Mathematics 10 Quarter 1 Module 9: Proving the Remainder and Factor Theorems PDF

10 Mathematics Quarter 1 – Module 9: Proving the Remainder and Factor Theorems CO_Q1_Mathematics 10_ Module 9 Mathematics – Grade 10 Alternative Delivery Mode Quarter 1 – Module 9: Proving the Remainder and Factor Theorems First Edition, 2019 Repu...

10 Mathematics Quarter 1 – Module 9: Proving the Remainder and Factor Theorems CO_Q1_Mathematics 10_ Module 9 Mathematics – Grade 10 Alternative Delivery Mode Quarter 1 – Module 9: Proving the Remainder and Factor Theorems First Edition, 2019 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed material (i.e. songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials form their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio Development Team of the Module Writer’s Name: Josefa H. Pugong Editor’s Name: Melchor B. Ticag; Jim D. Alberto Reviewer’s Name: Bryan A. Hidalgo; Heather G. Banagui; Laila B. Kiw-isen; Selalyn Maguilao Management Team:May B. Eclar Benedicta B. Gamatero Carmel F. Meris Marciana M. Aydinan Ethielyn E. Taqued Edgar H. Madlaing Lydia I. Belingon Printed in the Philippines by: Department of Education – Cordillera Administrative Region Office Address: Wangal, La Trinidad, Benguet Telefax: (074) 422-4074 E-mail Address: car@deped.gov.ph  10 Mathematics Quarter 1 – Module 9: Proving the Remainder and Factor Theorems Introductory Message This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson. Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you. Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these. In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning. Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task. If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Thank you.  What I need to Know This module was designed and written with you in mind. It is here to help you understand better on how to prove the remainder and factor theorems. The scope of this module permits it to be used in many different learning situations. The lessons are done to follow the standard sequence of the course. LEARNING OBJECTIVES: After going through this module, the learner should be able to: 1. prove the remainder and factor theorems, 2. find the remainder using synthetic division or the remainder theorem, and 3. solve word problems using the remainder and factor theorem 1 CO_Q1_Mathematics 10_ Module 9  What I know DIRECTION: Let us determine how much you already know about the remainder theorem and factor theorem. Read and understand each item, then choose the letter of your answer and write it on your answer sheet. 1) Which of the following binomials is a factor of the P(x) = x3 – 7x + 5? a) x – 1 b) x + 1 c) x + 2 d) none of these 2) Is the first polynomial a factor of the second polynomial, x – 1; x2 + 2x + 5? a) yes b) no c) uncertain d) invalid 3) Which is the missing factor in the equation x2 – 4 = (x – 2) (_____)? a) x -2 b) x+ 2 c) x + 4 d) x – 4 4) Which of the following is NOT a factor of x3+ 5x2 – x – 5? a) x + 1 b) x – 1 c) x – 5 d) x + 5 5) What are the factors of x2 – 2x – 24? a) (x + 4) (x – 6) c) (x -12) (x+ 1) b) (x -8) (x + 3) d) (x + 12) (x – 12) 6) What is the remainder when x3 – 4x2 + x + 8 is divided by x – 2? a) 1 b) 2 c) 3 d) 4 7) When P(x) is divided by x – r and the remainder is equal to zero, it means _____. a) x – r is a factor of P(x) c) P(x) is the only factor of x – r b) P(x) is a factor of x – r d) x – r is the only factor of P(x) 1 8) Evaluate the polynomial x3 + x2 + x + 3 if the value of x = -2. a) 3 b) -3 c) 17 