many big square tiles are needed to cover the floor? c\. If each small tile costs Php. 37.00, and each big tile cost Php. 50.00. Which tile should Peter use? Explain why. Photo Credits: Homedepot. (2021). Merola Tile Artisan Damero Azul. \[Photograph\]. Pinterest. https://www.pinterest.ph/pin/183732859786084765/ cbp2024 \| G8SHMATH 35 Solution: a\. How many small square tiles are needed to cover the floor? The small tile has a measure of 12 inches on the side. To be able to know the number of tiles needed on the length and width of the floor, factor 12 from the dimensions. 180 by 240 is the same as 12 (15 by 20) If we multiply 15 and 20, it will give a result of 300. This means that Peter will need 300 small (12" x 12") tiles. b. How many big square tiles are needed to cover the floor? The big tile has a measure of 15 inches on the side. To be able to know the number of tiles needed on the length and width of the floor, factor 15 from the dimensions. 180 by 240 is the same as 15 (12 by 16) If we multiply 12 and 16, it will give a result of 192. This means that Peter will need 192 big (15" x 15") tiles. c. If each small tile costs Php. 37.00, and each big tile cost Php. 50.00. Which tile should Peter use? Explain why. Small tiles : 37 (300) = Php. 11, 100 Big tiles: 50 (192) = Php. 9, 600 The computation shows that tiling the floor using the big tiles is cheaper. Peter should choose then the square tile measuring 15 inches on the side. summatIve assessment 1 Let us now check your understanding of the essential concepts that you have learned. Do your best to solve the problems and get the correct answers. Write your answers first in this module. Double- check your answers before transferring them in the Quiz Assignment of the Google Classroom titled "Summative Assessment 1." A. Multiple Choice (2 points each) 1\. Find the factors of 81m2 -- 100n 4. a. (9m -- 10n 2 )2 c. (9m + 10n 2 ) (9m -- 10n 2 ) b\. (9m + 10n)2 (9m -- 10n)2 d. (9m -- 10n 2 ) (9m -- 10n 2 ) 2\. Factor completely: 6x 5y -- 24xy 3 a\. 6xy (x 4 -- 4y 2 ) c. 3xy (x 4 -- 8y 2 ) b\. 6xy (x 2 -- 2y) (x 2 + 2y) d. 3xy (x 2 -- 4y) (x 2 -- 4y) INTEGRATION (Transfer) cbp2024 \| G8SHMATH 36 3\. Factor completely: 8x 6 + 64 a\. 8 (x 2 + 2)3 c. 8 (x 2 + 2) (x 4 -- 2x 2 + 4) b\. 8 (x + 2)6 d. 8 (x 2 + 2) (x 4 + 2x 2 + 4) 4\. The area of a square is 4x 2 -- 12x + 9 square units. Which expression represents the length of the side? a\. (2x -- 3) units c. (4x -- 9) units b\. (3x -- 2) units d. (9x -- 4) units 5\. Factor completely 4n 3 + 32n 2 + 24n. a. 2 (2n 3 + 16n 2 + 12n) c. 4 (n 3 + 8n 2 + 6n) b\. 2n (2n 2 + 16n + 12) d. 4n (n 2 + 8n + 6) 6\. What are the factors of -- 196 -- x 2? a\. PRIME c. (--14 + x) (--14 -- x) b\. -- (196 + x 2 ) d. -- (14 + x) (14 + x) 7\. Factor x 2 -- 12x -- 36. a. (x -- 6) (x -- 6) c. (x -- 6) (x + 6) b\. (x + 6) (x + 6) d. NOT Factorable 8\. Which of the following are Perfect Square Trinomials? I. m2 -- 16m + 64 II\. 121w6 -- 33w3z + 9z 2 III\. 36y 4 + 84y 2 -- 49 IV\. 25x 2 + 20xy + 4y 2 a\. I. II and III only c. I and IV only b\. I, III and IV only d. III and IV only 9\. Find the product of (2x + 7) (2x -- 7). a. 4x 2 -- 14x + 49 c. 4x 2 + 49 b\. 4x 2 -- 28x + 49 d. 4x 2 -- 49 10\. What are the factors of (5n -- 13)2 -- 1? a\. (5n -- 14) and (5n -- 12) c. (5n -- 12) and (5n -- 14) b\. (5n -- 13) and (5n -- 1) d. (5n -- 13) and (5n -- 1) 11\. What are the factors of 64 -- 125x 3? a\. (4 -- 5x) (4 -- 5x) (4 -- 5x) c. (4 -- 5x) (16 + 40x + 25x 2 ) b\. (4 -- 5x) (16 + 25x 2 ) d. (4 -- 5x) (16 + 20x + 25x 2 ) 12\. What are the factors of 5y (x + 9) -- (x + 9)? a\. (x + 9) (5y) c. (x + 9) (--5y) b\. (x + 9) (5y -- 1) d. 5y -- x + 9 13\. Factor completely the polynomial 63m2 -- 42mn + 7n 2. a. 9 (7m -- n) (7m -- n) c. 7 (3m + n) (3m + n) b\. 7 (3m -- n) (3m + n) d. 7 (3m -- n) (3m -- n) cbp2024 \| G8SHMATH 37 14\. What is the area of the square whose side measures 5xy -- 3? a\. 25x 2y 2 + 9 b\. 25x 2y 2 --15xy + 9 c\. 25x 2y 2 -- 30xy + 9 d\. 25x 2y 2 -- 30xy -- 9 15\. Which of the following is NOT true? a\. 25x 2 -- 121 = (5x + 11) (5x -- 11) c. 25x 2 -- 110x + 121 = (5x -- 11)2 b\. 25x 2 + 121 = (5x + 11) (5x + 11) d. 25x 2 + 110x + 121 = (5x + 11)2 B. Solving Find the factors of the following polynomials using the grouping method. Show your solution. 1. 2n 3 -- 4n 2 + 3n -- 6 (3 points) 2\. 14xy +77x -- 2y -- 11 (3 points) 3\. (10c + 7) (d -- 1) -- (d -- 1) (5c -- 6) (4 points) C. Application (10 points) John wants to make his square pool rectangular in shape by increasing its length by 3 meters and decreasing its width by 3 meters. John asked your expertise to help him decide on some issues. a. Represent the new length of John's pool. b. Represent the new width of John's pool. c. Compute the new area of John's pool using the new dimensions. d. If John does not want the area of his pool to decrease, will he pursue his plan? Explain your answer. Rubric: a -- c → 2 points each d → 4 points 4 → The explanation shows a commendable mastery of the concepts. 3 → The explanation shows a good understanding of the concepts. 2 → The explanation shows an adequate understanding of the concepts. 