d) -17 9) Which polynomial gives a remainder of 0 when divided by 3x – 2? a) 12x2 + 15x – 18 c) 12x2 + 19x – 18 b) 12x2 + 18x + 7 d) 12x2 + 8x + 7 10) Which of the following is a factor of 2x2 – 5x + 3? a) x – 3 b) 2x + 3 c) x – 1 d) 3x + 2 11) What is the remainder if x2 – 7x – 4 is divided by x – 2? a) -14 b) 5 c) 6 d) 7 12) Which statement is true? a) The quotient multiplied by the dividend plus the remainder equals the divisor. b) If x2 + 5x + 7 is divided by x + 2, the remainder is 1. c) If the remainder is 0, then the dividend is a factor of the divisor. d) The remainder is a factor of the dividend if the quotient is 0. 2 CO_Q1_Mathematics 10_ Module 9 13) Which is the remainder if 2x3 – 7x2 – 19x + 20 is divided by x – 5? a) -1 b) 0 c) 1 d) 2 14) x – 3 is a factor of x4 – 3x3 – x +3. a) false b) true c) uncertain d) invalid 15) Find another factor of x3 – 7x2 + 4x – 28 if x – 7 is a factor. a) x2 + x + 4 b) x2 – 2x – 4 c) x2 – x – 4 d) x2 + 4 Lesson The Remainder Theorem 1 and the Factor Theorem What’s In In the earlier lesson, you have learned how to divide polynomials using long division or by synthetic division. Read and understand the discussion below, then investigate. 1. Divide the polynomial function f(x) = x3 – 2x2 + x – 3 by x – 2 using synthetic division. Solution: 2 1 -2 1 -3 2 0 2 1 0 1 -1 2. Since the divisor is x – 2, evaluate the 2 above function at x = 2. Solution: f(x) = x3 – 2x2 + x – 3 f(2) = 23 – 2(22) + 2 – 3 Substitute x by 2 f(2) = 8 – 8 + 2 – 3 Simplify f(2) = -1 Perform the operations Guide Question: What can you say about the results of the two separate solutions? Notice: When the function was divided by x – 2, the remainder is -1. When the function was evaluated at x = 2, the result is -1. This leads us to the Remainder Theorem. 3 CO_Q1_Mathematics 10_ Module 9  What’s New Definition REMAINDER THEOREM If a polynomial P(x) is divided by x – a, then the remainder is r = P(a). Example:If f(x) = x3 – 2x2 + x – 3 divided by x – 2, then the remainder is r = P(a). Based from the solutions above, the remainder r = -1, and f(2) = -1. Thus, r = f(2). When a polynomial is divided by x – a, if the remainder is zero, we say that x – a is a factor of the polynomial. Through the remainder theorem, we now know that the remainder is related to evaluation of the polynomial at the point x = a. We are then led to the factor theorem. Definition FACTOR THEOREM If P(a) = 0, then x – a is a factor of P(x). Conversely, if x – a is a factor of P(x), then P(a) = 0. What Is It You have already learned the difference between the Remainder Theorem and the Factor Theorem. Now, you will learn how to use these theorems to solve problems. Example 1. Find the remainder when x3 + 2x2 – 5x + 2 is divided by x + 3. Solution: We can find the remainder in two methods: by synthetic division or by remainder theorem. Using the Remainder Theorem: P(x) = x3 + 2x2 – 5x + 2 At x = - 3 P(-3) = (-3)3 + 2(-3)2 – 5(-3) + 2 Substitute x by -3 P(-3) = -27 + 2(9) + 15 + 2 Simplify P(-3) = -27 + 18 + 15 + 2 P(-3) = 8 Perform the operations Therefore, the remainder is 8. 