1 → The explanation shows a limited understanding of the concepts. Photo Credits: Castiglione, A. (n.d.) Square Pool. \[Photograph\]. Pinterest. https://www.pinterest.ph/pin/573434965026732662/ 5xy -- 3 cbp2024 \| G8SHMATH 38 Factoring Trinomials of the Form x 2 + Bx + C In the previous lessons, you learned that there are patterns you can apply to get the product of polynomials without undergoing the usual way of multiplication. This makes these products special. Thus, they are called Special Products. Can you identify these special products? In this lesson, you still have to use the FOIL method in multiplying the binomials. Instead of coming out with a shortcut in multiplying binomials, you will discover patterns that will help you find the factors of trinomials easily. Find the product of the following polynomials using the FOIL Method. Do not forget to combine like terms if there are any. 1. (x + 3) (x + 2) 2\. (x -- 5) (x -- 3) 3\. (x + 6) (x -- 1) 4\. (x -- 2) (x + 4) 1.) What have you noticed with your answers? 2.) Can you see any pattern? What are those patterns? While it is true that all answers are trinomials, let's take a closer look at each of the examples above. 1. (x + 3) (x + 2) = (x 2 + 2x + 3x + 6) → x 2 + 5x + 6 2\. (x -- 5) (x -- 3) = (x 2 -- 3x -- 5x + 15) → x 2 -- 8x + 15 3\. (x -- 2) ( x + 6) = (x 2 + 6x -- 2x -- 12) → x 2 + 4x -- 12 4\. (x + 6) (x + 1) = (x 2 + x + 6x + 6) → x 2 + 7x + 6 Photo Credits: \[Untitled Image for Multiplying Two Binomials Using the FOIL method\]. Mesacc. http://www.mesacc.edu/\~scotz47781/mat120/notes/polynomials/foil\_met hod/foil\_method.html INTRODUCTION (Explore) Process Questions INTERACTION (Firm-up & Deepen) cbp2024 \| G8SHMATH 39 Let us use algebra tiles to further understand these connections. Problem Last Two Terms of the Binomial Middle Term of the Trinomial Answer Last Term of the Trinomial Answer 1\. (x + 3) (x + 2) 3 and 2 5x 6 2\. (x -- 5) (x -- 3) -- 5 and -- 3 -- 8x 15 3\. (x -- 2) ( x + 6) -- 2 and 6 4x -- 12 4\. (x + 6) (x + 1) 6 and 1 7x 6 Focus on the connection of the middle term to the last terms of the binomial. Look also at the connection of the last term of the trinomial with the last terms of the binomial. What do you observe? Example 1: Find the product of (x + 3) (x + 2). That is the same as getting the area of the rectangle. Example 2: Find the product of (x -- 5) (x -- 3). That is the same as getting the area of the rectangle. Algebra Tiles x 2 -- x 2 x -- x 1 --1 Area Based on number of Tiles 1 → x 2 5 → x 6 →1 Area is x 2 + 5x + 6. Area Computed algebraically (x + 3) (x + 2) (x 2 + 2x + 3x + 6) x 2 + 5x + 6 Area is x 2 + 5x + 6. Area Based on number of Tiles 1 → x 2 8 → -- x 15 →1 Area is x 2 -- 8x + 15. Area Computed algebraically (x -- 5) (x -- 3) (x 2 -- 3x -- 5x + 15) x 2 -- 8x + 15 Area is x 2 -- 8x + 15. cbp2024 \| G8SHMATH 40 cbp2024 \| G8SHMATH 41 Example 3: Find the product of (x -- 2) (x + 6). That is the same as getting the area of the rectangle. Were you able to see already the connection through the help of these algebra tiles? Noting these connections will help you factor trinomials of this form. Let us try to do the opposite operation. We should get back to the two binomial factors. 1. x 2 + 5x + 6 = (x \_\_\_) (x \_\_\_) Factor the first term of the of the trinomial. = (x \_\_\_) (x \_\_\_) Think of factors of 6 which will give a sum of 5 = (x + 3) (x + 2) \* Note that this can also be written as (x +2) (x+3) by Commutative Property of Multiplication 2\. x 2 -- 8x + 15 = (x \_\_\_) (x \_\_\_) Factor the first term of the of the trinomial. = (x \_\_\_) (x \_\_\_) Think of factors of 15 which will give a sum of -- 8 = (x -- 5) (x -- 3) \* Note that this can also be written as (x -- 3) (x -- 5) by Commutative Property of Multiplication Factors of 6 6 and 1 2 and 3 → The sum is 5. -6 and -1 -2 and -3 Factors of 15 15 and 1 5 and 3 -15 and -1 -5 and -3 → The sum is -- 8. Area Based on number of Tiles 1 → x 2 6 → x 2 → -- x 4x 6 →1 Area is x 2 + 4x -- 12. Area Computed algebraically (x -- 2) (x + 6) x 2 + 6x -- 2x -- 12 x 2 + 4x -- 12 Area is x 2 + 4x -- 12. Note: Will add up to zero (different signs) cbp2024 \| G8SHMATH 42 Note: x 2 + Bx + C = (x + a) (x + b) When B and C are both positive, it tells us that the number we are looking for are both positive. If B is negative and C is positive, it tells us that the two numbers we are looking for are both negative. When C is negative, it means that the two numbers have different signs. The sign of the middle term tells you the sign of the number which is more represented. 3\. x 2 + 4x -- 12 = (x \_\_\_) (x \_\_\_) Factor the first term of the of the trinomial. = (x \_\_\_) (x \_\_\_) Think of factors of -- 12 which will give a sum of 4 = (x + 6) (x -- 2) \* Note that this can also be written as (x -- 2) (x+6) by Commutative Property of Multiplication 4\. x 2 + 7x + 6 = (x \_\_\_) (x \_\_\_) Factor the first term of the of the trinomial. = (x \_\_\_) (x \_\_\_) Think of factors of 6 which will give a sum of 7 = (x + 6) (x + 1) \* Note that this can also be written as (x +1) (x+6) by Commutative Property of Multiplication Factor the following. 