4 CO_Q1_Mathematics 10_ Module 9  -3 1 2 -5 2 We can check the answer -3 3 6 using the Synthetic Division: 1 -1 -2 8 Example 2. Determine whether or not x + 2 is a factor of x3 – 2x2 – 5x + 6. Solution: By definition of Factor Theorem, x + 2 is a factor of x3 – 2x2 – 5x + 6 if and only if the remainder is zero. Again, we can use two methods in finding the remainder. Using the Remainder Theorem: P(x) = x3 – 2x2 – 5x + 6 At x = - 2 P(-2) = (-2)3 – 2(-2)2 – 5(-2) + 6 Substitute x by -2 P(-2) = -8 – 2(4) + 10 + 6 Simplify P(-2) = -8 – 8 + 10 + 6 P(-2) = 0 Perform the operations Since the remainder is zero, x + 2 is a factor of x3 – 2x2 – 5x + 6. Using the Synthetic Division: Note: It is not required to present -2 1 -2 -5 6 the solution in two methods. Use -2 8 -6 the method that you are most 1 -4 3 0 comfortable with. However, if you feel uncertain with your answer, you may use the two methods. Example 3. Show that x – 4 is a factor of P(x) = x3 – 6x2 + 5x + 12. Solution: Since x – 4 is a factor of P(x) = x3 – 6x2 + 5x + 12, the remainder must be zero. Using the Remainder Theorem: P(x) = x3 – 6x2 + 5x + 12 At x = 4 P(4) = (4)3 – 6(42) + 5(4) + 12 Substitute x by 4 P(4) = 64 – 6(16) + 20 + 12 Simplify P(4) = 64 – 96 + 20 + 12 P(4) = 0 Perform the indicated operations The remainder is 0. Example 4. Find k so that x + 5 is a factor of P(x) = x3 + 5x2 – kx – 20. Solution: Since x + 5 is a factor of P(x) = x3 + 5x2 – kx – 20, the remainder is zero. 5 CO_Q1_Mathematics 10_ Module 9  Using the Remainder Theorem: P(x) = x3 + 5x2 – kx – 20 At x = -5 P(-5) = (-5)3 + 5(-5)2 – k(-5) – 20 Substitute x by -5 0 = (-5)3 + 5(-5)2 – k(-5) – 20 Change P(-5) by 0 (remainder is 0) 0 = -125 + 125 + 5k – 20 Simplify 0 = 5k – 20 5k = 20 Solve for k k=4 Divide both sides by 5 So that x + 5 is a factor of P(x) = x3 + 5x2 – kx – 20, k = 4. What’s More Exercise 1. Answer what is asked. A. Use the Remainder Theorem or synthetic division to find each function value. 1 1. f(x) = x3 – 5x2 – 7x + 4 a) f(1) b) f(-2) c) f(2) 2. g(x) = 2x6 + 3x4 – x2 + 3 a) g(2) b) g(3) c) g(-1) 3. h(x) = 2x3 – 7x + 3 a) h(-3) b) h(5) c) h(-10) B. Solve. 4. Determine if x – 3 is a factor of 𝑃(𝑥) where P(x) = x4 – 3x3 – x + 3. 5. Determine if x – 1 is a factor of 𝑃(𝑥) where P(x) = x25 – 4. 6. Find k so that x – 2 is a factor of P(x) = x3 – kx2 – 4x + 20. Exercise 2. Answer what is asked. 1. Use the remainder theorem to find P(2) in P(x) = x4 + 4x3 – x2 – 16x – 4. 2. Prove that y – 3 is a factor of 3y3 – 7y2 – 20 using the remainder theorem. 3. Evaluate P(4) in P(y) = 3y3 – 7y2 – 20. 4. Use the factor theorem to determine whether x – 1 is a factor of P(x) = x2 + 2x + 5. 5. Use the remainder theorem to find the remainder R in (3x 2 + 5x3 – 8) ÷ (x – 4). 6 CO_Q1_Mathematics 10_ Module 9  What I have learned 1. How will you determine that x –r is a factor of P(x)? 2. In the remainder theorem, what will you substitute in the polynomial expression? 3. What are the two ways on how to find the remainder when P(x) is divided by x – r? What I can do A. Fill the blanks to complete the statement. Write your answer in the separate sheet of pad paper. 1. x – r is a factor of P(x) if and only if the remainder R when P(x) ÷ (x – r) is _____. 2. By the Remainder Theorem, R = 0 if and only if _____. 3. Thus, x – r is a factor of P(x) if and only if _____. B. Solve this problem correctly: The volume of a rectangular solid is (x3 + 3x2 + 2x) cubic cm, and its height is (x+1) cm. What is the area of the base? Assessment DIRECTION: Let us determine how much you have learned from this module. Read and understand each item, then choose the letter of your answer and write it on your answer sheet. 1. What is the remainder if x2 – 7x – 4 is divided by x – 2? a) 4 b) -14 c) 2 d) -2 2. Which of the following statements is true? a) The quotient multiplied by the dividend plus the remainder equals the divisor. b) If x2 + 5x + 7 is divided by x + 2, the remainder is not 1. c) If the remainder is 0, then the dividend is a factor of the divisor. d) The quotient is a factor of the dividend if the remainder is 0. 