1. x 2 + 12x + 20 5. x 2 -- 2x -- 24 2\. x 2 + 9x + 18 6. x 2 -- 7x -- 18 3\. x 2 -- 7x + 10 7. x 2 + 29x -- 30 4\. x 2 -- 11x + 18 8. x 2 + 6x -- 16 Factoring Trinomials of the Form Ax 2 + Bx + C In the previous lesson, we used what we call as Trial and Error Method to be able to find two binomials, which are factors of trinomials in the Form x 2 + Bx + C. What is the difference of the trinomials of the form x 2 + Bx + C to trinomials of the form Ax 2 + Bx + C? Can we apply the pattern we discovered earlier in factoring trinomials of the form Ax 2 + Bx + C? Why? Why not? When the coefficient of the x 2 term is not 1, there are more possibilities to consider. Try This 13 cbp2024 \| G8SHMATH 43 Example 1: Factor 5x 2 + 21x + 4 1\. Write the factors of the first term on the left boxes 2\. Write the factors of the last term on the right boxes 3\. Look for factors in (1) and (2) such that the sum of their products is the middle term by a.) rearranging their positions in the grid b.) exchanging their signs c.) look for another pair of factors The sum did not match the middle term. Repeat Step 3. Let's interchange the factors of 4 The sum already matched the middle term. Example 2: Factor 7x 2 -- 33x -- 10 Example 3: Factor 10x 2 -- 43x -- 9 5x 4 x 1 5x 4x 9x Product Sum 5x 1 x 4 20x x 21x Product Sum The factors then are (5x + 1) (x + 4) \*Check by using FOIL Method 7x 2 x -- 5 -- 35x 2x --33x Product Sum The factors then are (7x + 2) (x -- 5) \*Check by using FOIL Method 5x -- 1 2x 9 45x -- 2x \- 2 43x Product Sum The sum did not match the middle term which is supposed to be -- 43x. Repeat Step 3. cbp2024 \| G8SHMATH 44 Let us interchange the signs of the factors of -9. The sum already matched the middle term. Factor the following. 1. 6x 2 -- 29x + 35 2\. 4x 2 -- xy -- 5y 2 3\. 9x 2 + 23x + 10 4\. 6x 2 + x -- 12 Now that you already know all the factoring techniques, how will you know which to apply if you are given different problems to solve? Factoring Polynomials Completely Here is a concept map that will guide you on when to apply the different factoring techniques. Factoring Polynomials Completely Apply Greatest Common Monomial Factoring (GCMF) Count the number of terms in the other factor to know which technique is applicable. 5x 1 2x --9 --45x 2x --43x Product Sum The factors then are (5x + 1) (2x -- 9) \*Check by using FOIL Method Try This 14 cbp2024 \| G8SHMATH 45 Example 1: Factor x 4 + 7x 3 + 12x 2 completely. Factor out first the GCMF if there is. x 2 (x 2 + 7x + 12) x 2 is the GCMF x 2 (x \_\_\_) (x \_\_\_) x 2 (x + 3) (x + 4) x 2 (x + 3) (x + 4) Answer Example 2: Factor 3q 3 -- 12q 2 + 12q completely. Factor out first the GCMF if there is. 3q (q 2 -- 4x + 4) 3q is the GCMF 3q (\_\_\_ \_\_\_)2 3q (q -- 2)2 Answer Sign at the middle is negative since the middle term of the trinomial is negative. Example 3: Factor 72m4 + 58m2 -- 40 completely. Factor out first the GCMF if there is. 2 (36m4 + 29m2 -- 20) 2 is the GCMF 2 (4m2 + 5) (9m2 -- 4) 2 (4m2 + 5) (3m + 2) (3m -- 2) Answer Example 4: Factor 40m4p -- 135mp 7 completely. Factor out first the GCMF if there is. 5mp (8m3 -- 27p 6 ) 5mp is the GCMF 5mp (2m -- 3p 2 ) (\_\_\_ + \_\_\_ + \_\_\_) 5mp (2m -- 3p 2 ) (4m2 + 6mp 2 + 9p 4 ) Answer. This has 3 terms. There are three possibilities under 3 Terms but this can be factored by Trial and Error 1 Think of factors of 12 which will give a sum of 7. These are 3 and 4. This has 3 terms. There are three possibilities under 3 Terms but this can be factored by PST. Square root of 4 Square root of p 2. This has 3 terms. There are three possibilities under 3 Terms but this can be factored by Trial and Error 2. Two terms and still factorable by Difference of 2 squares This has 2 terms but both terms are perfect cubes. This can be factored by Difference of 2 Cubes. Find the terms of the binomial by getting the cube roots of terms above. cbp2024 \| G8SHMATH 46 Example 5: Factor 12xyz -20xz + 6yz -- 10z completely. Factor out first the GCMF if there is. z (12xy -- 20x + 6y -- 10) z is the GCMF z \[(12xy -- 20x) + (6y -10)\] Factor by Grouping z \[ 4x (3y -- 5) + 2 (3y -- 5)\] Factor the common binomial z (3y -- 5) (4x + 2) Answer Factor the following polynomials completely. 1. 105n 3 + 175n 2 -- 75n -- 125 2\. 10y 2 -- 14y -- 12 3\. 4bc 2 + 20bc + 24b 4\. x 2y 2 -- 4x 2y 4 5\. 3m3 + 81n 9 6\. 100x 2 + 80x + 16 7\. 8x 3 + 16x 2 -- 18x -- 36 Let us go back to the Mental Math Problems given to you in the earlier sections of this module. What special product and factors did you apply to solve these problems without using paper and pencil? a\. 98 x 102 b. 34 x 34 c. 47 x 47 This has 4 terms which can only be factored by grouping. Find the terms of the binomial by getting the cube roots of terms above. 98 x 102 (100 -- 2) (100 + 2) 10 000 -- 4 9996 34 x 34 (30 + 4) (30 + 4) 900 + 2 (30) (4) + 16 1 156 47 x 47 (50 -- 3) (50 -- 3) 2500 -- 2 (50) (3) + 9 2 209 Try This 15 cbp2024 \| G8SHMATH 47 Applications of Special Products and Factors You learned that special products and factors can be used in solving some mathematical problems mentally and in a short period of time. Where else can special products and factors applicable? Try to solve this problem. A garden that is 4 meters wide and 6 meters long is to have a uniform border such that the area of the border is the same as the area of the garden. Find the width of the border. Hint: Area of New Rectangle = Twice the Area of Original Rectangle Photo Credits: Archibald, A. (n.d.). The Nature of Garden Giving. \[Photograph\]. Mother Earth News. https://www.motherearthnews.com/organic-gardening/urban- garden-share-zb0z10zarc Geometric problems like this, which involve unknown values, can easily be solved by using special products and factors. Let us represent the width of the border as x. Then, (2x + 6) (2x + 4) = 2 (24) By FOIL Method 4x 2 + 8x + 12x + 24 = 2 (24) 4x 2 + 20x + 24 = 2 (24) Collect all terms on one side of the equation. 4x 2 + 20x + 24 -- 2 (24) = 0 4x 2 + 20x -- 24 = 0 Simplify the equation by dividing all terms by the GCF. x 2 + 5x -- 6 = 0 Factor the Trinomial (x + 6) (x -- 1) = 0 Equate each Factor to zero then solve the equations. x + 6 = 0 x -- 1 = 0 x = -- 6 x = 1 Note: Area = Length x Width INTRODUCTION (Explore) INTERACTION (Firm-up & Deepen) We need to disregard the negative value since there are no negative distances. In this case, the width of the border is 1 meter. cbp2024 \| G8SHMATH 48 Example 1: A square painting is surrounded by a 3 -- centimeter -- wide frame. If the total area of the painting plus frame is 961 cm2 , find the dimensions of the painting. Solution: a) Understand the problem. b.) Write the equation. Area of square = s 2 961 = (x + 6)2 c.) Solve the equation. (x + 6)2 = 961 x 2 +12x + 36 = 961 Square the binomial x 2 + 12x + 36 -- 961 = 0 Collect all terms on one side of the equation x 2 + 12x -- 925 = 0 Simplify (x -- 25) (x + 37) = 0 Factor x -- 25 = 0 x + 37 = 0 x = 25 x = -- 37 d.) Answer what is asked in the problem. The length of the side cannot be negative, so we discard x = --37. Hence, the length of each side of the painting is 25 cm. Example 2: While Mang Rico is calculating the amount of fertilizer he needs for his plants in his square flower gardens, he observed that the side of one garden is 2 meters longer than the side of the other. If the total area of the two gardens is 244 square meters, what is the area of each garden? Solution: a) Understand the problem. b.) Write the equation. Areasmaller garden + Areabigger garden = Total Area (x)2 + (x + 2)2 = 244 c.) Solve the equation. (x)2 + (x + 2)2 = 244 Expand the binomial being squared x 2 + x 2 + 4x + 4 = 244 Combine like terms 2x 2 + 4x + 4 = 244 Collect all terms on one side of the equation by APE or by transposition 2x 2 + 4x + 4 -- 244 = 0 Combine like terms x x 3 cm x + 6 x + 6 Let: x = the length of each side of the painting x + 6 = the length of each side of the frame Let: x = side of the smaller garden x + 2 = side of the bigger garden x x + 2 cbp2024 \| G8SHMATH 49 2x 2 + 4x -- 240 = 0 Simplify the equation by dividing it with 2. x 2 + 2x -- 120 = 0 Factor the trinomial by Trial and Error. (x + 12) (x -- 10) = 0 Equate each factor to zero x + 12 = 0 x -- 10 = 0 Solve each equation x = --12 x = 10 The side of the smaller garden cannot be negative, so we discard x = -- 12. Hence, the side of the smaller garden then is 10 ft. d.) Answer what is asked in the problem. side of the smaller garden (x) = 10 feet side of the bigger garden (x + 2) = 12 feet Example 3: A garden is rectangular with a width of 50 feet and a length of 40 feet. A walk of uniform width that surrounds it has an area of 576 square feet. Find the width of the walk. Solution: a) Understand the problem. b.) Write the equation. Total Area = Area of the garden + Area of the walk (2x + 50) (2x + 40) = (50) (40) + 576 c.) Solve the equation. (2x + 50) (2x + 40) = (50) (40) + 576 Apply FOIL Method 4x 2 + 80x + 100x + 2000 = 2000 + 576 Combine like terms 4x 2 + 180x + 2000 = 2576 Collect all terms on one side of the equation by APE or by transposition 4x 2 + 180x + 2000 -- 2576 = 0 Combine like terms 4x 2 + 180x -- 576 = 0 Simplify the equation by dividing both sides by 4 x 2 + 45x -- 144 = 0 Factor by Trial and Error 1 (x + 48) (x -- 3) = 0 Equate each factor to zero. x + 48 = 0 x -- 3 = 0 Solve each equation x = -- 48 x = 3 d.) Answer what is asked in the problem. The width of the walk cannot be negative, so we discard x = -- 48. Hence, the width of the walk is 3 feet. Let: 2x + 50 = length of garden including the walk 2x + 40 = width of garden including the walk cbp2024 \| G8SHMATH 50 Example 4: Archeologists discover a certain hydraulic concrete block with a volume of 330 cubic yards. Its length is eleven less than thirteen times the height. Its width is fifteen less than thirteen times its height. What are the dimensions of the block? Solution: a) Understand the problem. b\. Write the equation. Volume = (length) (width) (height) or (length) (width) (height) = volume c\. Solve the equation. (13x -- 11) (13x -- 15) (x) = 330 Apply FOIL Method in the 2 binomials (169x 2 -- 195x -- 143x + 165) (x) = 330 Combine like terms (169x 2 -- 338x + 165) (x) = 330 Multiply the trinomial with x. Use Distributive Property of Multiplication 169x 3 -- 338x 2 + 165x = 330 Collect all terms on one side of the equation by applying Addition Property of Equation commonly known as transposition. 169x 3 -- 338x 2 + 165x -- 330 = 0 There are four terms. This can be factored by Grouping Method (169x 3 -- 338x 2 ) + (165x -- 330) = 0 Factor the GCMF on both groups. 169x 2 (x -- 2) + 165 (x -- 2) = 0 Factor the common binomial (x -- 2) (169x 2 + 165) = 0 Equate each factor to zero. x -- 2 = 0 169x 2 + 165 = 0 Solve each equation x = 2 169x 2 = -- 165 Divide both sides of the equation by 169 169 2 169 = − 165 169 x 2 = − 165 169 Get the square root of both sides. 2= − 165 169 x = − 165 169 If we continue to solve the second equation, it will lead us to an imaginary number. This cannot be the height of the block. Therefore, the measure of the height of the block is 2 yards. d. Answer what is asked in the problem. Height (x) = 2 yards Length (13x -- 11) = 15 yards Width (13x -- 15) = 11 yards 13x -- 11 x + 6 x The result of the second equation is imaginary number. cbp2024 \| G8SHMATH 51 Solve the following problems. Show your solution. 1. Andy wants to buy a piece of land owned by his friend. His friend told him that the length of his land is 5 meters more than twice its width. The area of the land is 900 square meters. What are the dimensions of the land Andy wants to buy? 2\. Allen is buying glass for two square tabletops. One tabletop is one foot wider than the other. If he needs 41 square feet of glass, what were the dimensions of each tabletop? 3\. The local park measures 60 m by 80 m. Part of the park is torn up to install a sidewalk of uniform width about it, reducing the area of the park itself by 544 sq. m. How wide is the sidewalk? summatIve assessment 2 Let us now check your understanding of the essential concepts that you have learned. Do your best to solve the problems and get the correct answers. Write your answers first in this module. Double- check your answers before transferring them in the Quiz Assignment of the Google Classroom titled "Summative Assessment 2." A. Multiple Choice (2 points each) 1\. If 2x is one of the factors of 50x 3 -- 98x, what are the other factors? a\. (5x -- 7) (5x -- 7) c. (5x -- 49) (5x + 1) b\. (5x + 7) (5x -- 7) d. (25x -- 1) (x + 49) 2\. What are the factors of m2 -- 13m + 42? a\. (m -- 14) (m + 3) c. (m -- 21) (m -- 2) b\. (m -- 7) (m -- 6) d. (m -- 7) (m + 6) 3\. Find the dimensions of the rectangle whose area is represented as 27p 3 + 343q 6. a. (3p + 7q 2 ) and (9p 2 -- 42pq 2 -- 49q 4 ) c. (3p + 7q 2 ) and (9p 2 -- 42pq 2 + 49q 4 ) b\. (3p + 7q 2 ) and (9p 2 -- 21pq 2 -- 49q 4 ) d. (3p + 7q 2 ) and (9p 2 -- 21pq 2 + 49q 4 ) 4\. Find the prime factors of 5x 4 -- 20x 2y + 20y 2. a. (5x 2 -- 5y) (x 2 -- 4y) c. 5 (x 2 -- 2y) (x 2 -- 2y) b\. (x 2 -- y) (5x 2 -- 4y) d. 5 (x 2 -- y) (x 2 -- 4y) 5\. A garden is rectangular with a width of 80 feet and a length of 120 feet. A walk of uniform width that surrounds it has an area of 1,164 square feet. Which of the following equations can be used to find the width of the walk, which is represented as x? a\. (80 -- x) (120 -- x) = (80) (120) -- 1 164 c. (80 -- x) (120 -- x) = 1 164 b\. (80 -- 2x) (120 -- 2x) = (80) (120) -- 1 164 d. (80 -- 2x) (120 -- 2x) = 1 164 Try This 16 INTEGRATION (Transfer) cbp2024 \| G8SHMATH 52 6\. Elma needs table cloth for the two square tables she has at home. One table is 9 inches shorter than the other. Represent the total area of table cloth she will need. a. 2x 2 -- 18x + 81 c. x 2 -- 18x + 81 b\. 2x 2 -- 9x + 81 d. x 2 -- 9x + 81 7\. A cardboard measuring 12 inches by 10 inches is used to make a box by cutting a square from each corner and folding up the sides. What will be the length and width of the box if x represents the side of the square removed from each corner? a\. Length: 12 -- x, Width: 10 -- x c. Length: x -- 12 , Width: x -- 10 b\. Length: 12 -- 2x, Width: 10 -- 2x d. Length: 2x -- 12 , Width: 2x -- 10 8\. Factor completely: y 4 -- 7y 3 -- 18y 2 a\. y 2 (y -- 9) (y -- 2) c. y 2 (y + 9) (y -- 2) b\. y 2 (y -- 9) (y + 2) d. (y 2 -- 9) (y 2 + 2) 9\. Factor completely: 16x 4y 2 -- y 6 a\. y 2 (4x 2 -- 1) (4x 2 -- 1) c. y 2 (4x 2 + 1) (2x + 1) (2x -- 1) b\. y 2 (4x 2 + 1) (4x 2 -- 1) d. y 2 (2x + 1) (2x + 1) (2x -- 1) (2x -- 1) 10\. A vulcanizing shop with a square shape is affected by road widening. Its length has to increase by 2 meters, and its width has to decrease by 2 meters. What will be the new area of the shop? a\. x 2 -- 4 c. x 2 + 2x + 4 b\. 4 -- x 2 d. x 2 -- 2x + 4 11\. Factor completely: 40x 5 + 36x 4 -- 36x 3 a\. 4x 3 (5x -- 3) (4x + 6) c. 4x 3 (5x -- 3) (2x + 3) b\. 2x 3 (10x -- 6) (2x + 3) d. 4x 3 (2x -- 3) (5x + 3) 12\. What are the prime factors of 30x 3 + 15x 2 -- 18x -- 9? a\. (5x 2 -- 3) (6x + 1) c. (5x 2 -- 3) (6x + 3) b\. 3 (5x 2 + 3) (2x -- 1) d. 3 (5x 2 -- 3) (2x + 1) 13\. A portion from a square cardboard is removed. How large is the area remaining in the cardboard? a\. 25x 2 -- 6x -- 5 b\. 25x 2 -- 6x -- 7 c\. 25x 2 -- 14x -- 5 d\. 25x 2 -- 14x -- 7 14\. Write 4x2 -- 17x + 4 as product of its factors. a. (2x -- 2) (2x -- 2) c. 4 (x + 1) (x + 4) b\. (2x + 2) (2x -- 2) d. (x -- 4) (4x -- 1) 15\. The area of a rectangle is 3y 2 -- 16y -- 12. Find the dimensions. a. (3y + 2) and (y -- 6) c. (3y -- 12) and (y + 1) b\. (3y + 6) and (y -- 2) d. (3y -- 4) and (y + 3) 5x -- 1 5x -- 1 3 -- 2x 2 cbp2024 \| G8SHMATH 53 B. Problem Solving (10 points each) Solve the following problems on a short bond paper. Follow the steps in solving word problems, as illustrated in the examples given. The rubric below will be used in checking your work. Rubric: 2 points illustration with labels 2 points representation of the unknown/s 1 points equation 4 points solution 1 point answer with unit 1\. A playground has a length that is twenty-four meters shorter than thrice its width. The area of the playground is 720 square meters. What are the dimensions of the playground? 2\. Amanda wants her two square rooms to be tiled. One room is 8 feet wider than the other. The tiler said that she will be needing 832 pieces of 1-foot by 1-foot tiles. What are the dimensions of the two rooms? 3\. A mirror measuring 48 in by 36 in has to be installed in Bernie's room. Part of it has to be covered by wood, which will serve as its frame to keep it intact with the wall reducing the area of the mirror by 320 in 2. How wide is the frame? Now that you have a deeper understanding of the topic, you are ready to do the next task. This activity will give you insights into accomplishing your mini task 1. larGest tanK A sheet of metal 12 feet by 10 feet is to be used to make an open water tank. Squares of equal sides x are cut out of each corner then the sides are folded to make the box. Find the value of x that makes the volume maximum. a\. Represent the dimensions: Height → x Width → \_\_\_\_\_\_\_ Length → \_\_\_\_\_\_\_ b\. Represent the volume: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ x x Photo Credits: Garden Planter Galvanized Water Trough Raised Bed Garden Feature 6ft \[Photograph\]. (n.d.). eBay. https://www.ebay.co.uk/itm/184871005408?mkevt=1&mkcid=1&mkrid=710-53481- 19255-0&campid=5338722076&toolid=10001 cbp2024 \| G8SHMATH 54 c\. Complete the table below. Height (x) Width (\_\_\_\_\_\_\_) Length (\_\_\_\_\_\_\_) Volume (\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_) 1 1.5 2 2.5 3 3.5 4 d\. What dimensions of the tank can contain the most amount of water? After learning special products and factoring and their varied application to real-life situations, it is now time to showcase your learning in this module. Choose 1 from the three differentiated tasks. mInI tasK 1: paCKaGInG BOxes Instruction: The ABC packaging company is in search for the most appropriate packaging box for a new product that they will introduce in the market. You are a member of the design department of ABC Packaging Company. Your company is tapped to create the most appropriate packaging box using a provided rectangular shaped cardboard with a specified area. You need to present the different boxes you can make from the cardboard, and the corresponding volume of the box then choose the best size that can accommodate your product. You may put these on a table. You are to present your work to the Chief Executive Officer of the company and head of the ABC Packaging Department. Your work is evaluated according to the following: presentation of the lay-out, accuracy of computations, explanation of the computations, and appropriateness of the box to the product. Differentiated Task 1: Area of the cardboard to be used is 600 cm2 Differentiated Task 2: Area of the cardboard to be used is 500 cm2 Differentiated Task 3: Area of the cardboard to be used is 480 cm2 Criteria Excellent (4) Proficient (3) Progressing (2) Beginning (1) Presentation of the Lay-out (x2) Presentation of the lay- out is detailed and clear. All parts are clearly and accurately labeled Presentation of lay- out is clear. All parts are clearly labeled. Presentation of lay- out is a little difficult to understand but includes critical components. Presentation of the lay-out is difficult to understand and is missing several components. Accuracy of the Computations (x2) The computations done are accurate and show a deep understanding of the concepts of special products and factoring. There is an explanation for every computation made. The computations done are accurate and show proper use of the concepts of special products and factoring. The computations done are erroneous and show some use of the concepts of special products and factoring. The computations done are erroneous and do not show wise use of the concepts of special products and factoring. Appropriateness of the Box to the Product The product fits perfectly in the box. There is little space in the box that is not accommodated by the product. The product needs to be compressed to fit in the box. The product does not accommodate reasonable space. The box is too big or too small for the intended product. cbp2024 \| G8SHMATH 55 In -- Out Of the BOx revIsIted Now that you have completed doing the various activities in the module, on the space "Out of the box," you may revise your previous answer to the essential question: How can unknown quantities in geometric problems be solved? I can now say that\... VI\. POST-ASSESSMENT 1 It is time to evaluate your learning. In each item, choose the letter of the option that you think correctly answers the question. DO NOT leave any item unanswered. Do your best to solve the problems and get the correct answers. Write your answers first in this module. Double-check your answers before transferring them in the Quiz Assignment of the Google Classroom titled "POST -- ASSESSMENT 1." If you do well, it indicates that you are ready to move on to the next module. 1. What is the greatest common monomial factor of 4n 4 -- 32n 3 + 64n 2? a\. 2n c. 4n 2 b\. 2n 2 d. 4n 4 2\. Find the product of (3x+ 7y) (3x -- 7y). a. 9x 2 -- 42xy + 49y 2 c. 9x 2 -- 49y 2 b\. 9x 2 + 42xy + 49y 2 d. 9x 2 + 49y 2 3\. The side of a square is represented as (11x 3 -- y). What is the area of the square? a\. 121x 6 + y 2 c. 121x 6 -- 11x 3y + y 2 b\. 121x 6 -- y 2 d. 121x 6 -- 22x 3y + y 2 IN THE BOX I think\... OUT OF THE BOX cbp2024 \| G8SHMATH 56 4\. Find the product of (3m -- 1) (9m2 + 3m + 1). a. 27m3 + 1 c. 27m3 + 18m2 -- 6m + 1 b\. 27m3 -- 1 d. 27m3 -- 18m2 + 6m -- 1 5\. Factor x 2 + 4x -- 21. a. (x + 3) (x -- 7) c. (x + 3) (x + 7) b\. (x + 7) (x -- 3) d. (x -- 3) (x -- 7) 6\. What is the factored form of 6a 2 -- 17ab -- 14b 2 a\. (2a -- 2b) (3a -- 7b) c. (3a + 2b) (2a -- 7b) b\. (2a -- 2b) (3a + 7b) d. (3a -- 2b) (2a -- 7b) 7\. When factored completely, 56x 4 -- 7xy 6 is equal to \_\_\_\_\_. a. 7x (8x 3 -- y 6 ) c. 7x (2x -- y 2 ) (4x 2 + 4xy 2 + y 4 ) b\. 7x (2x -- y 2 ) (4x 2 + 2xy 2 + y 4 ) d. -- 7x (2x + y 2 ) (4x 2 -- 2xy 2 + y 4 ) 8\. The area of a square is equal to (25m2 + 70m + 49) square units. Find the binomial that represents the length of one side of the square. a. (5m + 7) units c. (25m + 49) units b\. (7m + 5) units d. (49m + 25) units 9\. The area of a rectangle is 3y 2 -- 16y -- 12. If the width is y -- 6, what is its length? a\. 3y + 2 c. 2y + 3 b\. 3y -- 2 d. 2y -- 3 10\. What are the prime factors of the 14ab -- 56a + 7b 2 -- 28b? a\. 7 (a -- b) (2b + 4) c. 7 (2a -- b) (b + 4) b\. 7 (a + b) (2b -- 4) d. 7 (2a + b) (b -- 4) 11\. Find the dimensions of the rectangle whose area is represented as 125p 3 + 27q 6. a. (5p -- 3q 2 ) and (25p 2 + 30pq 2 + 9q 4 ) c. (5p + 3q 2 ) and (25p 2 -- 30pq 2 + 9q 4 ) b\. (5p -- 3q 2 ) and (25p 2 + 15pq 2 + 9q 4 ) d. (5p + 3q 2 ) and (25p 2 -- 15pq 2 + 9q 4 ) 12\. Find the prime factors of 4x 3 + 4x 2y-- xy 2 -- y 3. a. (2x + y) (2x -- y) (x + y) c. (2x -- y)2 (x + y) b\. (2x -- y) (2x -- y) (x + y) d. (4x 2 -- y 2 ) (x + y) 13\. Which of the following is NOT true? a\. x 2 -- 25 = (x + 5) (x -- 5) c. x 2 -- 10x + 25 = (x -- 5)2 b\. x 2 + 25 = (x + 5) (x + 5) d. x 2 + 10x + 25 = (x + 5)2 14\. A garden is rectangular with a width of 60 feet and a length of 50 feet. A walk of uniform width that surrounds it has an area of 816 square feet. Which of the following equations can be used to find the width of the walk, which is represented as x? a\. (60 -- 2x) (50 -- 2x) + 816 = (60) (50) c. (60 -- 2x) (50 -- 2x) = 816 b\. (60 -- x) (50 -- x) + 816 = (60) (50) d. (60 -- x) (50 -- x) = 816 cbp2024 \| G8SHMATH 57 15\. The length of a rectangular piece of cardboard is twice its width. A 2-inch square is cut out from each corner, and the sides are turned up to make a box. The volume of the box formed is 192 in 3. How would you represent the length and width of the box? a\. Length: 2x -- 2, Width: x -- 2 b\. Length: 2x -- 4, Width: x -- 4 c\. Length: 2x + 2, Width: x + 2 d\. Length: 2x + 4, Width: x + 4 16\. How long and wide originally was the cardboard in number 13? a\. Length: 20 inches, Width: 10 inches b\. Length: 24 inches, Width: 12 inches c\. Length: 26 inches, Width: 13 inches d\. Length: 30 inches, Width: 15 inches 17\. A portion from a square yard is allotted for the building of a fountain. How large is the area remaining in the garden? a\. 16x 2 -- 6x -- 14 b\. 16x 2 -- 14x -- 14 c\. 16x 2 -- 18x -- 14 d\. 16x 2 + 2x -- 14 18\. A square of side x is to be made into a rectangle by increasing its length by 6 cm and decreasing its width by 6 cm. Which statement is true? a\. The area of the square is greater than the area of the rectangle. b. The area of the rectangle is greater than the area of the square. c. The area of the square is equal to the area of the rectangle. d. The relationship cannot be determined from the given information. 19. An antique shop makes rectangular boxes for its products. If the volume of each box is given by V = 3x 3 + 13x 2 -- 10x, which are the possible dimensions of the carton? a\. x (x -- 2) (3x + 5) c. x (x + 2) (3x -- 5) b\. x (3x -- 2) (x + 5) d. x (3x + 2) (x -- 5) 20\. Based on your answer in the previous number, which are the dimensions of the carton if x = 4? a\. 4 by 9 by 10 c. 4 by 9 by 12 b\. 4 by 10 by 12 d. 4 by 10 by 15 4x -- 1 4x -- 1 3 -- 2x 5 cbp2024 \| G8SHMATH 58 VII\. SELF-ASSESSMENT What activities did you enjoy in this module? Why? What do you suggest for the improvement of this module? Please do not hesitate to send your answers to the facebook messenger of your teacher, for this is NOT GRADED. Your teacher is surely delighted and thankful to know your responses VIII\. REFERENCES Books Berondo, Maria Sofie M. et al. (2017). C and J Mathematics: Creative and Interactive. Magallanes Publishing House, Sampaloc, Manila Ferrera, Shirley et al. (2017). Empowering through Mathematics 8. Ephesians Publishing Inc., Quezon City Grade 8 Learning Module -- Mathematics, Mathematics Teachers Guide, Fund for Assistance to Private Education, 2017. Nivera, Gladys C. (2018). Grade 8 Mathematics: Patterns and Practicalities. Salesiana Books by Don Bosco Press, Inc., Makati City Electronic Resources Archibald, A. (n.d.). The Nature of Garden Giving. \[Photograph\]. Mother Earth News. https://www.motherearthnews.com/organic-gardening/urban-garden-share-zb0z10zarc Berendsohn, R. (2016). Why are There Lines on My Newly Painted Wall? \[Photograph\]. Popular Mechanics. https://www.popularmechanics.com/home/interior-projects/a24336/faint-lines-newly- painted-wall/ Castiglione, A. (n.d.). Square Pool. \[photograph\]. Pinterest. https://www.pinterest.ph/pin/573434965026732662/ Clipart Library. (n.d.). Collection of Pictures of Boxes. \[Clip art\]. http://clipart-library.com/pictures-of- boxes.html eBay. (n.d.). Garden Planter Galvanized Water Trough Raised Bed Garden Feature 6ft. \[Photograph\]. eBay. https://www.ebay.co.uk/itm/184871005408?mkevt=1&mkcid=1&mkrid=710-53481-19255- 0&campid=5338722076&toolid=10001 Homedepot. (n.d). Merola Tile Artisan Damero Azul. \[Photograph\]. Pinterest. https://www.pinterest.ph/pin/183732859786084765/ Jenn. (2020). What is the FOIL Method? \[Online Image\]. Calcworkshop. https://calcworkshop.com/polynomials/foil-method/ Metrobank Foundation. (2020). \[Mathletes from Jose Rizal Elementary School in Pasay City during the first day of 2020 MMC\]. \[Photograph\]. Metrobank Found. https://twitter.com/hashtag/2020mmc Roberts, D. (n.d.). Factoring Perfect Square Trinomials. \[Online Image\]. MathBitsNotebook. https://mathbitsnotebook.com/Algebra1/Factoring/FCPerfSqTri.html cbp2024 \| G8SHMATH 59 Key Answers Pre -- Assessment 1 1\. D 6. C 11. B 16. C 2\. D 7. D 12. B 17. A 3\. D 8. B 13. C 18. A 4\. A 9. A 14. A 19. D 5\. C 10. A 15. C 20. D Try This 1 Polynomial Common Monomial Factor Remaining factor Factored Form Ex. 18x 3y 4 -- 12x 2y 5 6x 2y 4 3x -- 2y 6x 2y 4 (3x -- 2y) 1\. 5x + 10 5 x + 2 5 (x + 2) 2\. -- 8x -- 20 -- 4 2x + 5 -- 4 (2x + 5) 3\. 3x 5 -- 12x 4 -- 9x 3 3x 3 x 2 -- 4x -- 3 3x 3 (x 2 -- 4x -- 3) 4\. y 4 + 3y 2 y 2 y 2 + 3 y 2 (y 2 + 3) 5\. 2m5n 4 + mn 4 mn 4 2m4 + 1 mn 4 (2m4 + 1) 6\. 17x 2 + 34x + 51 17 x 2 + 2x + 3 17 (x 2 + 2x + 3) 7\. 18m2n 4 -- 12m2n 3 + 24m2n 2 6m2n 2 3n 2 -- 2n + 4 6m2n 2 (3n 2 -- 2n + 4) Note that in number 2, the negative sign is common to both terms. You can also factor this. Try This 2 1\. 5m3 + 5m2 -- m -- 1 (5m3 + 5m2 ) + (-- m -- 1) 5m2 (m + 1) -- 1( m + 1) (m + 1) (5m2 -- 1) 6\. 8m2 -- 4mn -- 6m + 3n (8m2 -- 4mn) + (-- 6m + 3n) 4m(2m -- n) -- 3 (2m -- n) (2m -- n) (4m -- 3) 2\. 3x 2 -- 7xy + 3x -- 7y (3x 2 -- 7xy) + (3x -- 7y) x (3x -- 7y) + 1(3x -- 7y) (3x -- 7y) (x + 1) 7\. 3p 2q + 6pq -- 5p -- 10 (3p 2q + 6pq) + (-- 5p -- 10) 3pq (p + 2) -- 5 (p + 2) (p + 2) (3pq -- 5) 3\. 8xy -- zw + 8xw -- yz (8xy + 8xw) + (-- zw -- yz) 8x (y + w) -- z (w + y) \*(y + w) is the same as (w + y) because of commutative property of addition. Meaning, 3 + 2 is the same as 2 + 3 (w + y) (8x -- z) 8\. (y 2 -- 2) (y -- 4) + (y + 3) (y -- 4) (y -- 4) \[(y 2 -- 2) + (y + 3)\] Remove the grouping signs inside (y -- 4) (y 2 -- 2 + y + 3) (y -- 4) (y 2 + y + 1) 4\. 6mx -- y + 2my -- 3x (6mx + 2my) + (-- y -- 3x) 2m (3x + y) --1(y + 3x) (3x + y) (2m -- 1) 9\. (5b + 6) (a -- 1) -- (a -- 1) (2b + 7) (a -- 1) \[(5b + 6) -- (2b + 7)\] Remove the grouping signs inside by using distributive property of multiplication. (a -- 1) (5b + 6 -- 2b -- 7) (a -- 1) (3b -- 1) cbp2024 \| G8SHMATH 60 5\. 10b 3 + 25b -- 4b 2 -- 10 (10b 3 + 25b) + (-- 4b 2 -- 10) 5b (2b 2 + 5) -- 2 (2b 2 + 5) (2b 2 + 5) (5b -- 2) 10\. (2x -- 5) (x + 2) -- (x + 2) (x -- 1) (x + 2) \[(2x -- 5) -- (x -- 1)\] Remove the grouping signs inside by using distributive property of multiplication. (x -- 2) (2x -- 5 -- x + 1) (x -- 2) (x -- 4) For numbers 1 - 7, there are several ways of forming binomial groups, which will lead you to different solutions but the same answer. One way is presented for each of the said items. For numbers 8 -- 10, there is only one solution. Try This 3 1\. 25d 2 -- 49 2\. 16m2 -- 169n 2 3\. 121 -- 81c 2 4\. x 2 -- 16y 4 5\. 4d 6 -- 36f2 Try This 4 a\. 9x 2 -- 16 The first term is a square. The second term is a square The operation involved is subtraction. This is a difference of two squares. b. -- 4 + 9x 6 -- 4 + 9x 6 can be written as 9x 6 -- 4 Both terms are squares. The operation involved is subtraction. This is a difference of two squares. c. x 4 -- 8 8 is not a perfect square. This is NOT a difference of two squares. d. x 3 -- 25 x 3 is not a perfect square since the exponent of the variable is an odd number. Hence, x 3 -- 25 is NOT a difference of two squares Try This 5 1\. (10x + 9) (10x -- 9) 2\. (x 3y 4 + 11z 2 ) ( x 3y 4 -- 11z 2 ) 3\. (x + 7y 5 ) (x -- 7y 5 ) 4\. (4a 3 + 5b) (4a 3 -- 5b) 5\. (ab 2 + 9) (ab 2 -- 9) Try This 6 1\. 5x (x + 1) (x -- 1) 2\. xy (6xy 2 + z 3 ) (6xy 2 -- z 3 ) 3\. \[(3m -- 4) + 8\] \[(3m -- 4) -- 8\] simplify (3m + 4) (3m -- 12) cbp2024 \| G8SHMATH 61