7 CO_Q1_Mathematics 10_ Module 9 3. Which is the remainder if 2x3 + 4x2 – x + 7 is divided by x – 2? a) 35 b) 36 c) 37 d) 38 4. Is x – 3 a factor of x4 – 3x3 – x +3? a) false b) true c) uncertain d) invalid 5. Find another factor of x3 – 7x2 + 4x – 28 if x – 7 is a factor. a) x2 + x +4 b) x2 – 2x – 4 c) x2 – x – 4 d) x2 + 4 6. Determine which of the following binomials is a factor of P(x) = x 3 -7x + 5. a) x – 1 b) x + 1 c) x + 2 d) none of these 7. Is the first polynomial a factor of the second polynomial, x – 1; x2 + 2x + 5? a) yes b) no c) uncertain d) invalid 8. Which is the missing factor in the equation x2 – 4 = (x – 2) (_____)? a) x – 2 b) x+ 2 c) x + 4 d) x – 4 9. Which of the following is a factor of x3+ 5x2 – x – 5? a) x + 1 b) x + 2 c) x – 5 d) none of these 10. What are the factors of x2 – 2x – 24? a) (x + 4) (x – 6) b) (x -8) (x + 3) c) (x -12) (x+ 1) d) (x + 12) (x – 12) 11. What is the remainder when x3 – 4x2 + x + 8 is divided by x – 2? a) 1 b) 2 c) 3 d) 4 12. When P(x) is divided by x – r and the remainder is equal to 0, it means _____. a) x – r is a factor of P(x) c. P(x) is a factor x – r b) P(x) is the only factor of x – r d) x- r is the only factor of P(x) 13. Evaluate the polynomial x3 + x2 + x + 3 at the given value of x = -2 a) 3 b) -3 c) 17 d) -17 14. Which polynomial gives a remainder of zero when divided by 3x – 2? a) 12x2 + 15x – 18 c) 12x2 + 19x – 18 b) 12x + 18x + 7 2 d) 12x2 + 8x + 7 15. Which of the following is a factor of 2x2 – 5x + 3? a) x – 3 b) 2x + 3 c) x – 1 d) 3x + 2 8 CO_Q1_Mathematics 10_ Module 9  Additional Activity DIRECTION. Perform each given task. Task A. Let P(x) = x3 – 64 a. Find P(4). b. Find the remainder when P(x) is divided by x – 4. Task B. Let P(x) = x3 – 64 a. Find P(-4). b. Find the remainder when P(x) is divided by (x + 4). Task C. Compare your answers. What do you observe? 9 CO_Q1_Mathematics 10_ Module 9 CO_Q1_Mathematics 10_ Module 9 10 What I Know 1. d 4. c 7. a 10. c 13. b 2. b 5. a 8. b 11. a 14. b 3. b 6. b 9. c 12. b 15. d What’s More Exercise 1 Exercise 2 (A) (B) 5 1) a = - 7 b = - 10 c=− 4) x – 3 is a factor 1) 8 3) Not 8 2) a = 175 b = 1695 c=7 5) x – 1 is not a factor 2) Not 4) 360 3) a = -30 b = 218 c = -1927 6) k = 5 3) 60 What I Can Do (A) 1) Zero 2) (x – a) is a factor of P(x) 3) the remainder of P(x) ÷ x – r is zero (B) x2 + 2x Assessment 1. b 4. b 7. b 10. a 13. b 2. d 5. d 8. b 11. b 14. c 3. c 6. d 9. a 12. a 15. c Additional Activity Task A Task B Task C a) 0 a) -128 The remainder and the value of the polynomial at r taken from x – r are equal. b) 0 b) -128 Answer Key References De Guzman, Santos, R. , De Guzman, T.,Ungriano, A, Yabut, E. Statistic. Manila, Philippines. Centro Escolar University Publishing House. 2005 Febre, Francisco Jr.. Introduction to Statistics. Quezon City, Philippines. Phoenix Publishing House, Inc. 1987 Manansala, T..Statistics. Manila, Philippines. Jimcy Publishing House. 2007 Mendenhall, W., Beaver, R., Beaver, B. Probability and Statistics. Thomson Learning Asia. 2006 Oronce, O., Mendoza, M. E-Math IV. Quezon City, Philippines. Rex Book Store, Inc. 2010 Young, Cynthia, Precalculus. Florida, USA. John Wiley & Sons, Inc. 2010 11 CO_Q1_Mathematics 10_ Module 9 For inquiries or feedback, please write or call: Department of Education - Bureau of Learning Resources (DepEd-BLR) Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600 Telefax: (632) 8634-1072; 8634-1054; 8631-4985 Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph: