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- 1. Mathematics Quarter 2 - Module 1 Illustrating Polynomial Functions Department of Education ● Republic of the Philippines 10
- 2. Mathematics- Grade 10 Alternative Delivery Mode Quarter 2 - Module 1: Illustrating Polynomial Functions First Edition, 2020 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 royalty. Borrowed materials (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 from 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 Development Team of the Module Author/s: Cristine Mae C. Baguio Reviewers: Rhodel A. Lamban, PhD Elbert R. Francisco, PhD Alicia P. Micayabas, PhD Illustrator and Layout Artist: Cristine Mae C. Baguio Regional Evaluator: Lourgen V. Maalam, MT- I Clarin National High School Management Team Chairperson: Arturo B. Bayocot, Ph. D, CESO III Regional Director Co-Chairpersons: Victor G. De Gracia Jr. Ph.D, CESO V Asst. Regional Director Randolph B. Tortola, PhD, CESO IV Schools Division Superintendent Shambaeh A. Usman, PhD Assistant Schools Division Superintendent Mala Epra B. Magnaong, Chief - CLMD Neil A. Improgo, Ph.D, EPS - LRMS Bienvenido U. Tagolimot, Jr., Ph.D., EPS-ADM Members Elbert R. Francisco, PhD, Chief - CID Rhodel A. Lamban, PhD, EPS - Mathematics Rejynne Mary L. Ruiz, PhD, LRMDS Manager Jeny B. Timbal, PDO II Shella O. Bolasco, Division Librarian II Printed in the Philippines by Department of Education – Division of Bukidnon Office Address: Fortich St. Sumpong, Malaybalay City Telephone: (088) 813-3634 E-mail Address: bukidnon@deped.gov.ph
- 3. Mathematics Quarter 2 - Module 1 ILLUSTRATING POLYNOMIAL FUNCTIONS This instructional material was collaboratively developed and reviewed by educators from public schools. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at bukidnon@deped.gov.ph. We value your feedback and recommendations. Department of Education-Division of Bukidnon ● Republic of the Philippines 10 i
- 4. Table of Contents PAGE COVER PAGE COPYRIGHT PAGE TITLE PAGE i TABLE OF CONTENTS ii WHAT THIS MODULE IS ABOUT iv Note to the Teacher/Facilitator Note to the Parents/Guardian Note to the Learner Module Icons WHAT I NEED TO KNOW 1 WHAT I KNOW (Pretest) 2 LESSON 1: Definition of Polynomial Function What I Need to Know 4 What I Know 5 What’s In 7 What’s New 7 What is it 8 What’s More 9 Guided/Controlled Practice Independent Practice What I Have Learned 10 What I Can Do 11 Assessment 12 Guided Assessment ii
- 5. Independent Assessment Additional Activities 14 LESSON 2: Writing Polynomial Functions in Standard Form What I Need to Know 15 What I Know 16 What’s In 18 What’s New 21 What is it 22 What’s More 26 Guided/Controlled Practice Independent Practice What I Have Learned 28 What I Can Do 28 Assessment 29 Guided Assessment Independent Assessment Additional Activities 31 SUMMARY 32 ASSESSMENT (Post-Test) 33 KEY TO ANSWERS 35 REFERENCES 39 iii
- 6. What This Module is About For the Facilitator: Welcome to the Mathematics Grade 10 Alternative Delivery Mode Module entitled “Illustrating Polynomial Functions”. This module was collaboratively designed, developed and reviewed by educators both from public and private institutions to assist you, the teacher or facilitator in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling. This learning resource hopes to engage the learners into guided and independent learning activities at their own pace and time. Furthermore, this also aims to help learners acquire the needed 21st century skills while taking into consideration their needs and circumstances. As a facilitator you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module. You may prepare your own related activities if you feel that the activities suggested here are not appropriate to the level and contexts of students (examples, slow/fast learners, and localized situations/examples). Notes to the Parents/Guardians: Welcome to the Mathematics Grade 10 Alternative Delivery Mode Module entitled “Illustrating Polynomial Functions”. This Module was designed and developed to cater the academic needs of the learners in this trying time. Teaching and learning process do not only happen inside the four corners of the classroom but also in your respective homes. We hope that you will cooperate, provide encouragement and show full support to your children in answering all the activities found in this module. iv
- 7. Notes to the Learners: Welcome to the Mathematics Grade 10 Alternative Delivery Mode Module entitled “Illustrating Polynomial Functions”. This module was intended to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. This module was designed and written with you in mind. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using. This module has the following parts and corresponding icons: What I Need to Know This will give you an idea of the skills or competencies you are expected to learn in the module. What I Know This part includes an activity that aims to check what you already know about the lesson to take. If you get all the answers correct (100%), you may decide to skip this module. What’s In This is a brief drill or review to help you link the current lesson with the previous one. What’s New In this portion, the new lesson will be introduced to you in various ways such as a story, a song, a poem, a problem opener, an activity or a situation. What is It This section provides a brief discussion of the lesson. This aims to help you discover and understand new concepts and skills. What’s More This comprises activities for independent practice to solidify your understanding and skills of the topic. You 156 What I Have Learned This includes questions or blank sentence/paragraph to be filled into process what you learned from the lesson. v
- 8. What I Can Do This section provides an activity which will help you transfer your new knowledge or skill into real life situations or concerns. Assessment This is a task which aims to evaluate your level of mastery in achieving the learning competency. Additional Activities In this portion, another activity will be given to you to enrich your knowledge or skill of the lesson learned. This also tends retention of learned concepts. Answer Key This contains answers to all activities in the module. At the end of this module you will also find: The following are some reminders in using this module: 1. Use the module with care. Do not put unnecessary mark/s on any part of the module. Use a separate sheet of paper in answering the exercises. 2. Don’t forget to answer What I Know before moving on to the other activities included in the module. 3. Read the instruction carefully before doing each task. 4. Observe honesty and integrity in doing the tasks and checking your answers. 5. Finish the task at hand before proceeding to the next. 6. Return this module to your teacher/facilitator once you are through with it. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. Remember, your academic success lies in your own hands! You can do it! References This is a list of all sources used in developing this module. vi
- 9. What I Need to Know In this module, you need to recall what you have learned about polynomials like the degree, coefficients, constant terms, factoring, etc. The module is divided into two lessons, namely: • Lesson 1: Definition of Polynomial Function • Lesson 2: Writing Polynomial Functions in Standard Form and in Factored Form After you go through this module, you are expected to: 1. illustrates polynomial functions (M10AL-IIa-1); 2. write polynomial function in standard form and in factored form. 1
- 10. What I Know Directions: Choose the letter that best answers each question. 1. Which of the following is the value of 𝑛 in 𝑓(𝑥) = 𝑥𝑛 if f is a polynomial function? A. √2 B. 2 C. −2 D. 1 2 2. Which of the following is NOT a polynomial function? A. 𝑓(𝑥) = 0 B. 𝑓(𝑥) = 1 C. 𝑓(𝑥) = 𝑥2 + 𝑥 + 1 D. 𝑓(𝑥) = − 1 2𝑥 3. Which of the following is a polynomial function? i. 𝑓(𝑥) = 𝑥−3 + 2𝑥 + 1 ii. 𝑓(𝑥) = 𝑥2 + 𝑥 + 1 iii. 𝑓(𝑥) = √2 𝑥2 + √𝑥 A. i only B. ii only C. i and ii D. i and iii 4. What is the leading coefficient of 𝑓(𝑥) = 𝑥2 + 4𝑥3 + 1? A. 1 B. 2 C. 3 D. 4 5. What is the constant term of the polynomial function in number 4? A. 1 B. 2 C. 3 D. 4 6. What is the standard form of 𝑓(𝑥) = (5𝑥 − 3)(25𝑥2 + 15𝑥 + 9)? A. −125𝑥3 − 27 B. 125𝑥3 − 27 C. −125𝑥3 + 27 D. 125𝑥3 + 27 7. What is the leading term of number 6? A. −27 B. 27 C. 125𝑥3 D. −125𝑥3 2
- 11. 8. What is the constant term of the polynomial in number 6? A. −3 B. −9 C. 27 D. −27 9. Given that 𝑓(𝑥) = 2𝑥−2𝑛 + 8𝑥2 , what value should be assigned to 𝑛 to make 𝑓 a function of degree 3? A. − 2 3 B. − 3 2 C. 2 3 D. 3 2 10.How should the polynomial function 𝑓(𝑥) = 𝑥4 − 8𝑥2 + 𝑥 2 + 4𝑥3 + 1 2 be written in standard form? A. 𝑓(𝑥) = −8𝑥2 + 4𝑥3 + 1 2 + 𝑥4 + 𝑥 2 B. 𝑓(𝑥) = 𝑥 2 + 1 2 − 8𝑥2 + 4𝑥3 + 𝑥4 C. 𝑓(𝑥) = 𝑥4 + 4𝑥3 − 8𝑥2 + 𝑥 2 + 1 2 D. 𝑓(𝑥) = 1 2 + 4𝑥3 − 8𝑥2 + 𝑥 2 + 𝑥4 11. What is the leading coefficient of number 10? A. −8 B. 1 C. 1 2 D. −4 12.What is the constant term of the polynomial in number 10? A. −8 B. 1 C. 1 2 D. -4 13. How should 𝑓(𝑥) = 𝑥4 + 𝑥3 + 𝑥2 + 𝑥 be written in factored form? A. 𝑓(𝑥) = 𝑥(𝑥 + 1)(𝑥2 + 1) B. 𝑓(𝑥) = 𝑥(1)(𝑥2 + 1) C. 𝑓(𝑥) = 𝑥(𝑥 − 1)(𝑥2 + 1) D. 𝑓(𝑥) = 𝑥(−1)(𝑥2 + 1) 14. What is the factored form of 𝑓(𝑥) = 𝑥3 + 3𝑥2 − 4𝑥 − 12? A. 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 2)(𝑥 + 3) B. 𝑓(𝑥) = (𝑥 + 2)(𝑥 + 2)(𝑥 + 3) C. 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 2)(𝑥 + 3) D. 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 2)(𝑥 − 3) 15.What is the factored form of 𝑦 = 9𝑥3 − 3𝑥2 + 81𝑥 − 27? A. 𝑦 = −3(𝑥2 + 9)(3𝑥 − 1) B. 𝑦 = 3(𝑥2 + 9)(3𝑥 − 1) C. 𝑦 = 3(𝑥2 − 9)(3𝑥 − 1) D. 𝑦 = 3(𝑥2 + 9)(3𝑥 + 1) 3
- 12. Lesson 1 DEFINITION OF POLYNOMIAL FUNCTIONS What I Need to Know This lesson is good for one (1) day. You may skip this if you can get a perfect score in What I Know. At the end of the lesson, you should be able to: 1. illustrates polynomial function; 2. identify polynomial function; and 3. determine the degree, the leading term and coefficient and the constant term. 4
- 13. What I Know Directions: Choose the letter that best answers each question. 1. Which of the following is a monomial or a sum of monomials? A. constant term B. degree C. leading term D. polynomial 2. What function is 𝑦 = 𝑥3 + 2𝑥 + 1? A. Linear Function B. Polynomial Function C. Quadratic Function D. Rational Function 3. What is the value of 𝑛 in 𝑓(𝑥) = 𝑥𝑛 if f is a polynomial function? A. √3 B. 3 C. −3 D. 1 3 4. Which of the following is NOT a polynomial function? A. 𝑃(𝑥) = 𝑎𝑥 + 𝑏 B. 𝑃(𝑥) = 𝑝(𝑥) 𝑞(𝑥) C. 𝑃(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 D. 𝑃(𝑥) = 𝑎𝑥4 + 𝑏𝑥3 + 𝑐𝑥4 + 𝑑𝑥 + 𝑒 For numbers 5-8, use the given function 𝑓(𝑥) = 3𝑥3 + 6𝑥2 + 𝑥 2 + 2𝑥4 and choose your answers below: A. 0 B. 2 C. 4 D. 6 5. Which of the choices is the leading coefficient of the function? 6. What is the constant term of the function? 7. What is the degree of the function? 8. Which is not a coefficient of the function? 5
- 14. 9. What type of polynomial function is 𝑓(𝑥) = 3𝑥3 + 6𝑥2 + 𝑥 2 + 2𝑥4 ? A. Cubic Polynomial Function B. Quadratic Polynomial Function C. Quartic Polynomial Function D. Zero Polynomial Function 10. What type of polynomial function is 𝑃(𝑥) = (𝑥 + 2)(𝑥 − 2)? A. Cubic Polynomial Function B. Quadratic Polynomial Function C. Quartic Polynomial Function D. Zero Polynomial Function For numbers 11-14, use the polynomial function in number 10. 11.What is the leading term of the function? A. 𝑥2 B. 2𝑥2 C. 3𝑥2 D. 4𝑥2 12.What is the constant term of the function? A. – 4 B. – 2 C. 0 D. 2 13.What is the degree of the function? A. 0 B. 1 C. 2 D. 3 14.Which is the leading coefficient of the function? A. – 4 B. – 2 C. 0 D. 1 15.Given that 𝑓(𝑥) = 𝑥−3𝑛 + 2𝑥2 , what value should be assigned to 𝑛 to make 𝑓 a polynomial function of degree 4? A. − 4 3 B. − 3 4 C. 2 3 D. 3 2 6
- 15. What’s In You have learned in the last module that to solve problems involving polynomials, you must follow steps to have an easy solution. Start this module by recalling your knowledge on the concept of polynomial expressions. • The word polynomial is derived from Greek words “poly” which means many and “nominal” which means terms, so polynomial means many terms. • Polynomials are composed of constants (numbers), variables (letters) and exponents such as 2 in x2 . The combination of numbers, variables and exponents is called terms. • Example: 2𝑥3 + 𝑥2 + 1 There are three (3) terms in this expression: 2𝑥3 , 𝑥2 & 1, where 1 is the constant, x is the variable and 3 and 2 are the exponents. This knowledge will help you understand the formal definition of polynomial function. What’s New Let’s explore! Directions: Complete the table below. State your reason if it is not a polynomial. Expression Polynomial or Not Reason/s 1. 10𝑥 2. 𝑥3 − 2√5𝑥 + 𝑥 3. −2020𝑥 4. 𝑥 2 3 + 3𝑥 + 1 5. 1 𝑥2 + 2 𝑥3 + 3 𝑥4 6. 𝜋 7. 3𝑥√2 + √3𝑥2 8. 𝑥3 + 2𝑥 + 1 9. −2𝑥−3 + 𝑥3 10.1 − 4𝑥2 Did you complete the table correctly? Do you remember when an expression is a polynomial? A polynomial is an expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variable raised to a nonnegative integral power. 7
- 16. What Is It A polynomial function is a function of the form 𝑃(𝑥) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + 𝑎𝑛−2𝑥𝑛−2 + ⋯ + +𝑎1𝑥 + 𝑎0, 𝑎𝑛 ≠ 0, where 𝑛 is a nonnegative integer , 𝑎0, 𝑎1, … , 𝑎𝑛 are real numbers called coefficients (numbers that appear in each term) , 𝑎𝑛𝑥𝑛 is the leading term, 𝑎𝑛 is the leading coefficient, and 𝑎0 is the constant term (number without a variable). The highest power of the variable of 𝑃(𝑥) is known as its degree. There are various types of polynomial functions based on the degree of the polynomial. The most common types are: • Zero Polynomial Function (degree 0): 𝑃(𝑥) = 𝑎𝑥0 = 𝑎 • Linear Polynomial Function (degree 1): 𝑃(𝑥) = 𝑎𝑥1 + 𝑏 = 𝑎𝑥 + 𝑏 • Quadratic Polynomial Function (degree 2): 𝑃(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 • Cubic Polynomial Function (degree 3): 𝑃(𝑥) = 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑 • Quartic Polynomial Function (degree 4): 𝑃(𝑥) = 𝑎𝑥4 + 𝑏𝑥3 + 𝑐𝑥2 + 𝑑𝑥 + 𝑒 where 𝑎, 𝑏, 𝑐, 𝑑 and 𝑒 are constants. Other than 𝑃(𝑥) , a polynomial function can be written in different ways, like the following: 𝑓(𝑥) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + 𝑎𝑛−2𝑥𝑛−2 + ⋯ + 𝑎1𝑥 + 𝑎0, 𝑦 = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + 𝑎𝑛−2𝑥𝑛−2 + ⋯ + 𝑎1𝑥 + 𝑎0, Example: Degree of the Polynomial Type of Function Leading Term Leading Coefficient Constant Term 1. 𝑦 = 8𝑥4 − 4𝑥3 + 2𝑥 + 22 4 Quartic 8𝑥4 8 22 2. 𝑦 = 3𝑥2 + 6𝑥3 + 2𝑥 3 Cubic 6𝑥3 6 0 8
- 17. What’s More Let’s do this… A. Directions: Complete the table below. If the given is a polynomial function, give the degree, leading coefficient and its constant term. If it is not, then just give the reason. Polynomial Function or Not Reason Degree Leading Coefficient Constant Term 1. 𝑓(𝑥) = 0 2. 𝑓(𝑥) = 𝑥2 − √2𝑥 + 𝑥 3. 𝑓(𝑥) = −𝑥 4. 𝑓(𝑥) = 𝑥 3 4 + 2𝑥 + 2 5. 𝑓(𝑥) = 3 √3𝑥 6. 𝑦 = √5𝑥 7. 𝑦 = 3𝑥 + 𝑥2 8. 𝑦 = −𝑥−1 9. 𝑦 = 1 + 2𝑥 + 𝑥3 10.𝑦 = 1 − 4𝑥2 11.𝑃(𝑥) = 2020 12.𝑃(𝑥) = −√𝑥 + 𝑥 13.𝑃(𝑥) = 3𝑥 √4 14.𝑃(𝑥) = 𝑥 + 2 15.𝑃(𝑥) = 3 𝑥−1 9
- 18. B. Directions: Identify whether the following is a polynomial function or not. If the given is a polynomial function, give the degree of polynomial, the type of polynomial function, the leading term and its constant term. 1. 𝑦 = 𝑥 2. 𝑦 = 3𝑥 + 4𝑥2 3. 𝑦 = −𝑥−10 4. 𝑦 = 12 + 6𝑥 + 𝑥2 5. 𝑦 = 10 − 5𝑥2 6. 𝑃(𝑥) = 1 2 7. 𝑃(𝑥) = −√𝑥 + 3𝑥2 8. 𝑃(𝑥) = 1 2 𝑥2 − 3 What I Have Learned A. Directions: Fill in the blank with the choices provided in the box. A __________(1)__________ is a function which involves only ________(2)____________ integer powers or only positive integer exponents. The _________(3)_______ of any polynomial is the highest power present in it. In the ____(4)_____ polynomial function 𝑦 = 4 + 2𝑥 + 𝑥3 , __(5)_____ is the leading term, 4 is the ___(6)_____, 1 is the ___(7)______, and ___(8)____ is the degree. polynomial function cubic nonnegative constant term leading coefficient degree 3 1 𝑥3 9. 𝑃(𝑥) = 𝑥5 − 𝑥4 − 𝑥 + 2 10.𝑃(𝑥) = 1 4𝑥 + 3 11.𝑓(𝑥) = 1 2 √𝑥 12.𝑓(𝑥) = 5 8 𝑥 13.𝑓(𝑥) = 2𝑥+2 3 14.𝑓(𝑥) = 𝑥2 15.𝑓(𝑥) = 𝑥−3 𝑥+2 10
- 19. B. Directions: Complete the table below. If the given is a polynomial function, give the degree, leading coefficient and its constant term. If it is not, then just give the reason. Polynomial Function or Not Reason Degree Leading Term, Coefficient Constant Term 9. 𝑦 = 20 10.𝑦 = √𝑥 + 18 11.𝑓(𝑥) = −1991𝑥 12.𝑓(𝑥) = 𝑥 1 2 + 𝑥 − 1 13.𝑓(𝑥) = 5 √5𝑥 14.𝑦 = √4𝑥 15.𝑦 = 20 − 𝑥 + 𝑥2 What I Can Do Directions: Give five polynomial functions of different degree of polynomial. Identify the degree of polynomial, the type of polynomial, the leading coefficient and its constant term. Polynomial Functions Degree of Polynomial Type of Polynomial Leading Coefficient Constant Term 1. 2. 3. 4. 5. 11
- 20. Assessment Directions: Choose the letter that best answers each question. 1. Which of the following is the term with number without variable? A. constant term B. degree C. leading term D. polynomial 2. What function is 𝑦 = 𝑥4 + 1? A. Linear Function B. Quadratic Function C. Quartic Function D. Rational Function 3. What is the value of 𝑛 in 𝑓(𝑥) = 𝑥𝑛 if f is a polynomial function? A. √3 B. 3 C. −3 D. 1 3 4. Which of the following is NOT a polynomial function? A. 𝑃(𝑥) = 𝑎𝑥 + 𝑏 B. 𝑃(𝑥) = 𝑝(𝑥) 𝑞(𝑥) C. 𝑃(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 D. 𝑃(𝑥) = 𝑎𝑥4 + 𝑏𝑥3 + 𝑐𝑥4 + 𝑑𝑥 + 𝑒 5. Given that 𝑓(𝑥) = 𝑥−3𝑛 + 2𝑥2 , what value should be assigned to 𝑛 to make 𝑓 a polynomial function of degree 4? A. − 4 3 B. − 3 4 C. 4 3 D. 3 4 12
- 21. For numbers 6-9, use the given function 𝑓(𝑥) = 5𝑥3 + 𝑥2 + 3𝑥 + 15 and choose your answers below: A. 1 B. 3 C. 5 D. 15 6. Which of the choices is the leading coefficient of the function? 7. What is the constant term of the function? 8. What is the degree of the function? 9. Which is not a coefficient of the function? 10.What type of polynomial function is 𝑓(𝑥) = 5𝑥3 + 𝑥2 + 15 ? A. Cubic Polynomial Function B. Quadratic Polynomial Function C. Quartic Polynomial Function D. Zero Polynomial Function 11.What type of polynomial function is 𝑓(𝑥) = (𝑥 + 2)(2𝑥 − 8)? A. Cubic Polynomial Function B. Quadratic Polynomial Function C. Quartic Polynomial Function D. Zero Polynomial Function For numbers 12-15, use the polynomial function in number 11. 12.What is the leading term of the function? A. 𝑥2 B. 2𝑥2 C. 4𝑥2 D. 8𝑥2 13.What is the constant term of the function? A. – 24 B. – 16 C. – 8 D. – 4 14.What is the degree of the function? A. 0 B. 1 C. 2 D. 3 15.Which is the leading coefficient of the function? A. 2 B. 4 C. 6 D. 8 13
- 22. Additional Activity Directions: Give two examples for each type of polynomials. Identify the degree of polynomial, the leading term and the constant term. 14
- 23. Lesson Writing Polynomial Functions 2 In Standard Form and in Factored Form What I Need to Know This lesson is good for one (1) day. You may skip this if you can get a perfect score in What I Know. At the end of the lesson, you should be able to: 1. write polynomial functions in standard form; and 2. write polynomial functions in factored form 15
- 24. What I Know Directions: Choose the letter that best answers each question. 1. What is the product of (𝑥 + 2)(𝑥 + 5)? A. 𝑥2 + 3𝑥 + 10 B. 𝑥2 − 3𝑥 + 10 C. 𝑥2 + 7𝑥 + 10 D. 𝑥2 + 3𝑥 − 3 2. What is the product of (𝑥 + 2)(𝑥2 − 2𝑥 + 4)? A. 𝑥3 − 8 B. 𝑥3 + 8 C. 𝑥3 − 4 D. 𝑥3 + 4 3. What term has the highest exponent in 𝑓(𝑥) = −2𝑥4 + 𝑥6 + 3𝑥 + 1? A. −2𝑥4 B. 𝑥6 C. 3𝑥 D. 1 4. What is the constant term in number 3? A. −2𝑥4 B. 𝑥6 C. 3𝑥 D. 1 5. What is the standard form of the polynomial function in number 3? A. 𝑓(𝑥) = 𝑥6 − 2𝑥4 + 3𝑥 + 1 B. 𝑓(𝑥) = 1 + 𝑥6 − 2𝑥4 + +3𝑥 C. 𝑓(𝑥) = 𝑥6 − 2𝑥4 + 1 + 3𝑥 D. 𝑓(𝑥) = −2𝑥4 + 3𝑥 + 𝑥6 + 1 6. What should be the order of terms of the polynomial function in standard form? A. constant term, term with highest exponent, term/s with lower exponent B. constant term, term/s with lower exponent, term with highest exponent C. term with highest exponent, constant term, term/s with lower exponent D. term with highest exponent, term/s with lower exponent, constant term 7. What is the standard form of 𝑦 = 8𝑥2 + 4𝑥 + 3𝑥6 + 3? A. 𝑦 = 3𝑥6 + 3 + 8𝑥2 + 4𝑥 B. 𝑦 = 3𝑥6 + 8𝑥2 + 3 + 4𝑥 C. 𝑦 = 3𝑥6 + 8𝑥2 + 4𝑥 + 3 D. 𝑦 = 3𝑥6 + 3 + 4𝑥 + 8𝑥2 16
- 25. 8. What is the standard form of 𝑦 = 20𝑥 + 14𝑥2 + 2𝑥3 ? A. 𝑦 = 2𝑥3 + 20𝑥 + 14𝑥2 B. 𝑦 = 14𝑥2 + 20𝑥 + 2𝑥3 C. 𝑦 = 2𝑥3 + 14𝑥2 + 20𝑥 D. 𝑦 = 14𝑥2 + 2𝑥3 + 20𝑥 9. What is the factored form of the polynomial function in number 8? A. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 − 2) B. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 + 2) C. 𝑦 = 5𝑥(𝑥 + 2)(𝑥 + 2) D. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 + 5) 10.How should the polynomial function 𝑓(𝑥) = 𝑥3 + 8 be written in factored form? A. 𝑓(𝑥) = (𝑥 + 2)(𝑥2 + 2𝑥 + 4) B. 𝑓(𝑥) = (𝑥 − 2)(𝑥2 + 2𝑥 + 4) C. 𝑓(𝑥) = (𝑥 + 2)(𝑥2 − 2𝑥 + 4) D. 𝑓(𝑥) = (𝑥 − 2)(𝑥2 − 2𝑥 + 4) 11. How should 𝑓(𝑥) = 𝑥3 − 64 be written in factored form? A. 𝑓(𝑥) = (𝑥 − 4)(𝑥2 + 4𝑥 + 16) B. 𝑓(𝑥) = (𝑥 + 4)(𝑥2 + 4𝑥 + 16) C. 𝑓(𝑥) = (𝑥 − 4)(𝑥2 − 4𝑥 + 16) D. 𝑓(𝑥) = (𝑥 + 4)(𝑥2 − 4𝑥 + 16) 12.What is the factored form of 𝑦 = 1 − 4𝑥2 ? A. 𝑦 = (1 + 2𝑥)(1 + 2𝑥) B. 𝑦 = (1 − 2𝑥)(1 + 2𝑥) C. 𝑦 = (2𝑥 + 1)(1 + 2𝑥) D. 𝑦 = (2𝑥 + 1)(2𝑥 − 1) 13.How should 𝑦 = −10 + 3𝑥 + 𝑥2 be written in standard form? A. 𝑦 = 𝑥2 + 3𝑥 − 10 B. 𝑦 = 𝑥2 −10 + 3𝑥 C. 𝑦 = −10 + 3𝑥 + 𝑥2 D. 𝑦 = 3𝑥 − 10 + 𝑥2 17
- 26. 14.How should 𝑦 = −10 + 3𝑥 + 𝑥2 be written in factored form? A. 𝑦 = (𝑥 + 5)(𝑥 + 2) B. 𝑦 = (𝑥 + 5)(𝑥 − 2) C. 𝑦 = (𝑥 − 5)(𝑥 + 2) D. 𝑦 = (𝑥 − 5)(𝑥 − 2) 15.What is the standard form of 𝑦 = (3𝑥 + 1)(2𝑥 − 7)? A. 𝑦 = 6𝑥2 + 19𝑥 − 7 B. 𝑦 = 6𝑥2 − 19𝑥 − 7 C. 𝑦 = 6𝑥2 − 23𝑥 − 7 D. 𝑦 = 6𝑥2 + 19𝑥 − 7 18
- 27. What’s In A polynomial function is a function of the form 𝑃(𝑥) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + 𝑎𝑛−2𝑥𝑛−2 + ⋯ + +𝑎1𝑥 + 𝑎0, 𝑎𝑛 ≠ 0. The terms of a polynomial may be written in any order. However, if they are written in decreasing powers of x, then the polynomial function is in standard form. Before you proceed, try to recall the following. Types of Special Products 1. Square of Binomial This special product results into Perfect Square Trinomial (PST). (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏2 (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏2 Example: (2𝑥 − 3)2 = 4𝑥2 − 12𝑥 + 9 2. Product of Sum and Difference of Two Terms This results to Difference of Two Squares. (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏2 Example: (𝑥 + 2)(𝑥 − 2) = 𝑥2 − 4 3. Square of Trinomial This would result to six (6) terms. (𝑎 + 𝑏 + 𝑐)2 = 𝑎2 + 𝑏2 + 𝑐2 + 2𝑎𝑏 + 2𝑎𝑐 + 2𝑏𝑐 Example: (2𝑥 + 3𝑦 + 4𝑧)2 = 4𝑥2 + 9𝑦2 + 16𝑧2 + 12𝑎𝑏 + 16𝑎𝑐 + 24𝑏𝑐 4. Product of Binomials The result is a General Trinomial. F.O.I.L (First, Outer, Inner, Last) method is usually used. (𝑎 + 𝑏)(𝑐 + 𝑑) = 𝑎𝑐 + (𝑏𝑐 + 𝑎𝑑) + 𝑏𝑑 Example: (𝑥 + 2)(𝑥 + 3) = 𝑥2 + (2𝑥 + 3𝑥) + 6 = 𝑥2 + 5𝑥 + 6 19
- 28. 5. Product of Binomial and Trinomial The result is a Sum or Difference of Two Cubes. (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2 ) = 𝑎3 + 𝑏3 (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2) = 𝑎3 − 𝑏3 Example: (𝑥 + 2)(𝑥2 − 2𝑥 + 4) = 𝑥3 + 8 Methods of Factoring Method When is it Possible Example 1. Factoring out the Greatest Common Factor (GCF) If each term in the polynomial has a common factor. 2𝑥2 + 8𝑥 The common factor of both terms is 2x. 2𝑥2 + 8𝑥 = 𝟐𝒙(𝒙 + 𝟒) 2. The Sum- Product Pattern (A-C Method) If the polynomial is of the form 𝑥2 + 𝑏𝑥 + 𝑐 and there are factors of 𝑐 that if added will get 𝑏. 𝑥2 + 5𝑥 + 6 The factors of 6 that if added will get 5 are 2 and 3. 𝑥2 + 5𝑥 + 6 = (𝒙 + 𝟐)(𝒙 + 𝟑) 3. Grouping Method If the polynomial is of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 and there are factors of 𝑎𝑐 that if added will get 𝑏. Steps: • Split up middle term. • Group the terms. • Factor out GCFs of each group. • Factor out the common binomial. 2𝑥2 + 9𝑥 − 5 The factors of 𝑎𝑐 = (2)(−5) = −10 that if added will get 9 are 10 and −1. • Split up middle term 2𝑥2 + 9𝑥 − 5 = 2𝑥2 + 10𝑥 − 1𝑥 − 5 • Group the terms (make sure to group the terms with common factors) = (2𝑥2 − 1𝑥) + (10𝑥 − 5) • Factor out GCFs of each group = 𝑥(2𝑥 − 1) + 5(2𝑥 − 1) • Factor out the common binomial = (𝟐𝒙 − 𝟏)(𝒙 + 𝟓) 20
- 29. 4. Perfect Square Trinomials If the first and last terms are perfect squares and the middle term is twice the product of their roots. 4𝑥2 + 12𝑥 + 9 The first and last terms are perfect squares: √4𝑥2 = 2𝑥 √9 = 3 The middle term is twice the product of their roots: 2(2𝑥)(3) = 12𝑥 4𝑥2 + 12𝑥 + 9 = (𝟐𝒙 − 𝟑)𝟐 5. Difference of Squares If the expression represents a difference of two squares 𝑥2 − 4 Square roots of the terms: √𝑥2 = 𝑥 √4 = 2 𝑥2 − 4 = (𝒙 + 𝟐)(𝒙 − 𝟐) What’s New Directions: Complete the table below. Polynomial Function Term with highest exponent Term/s with lower exponents in descending order Constant term 1. 𝑦 = −4𝑥2 + 𝑥4 − 45 2. 𝑦 = 6𝑥2 + 4𝑥 + 3𝑥3 3. 𝑦 = 5𝑥4 − 5 − 2𝑥 + 𝑥3 4. 𝑦 = 9𝑥2 − 11𝑥4 + 2 5. 𝑦 = −8𝑥2 + 2𝑥3 + 6𝑥 21
- 30. What Is It Writing Polynomial Function in Standard Form When giving a final answer, you must write the polynomial function in standard form. Standard form means that you write the terms by decreasing exponents. Here’s what to do: 1. Write the term with the highest exponent first. 2. Write the terms with lower exponents in descending order. 3. Remember that a variable with no exponent has an understood exponent of 1. 4. A constant term always comes last. Examples: Write the following polynomial functions in standard form. 1. 𝑦 = 1 + 2𝑥 + 𝑥5 − 4𝑥3 + 2𝑥4 + 5𝑥2 Term with highest exponent Term/s with lower exponents in descending order Constant term Standard form 𝑥5 2𝑥4 , −4𝑥3 , 5𝑥2 , 2𝑥 1 𝒚 = 𝒙𝟓 + 𝟐𝒙𝟒 − 𝟒𝒙𝟑 + 𝟓𝒙𝟐 + 𝟐𝒙 + 𝟏 2. 𝑓(𝑥) = 5𝑥 + 9𝑥2 − 3𝑥8 Often, the polynomial function does not contain all of the exponents. You still follow the same procedure listing the highest exponent first (8) then the next (2) and finally the term with just a variable (understood exponent of 1). Term with highest exponent Term/s with lower exponents in descending order Constant term Standard form −3𝑥8 9𝑥2 , 5𝑥 0 𝒇(𝒙) = −𝟑𝒙𝟖 + 𝟗𝒙𝟐 + 𝟓𝒙 22
- 31. 3. 𝑦 = 𝑥( 𝑥2 − 5) With a factored form of a polynomial function, you must find the product first. In finding the product of a monomial and a binomial, recall the Distributive Property. Multiply the monomial to the first term of the binomial 𝑥( 𝑥2) = 𝑥1+2 = 𝑥3 Multiply the monomial to the second term of the binomial 𝑥( −5) = −5𝑥 Arrange the exponents in descending order. Therefore, the standard form is 𝒚 = 𝒙𝟑 − 𝟓𝒙 4. 𝑓(𝑥) = −𝑥( 𝑥 − 4)( 𝑥 + 4) Use the special product, Sum and Difference of two terms, in answering this function. Get the product of the sum and difference of two terms. ( 𝑥 − 4)( 𝑥 + 4) = 𝑥2 − 16 Multiply -x to the product. −𝑥(𝑥2 − 16) = 𝑥3 + 16𝑥 Thus, the polynomial function in standard form becomes 𝒇(𝒙) = 𝒙𝟑 + 𝟏𝟔𝒙. 23
- 32. Writing Polynomial Function in Factored Form We will focus on polynomial functions of degree 3 and higher, since linear and quadratic functions were already taught in previous grade levels. The polynomial function must be completely factored. Examples: Write the following polynomial functions in factored form. 1. 𝑦 = 64𝑥3 + 125 This is of the form 𝑎3 + 𝑏3 which is called the sum of cubes. The factored form of 𝑎3 + 𝑏3 is (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2 ). To factor the polynomial function follow the steps below: Find 𝑎 and 𝑏 (𝑎 is the cube root of the first term) (𝑏 is the cube root of the second term) 𝑎 = 4𝑥 𝑏 = 5 Substitute the values of 𝑎 and 𝑏 in (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2 ) 𝑦 = (4𝑥 + 5)[(4𝑥)2 − (4𝑥)(5) + (5)2 ] So the factored form is 𝒚 = (𝟒𝒙 + 𝟓)(𝟏𝟔𝒙𝟐 − 𝟐𝟎𝒙 + 𝟐𝟓) 2. 𝑦 = 3𝑥3 + 6𝑥2 + 4𝑥 + 8 This is of the form 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑. This can be easily factored if 𝑎 𝑏 = 𝑐 𝑑 . To factor the polynomial function, follow the steps below: Group the terms (𝑎𝑥3 + 𝑏𝑥2 ) + (𝑐𝑥 + 𝑑) 𝑦 = (3𝑥3 + 6𝑥2 ) + (4𝑥 + 8) Factor 𝑥2 out of the first group of terms. Factor the constants out of both groups. 𝑦 = 𝑥2 (3𝑥 + 6) + (4𝑥 + 8) 𝑦 = 3𝑥2 (𝑥 + 2) + 4(𝑥 + 2) Add the two terms by adding the coefficients 𝑦 = 3𝑥2 (𝑥 + 2) + 4(𝑥 + 2) 𝑦 = (3𝑥2 + 4)(𝑥 + 2) So, the factored form is 𝒚 = (𝟑𝒙𝟐 + 𝟒)(𝒙 + 𝟐) 24
- 33. 3. 𝑦 = 45𝑥3 + 18𝑥2 − 5𝑥 − 2 This is of the form 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑. Follow the steps below: Group the terms (𝑎𝑥3 + 𝑏𝑥2 ) + (𝑐𝑥 + 𝑑) 𝑦 = (45𝑥3 + 18𝑥2 ) + (−5𝑥 − 2) Factor 𝑥2 out of the first group of terms. Factor the constants out of both groups. 𝑦 = 𝑥2(45𝑥 + 18) + (−5𝑥 − 2) 𝑦 = 9𝑥2(5𝑥 + 2) − (5𝑥 + 2) Add the two terms by adding the coefficients 𝑦 = 9𝑥2(5𝑥 + 2) − 1(5𝑥 + 2) 𝑦 = (9𝑥2 − 1)(5𝑥 + 2) This can be further factored as a difference of two squares 𝑦 = (3𝑥 + 1)(3𝑥 − 1)(5𝑥 + 2) So, the factored form is 𝒚 = (𝟑𝒙 + 𝟏)(𝟑𝒙 − 𝟏)(𝟓𝒙 + 𝟐) 4. 𝑦 = 81𝑥4 − 16 This is of the form 𝑎4 − 𝑏4 . We can factor a difference of fourth powers (and higher powers) by treating each term as the square of another base, using the power to a power rule. Follow the steps below: Treat 𝑎4 as (𝑎2 )2 and 𝑏4 as (𝑏2 )2 (𝑎2 )2 − (𝑏2 )2 𝑦 = (9𝑥2 )2 − (4)2 It shows difference of two squares, factor it. (𝑎2 )2 − (𝑏2)2 = (𝑎2 + 𝑏2 )(𝑎2 − 𝑏2 ) 𝑦 = (9𝑥2 + 4)(9𝑥2 − 4) This can be further factored as a difference of squares 𝑦 = (9𝑥2 + 4)(3𝑥 + 2)(3𝑥 − 2) So, the factored form is 𝒚 = (𝟗𝒙𝟐 + 𝟒)(𝟑𝒙 + 𝟐)(𝟑𝒙 − 𝟐) 5. 𝑦 = 𝑥4 − 4𝑥2 − 45 This is of the form 𝑎𝑥4 + 𝑏𝑥2 + 𝑐. In similar manner, we can factor some trinomials of degree four by treating 𝑥4 as (𝑎2 )2 . Follow the steps below: Treat 𝑎4 as (𝑎2 )2 (𝑥2 )2 − 𝑏(𝑥2 ) − 𝑐 𝑦 = (𝑥2 )2 − 4(𝑥2 ) − 45 Let 𝑥2 = 𝑥, thus, it shows a quadratic trinomial: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 Factor it: 𝑦 = 𝑥2 − 4𝑥 − 45 𝑦 = (𝑥 − 9)(𝑥 + 5) Put it back. (Substitute 𝑥 = 𝑥2 ) 𝑦 = (𝑥2 − 9)(𝑥2 + 5) This can be further factored as a difference of squares 𝑦 = (𝑥 + 3)(𝑥 − 3)(𝑥2 + 5) So, the factored form is 𝒚 = (𝒙 + 𝟑)(𝒙 − 𝟑)(𝒙𝟐 + 𝟓) 25
- 34. What’s More A. Directions: Complete the table below. Polynomial Function Term with highest exponent Term/s with lower exponents in descending order Constant term Standard form 1. 𝑓(𝑥) = 4 + 4𝑥4 + 8𝑥 2. 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 2) 3. 𝑦 = 1 + 2𝑥 + 𝑥3 4. 𝑦 = −5 + 5𝑥10 + 5𝑥5 5. 𝑓(𝑥) = 𝑥2 − 9𝑥5 + 6 B. Directions: Write the factored form of the following polynomial functions by completing the table: 1. 𝑦 = 343𝑥3 + 27 Find 𝑎 and 𝑏 (𝑎 is the cube root of the first term) (𝑏 is the cube root of the second term) 𝑎 = _____ 𝑏 = _____ Substitute the values of 𝑎 and 𝑏 in (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2 ) 𝑦 = (__ + __)[(__)2 − 2(__)(__) + (__)2 ] So, the factored form is 𝒚 = (__ + __)(__𝟐 − __ + __) 2. 𝑦 = 27𝑥3 − 8 Find 𝑎 and 𝑏 (𝑎 is the cube root of the first term) (𝑏 is the cube root of the second term) 𝑎 = _____ 𝑏 = _____ Substitute the values of 𝑎 and 𝑏 in (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2 ) 𝑦 = (__ − __)[(__)2 + 2(__)(__) + 1(__)2 ] So, the factored form is 𝒚 = (__ − __)(__𝟐 + __ + __) 26
- 35. 3. 𝑦 = 𝑥3 + 3𝑥2 − 4𝑥 − 12 Group the terms (𝑎𝑥3 + 𝑏𝑥2 ) + (𝑐𝑥 + 𝑑) 𝑦 = (__3 + __2 ) + (__ − __) Factor 𝑥2 out of the first group of terms. Factor the constants out of both groups. 𝑦 = 𝑥2(__ + __) − __(__ + __) Add the two terms by adding the coefficients 𝑦 = (__2 − __)(__ + __) This can be further factored as a difference of squares 𝑦 = (__ + __)(__ − __)(__ + __) So, the factored form is 𝑦 = (__ + __)(__ − __)(__ + __) 4. 𝑦 = 𝑥4 − 5𝑥2 + 4 Treat 𝑎4 as (𝑎2 )2 (𝑥2 )2 − 𝑏(𝑥2 ) + 𝑐 𝑦 = (__)2 − __(__2 ) + __ Let 𝑥2 = 𝑥 , thus, it shows a quadratic trinomial: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 Factor it: 𝑦 = 𝑥2 − __ + __ 𝑦 = (𝑥 − __)(𝑥 − __) Put it back. (Substitute 𝑥 = 𝑥2 ) 𝑦 = (𝑥2 − __)(𝑥2 − __) This can be further factored as a difference of Two squares 𝑦 = (𝑥 + __)(𝑥 − __)(𝑥 + __)(𝑥 − __) So, the factored form is 𝒚 = (𝒙 + __)(𝒙 − __)(𝒙 + __)(𝒙 − __) C. Directions: Match the following polynomial functions into its standard/factored forms. Numbers 6-10 have two answers which is it’s standard and factored form. Column A ____1. 𝑓(𝑥) = 2 − 𝑥4 + 8𝑥 ____2. 𝑓(𝑥) = (𝑥 + 5)(𝑥 + 1) ____3. 𝑓(𝑥) = 6 − 2𝑥 ____4. 𝑓(𝑥) = −16 + 5𝑥8 − 5𝑥3 ____5. 𝑓(𝑥) = 𝑥2 − 9𝑥5 + 6 ____6. 𝑦 = 𝑥 − 2𝑥2 + 𝑥3 ____7. 𝑦 = −100 + 𝑥2 ____8. 𝑓(𝑥) = 4 + 5𝑥 + 𝑥2 ____9. 𝑦 = 16 + 𝑥2 + 8𝑥 ____10. 𝑦 = 1 − 4𝑥2 Column B A. 𝑓(𝑥) = −9𝑥5 + 𝑥2 + 6 B. 𝑓(𝑥) = −𝑥4 + 8𝑥 + 2 C. 𝑓(𝑥) = 5𝑥8 − 5𝑥3 − 16 D. 𝑓(𝑥) = −2𝑥 + 6 E. 𝑓(𝑥) = 𝑥2 + 6𝑥 + 5 F. 𝑦 = 𝑥3 − 2𝑥2 + 𝑥 G. 𝑦 = −4𝑥2 + 1 H. 𝑦 = 𝑥(1 − 𝑥)(1 − 𝑥) I. 𝑦 = 𝑥2 + 5𝑥 + 4 J. 𝑦 = (𝑥 − 10)(𝑥 + 10) K. 𝑦 = 𝑥2 + 8𝑥 + 16 L. 𝑦 = (1 − 2𝑥)(1 + 2𝑥) M. 𝑦 = 𝑥2 − 100 N. 𝑦 = (𝑥 + 4)2 O. 𝑓(𝑥) = (𝑥 + 4)(𝑥 + 1) 27
- 36. What I Have Learned A. Directions: Fill in the blanks with the correct word/s to complete each statement. _______(1)________ means that you write the terms by decreasing exponents. Steps in writing this form: 1. Write the term with the ____(2)_________ first. 2. Write the terms with lower exponents in ____(3)_________ order. 3. Remember that a variable with no exponent has an understood exponent of (4). 4. A ______(5)_________ always comes last. B. Direction: Factor the following: 1. 𝑦 = 𝑥4 − 512𝑥 2. 𝑦 = 9𝑥3 − 36𝑥2 + 4𝑥 − 16 What I Can Do Directions: Write the standard form of the polynomial functions that is found in nature. 1. The intensity of light emitted by a firefly can be determined by 𝐿(𝑡) = 10 + 0.3𝑡 + 0.4𝑡2 − 0.01𝑡3 . 2. The total number of hexagons in a honeycomb can be modeled by the function 𝑓(𝑟) = 1 + 3𝑟2 − 3𝑟. 28
- 37. Assessment Directions: Choose the letter that best answers each question. 1. What is the product of (𝑥 + 3)(𝑥 + 3)? A. 𝑥2 + 3𝑥 + 9 B. 𝑥2 − 3𝑥 + 9 C. 𝑥2 + 6𝑥 + 9 D. 𝑥2 − 6𝑥 + 9 2. What is the product of (𝑥 − 2)(𝑥2 + 2𝑥 + 4)? A. 𝑥3 + 8 B. 𝑥3 − 8 C. 𝑥3 − 4 D. 𝑥3 + 4 3. What term has the highest exponent in 𝑓(𝑥) = 𝑥4 + 5𝑥7 + 3𝑥? A. 𝑥4 B. 5𝑥7 C. 3𝑥 D. 0 4. What is the constant term in number 3? A. 𝑥4 B. 5𝑥7 C. 3𝑥 D. 0 5. What is the standard form of the polynomial function in number 3? A. 𝑓(𝑥) = 5𝑥7 + 𝑥4 + 3𝑥 B. 𝑓(𝑥) = 5𝑥7 + 3𝑥 + 𝑥4 C. 𝑓(𝑥) = 𝑥4 + 5𝑥7 + 3𝑥 D. 𝑓(𝑥) = 3𝑥 + 5𝑥7 + 𝑥4 6. What should be the order of terms of the polynomial function in standard form? A. term with highest exponent, term/s with lower exponent, constant term B. term with highest exponent, constant term, term/s with lower exponent C. constant term, term with highest exponent, term/s with lower exponent D. constant term, term/s with lower exponent, term with highest exponent 7. What is the standard form of 𝑦 = 8𝑥2 + 4𝑥 + 3𝑥6 + 3? A. 𝑦 = 3𝑥6 + 3 + 8𝑥2 + 4𝑥 B. 𝑦 = 3𝑥6 + 8𝑥2 + 4𝑥 + 3 C. 𝑦 = 3𝑥6 + 8𝑥2 + 3 + 4𝑥 D. 𝑦 = 3𝑥6 + 3 + 4𝑥 + 8𝑥2 29
- 38. 8. What is the standard form of 𝑦 = 6𝑥 + 12𝑥2 + 2𝑥3 ? A. 𝑦 = 2𝑥3 + 6𝑥 + 12𝑥2 B. 𝑦 = 12𝑥2 + 6𝑥 + 2𝑥3 C. 𝑦 = 2𝑥3 + 12𝑥2 + 6𝑥 D. 𝑦 = 12𝑥2 + 2𝑥3 + 6𝑥 9. What is the factored form of the polynomial function 𝑦 = 2𝑥3 + 14𝑥2 + 20𝑥 A. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 − 2) B. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 + 2) C. 𝑦 = 5𝑥(𝑥 + 2)(𝑥 + 2) D. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 + 5) 10.How should 𝑓(𝑥) = 𝑥3 − 64 be written in factored form? A. 𝑓(𝑥) = (𝑥 − 4)(𝑥2 + 4𝑥 + 16) B. 𝑓(𝑥) = (𝑥 + 4)(𝑥2 + 4𝑥 + 16) C. 𝑓(𝑥) = (𝑥 − 4)(𝑥2 − 4𝑥 + 16) D. 𝑓(𝑥) = (𝑥 + 4)(𝑥2 − 4𝑥 + 16) 11.How should the polynomial function 𝑓(𝑥) = 𝑥3 + 8 be written in factored form? A. 𝑓(𝑥) = (𝑥 + 2)(𝑥2 + 2𝑥 + 4) B. 𝑓(𝑥) = (𝑥 − 2)(𝑥2 + 2𝑥 + 4) C. 𝑓(𝑥) = (𝑥 + 2)(𝑥2 − 2𝑥 + 4) D. 𝑓(𝑥) = (𝑥 − 2)(𝑥2 − 2𝑥 + 4 12.What is the factored form of 𝑦 = −4𝑥2 + 1? A. 𝑦 = (1 + 2𝑥)(1 + 2𝑥) B. 𝑦 = (1 − 2𝑥)(1 + 2𝑥) C. 𝑦 = (2𝑥 + 1)(1 + 2𝑥) D. 𝑦 = (2𝑥 + 1)(2𝑥 − 1) 30
- 39. 13.How should 𝑦 = −10 + 3𝑥 + 𝑥2 be written in standard form? A. 𝑦 = 𝑥2 + 3𝑥 − 10 B. 𝑦 = 𝑥2 −10 + 3𝑥 C. 𝑦 = −10 + 3𝑥 + 𝑥2 D. 𝑦 = 3𝑥 − 10 + 𝑥2 14.How should 𝑦 = 𝑥2 −10 + 3𝑥 be written in factored form? A. 𝑦 = (𝑥 + 5)(𝑥 + 2) B. 𝑦 = (𝑥 + 5)(𝑥 − 2) C. 𝑦 = (𝑥 − 5)(𝑥 + 2) D. 𝑦 = (𝑥 − 5)(𝑥 − 2) 15.What is the standard form of 𝑦 = (3𝑥 + 1)(2𝑥 − 7)? A. 𝑦 = 6𝑥2 − 19𝑥 − 7 B. 𝑦 = 6𝑥2 + 19𝑥 − 7 C. 𝑦 = 6𝑥2 − 23𝑥 − 7 D. 𝑦 = 6𝑥2 + 19𝑥 − 7 Additional Activity Directions: Give 3 situations where polynomial function is found and write their standard form. 31
- 40. Summary A polynomial function is a function of the form 𝑃(𝑥) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + 𝑎𝑛−2𝑥𝑛−2 + ⋯ + +𝑎1𝑥 + 𝑎0, 𝑎𝑛 ≠ 0, where 𝑛 is a nonnegative integer , 𝑎0, 𝑎1, … , 𝑎𝑛 are real numbers called coefficients (numbers that appear in each term) , 𝑎𝑛𝑥𝑛 is the leading term (has the highest degree), 𝑎𝑛 is the leading coefficient, and 𝑎0 is the constant term (number without a variable). The highest power of the variable of 𝑃(𝑥) is known as its degree. When giving a final answer, you must write the polynomial function in standard form. Standard form means that you write the terms by decreasing exponents. Here’s what to do: 1. Write the term with the highest exponent first. 2. Write the terms with lower exponents in descending order. 3. Remember that a variable with no exponent has an understood exponent of 1. 4. A constant term always comes last. In writing polynomial function in Factored Form, make sure that it is factored completely. The following questions might help you to factor the polynomial functions completely. 1. Is there a common factor? 2. Is there a difference of squares? 3. Is there a perfect square trinomial? 4. Is there an expression of the form 𝑥2 + 𝑏𝑥 + 𝑐? 5. Are there factors of 𝑎𝑐 that add up to 𝑏? 32
- 41. Assessment: (Post-Test) Directions: Choose the letter that best answers each question. 1. Which of the following is the value of 𝑛 in 𝑓(𝑥) = 𝑥𝑛 if 𝑓 is a polynomial function? A. √2 B. −2 C. 2 D. 1 2 2. Which of the following is NOT a polynomial function? A. 𝑓(𝑥) = 2021 B. 𝑓(𝑥) = 19 C. 𝑓(𝑥) = 𝑥2 − 𝑥 D. 𝑓(𝑥) = √3 𝑥2 3. Which of the following is a polynomial function? i. 𝑓(𝑥) = 𝑥3 + 2𝑥 + 1 ii. 𝑓(𝑥) = 𝑥2 + 𝑥 + 1 iii. 𝑓(𝑥) = √2 𝑥2 + √𝑥 A. i only B. ii only C. i and ii D. i and iii 4. What is the leading term of 𝑓(𝑥) = 𝑥2 + 4𝑥3 + 1? A. x B. 2 C. 3 D. 4𝑥3 5. What is the constant term of the polynomial function in number 4? A. 1 B. 2 C. 3 D. 4 6. What is the standard form of 𝑓(𝑥) = (5𝑥 + 3)(25𝑥2 − 15𝑥 + 9)? A. −125𝑥3 − 27 B. 125𝑥3 − 27 C. −125𝑥3 + 27 D. 125𝑥3 + 27 7. What is the leading term of number 6? A. 27 B. −27 C. 125𝑥3 D. −125𝑥3 8. What is the constant term of the polynomial in number 6? A. 27 B. −27 C. 125𝑥3 D. −125𝑥3 33
- 42. 9. Given that 𝑓(𝑥) = 2𝑥−2𝑛 + 8𝑥2 , what value should be assigned to 𝑛 to make 𝑓 a function of degree 5? A. − 2 5 B. − 5 2 C. 2 5 D. 5 2 10. How should the polynomial function 𝑓(𝑥) = 𝑥4 − 8𝑥2 + 𝑥 2 + 1 2 + 4𝑥3 be written in standard form? A. 𝑓(𝑥) = −8𝑥2 + 1 2 + 4𝑥3 + 𝑥4 + 𝑥 2 B. 𝑓(𝑥) = 𝑥 2 − 8𝑥2 + 1 2 + 4𝑥3 + 𝑥4 C. 𝑓(𝑥) = 𝑥4 + 4𝑥3 − 8𝑥2 + 𝑥 2 + 1 2 D. 𝑓(𝑥) = 4𝑥3 + 1 2 − 8𝑥2 + 𝑥 2 + 𝑥4 11. What is the leading coefficient of number 10? A. −8 B. 1 C. 1 2 D. −4 12. What is the constant term of the polynomial in number 10? A. −8 B. 1 C. 1 2 D. −4 13. What is the factored form of 𝑦 = 9𝑥3 − 3𝑥2 + 81𝑥 − 27? A. 𝑦 = −3(𝑥2 + 9)(3𝑥 − 1) B. 𝑦 = 3(𝑥2 + 9)(3𝑥 − 1) C. 𝑦 = 3(𝑥2 − 9)(3𝑥 − 1) D. 𝑦 = 3(𝑥2 + 9)(3𝑥 + 1) 14. How should 𝑓(𝑥) = 𝑥4 + 𝑥3 + 𝑥2 + 𝑥 be written in factored form? A. 𝑓(𝑥) = 𝑥(𝑥 + 1)(𝑥2 + 1) B. 𝑓(𝑥) = 𝑥(1)(𝑥2 + 1) C. 𝑓(𝑥) = 𝑥(𝑥 − 1)(𝑥2 + 1) D. 𝑓(𝑥) = 𝑥(−1)(𝑥2 + 1) 15. What is the factored form of 𝑓(𝑥) = 𝑥3 + 3𝑥2 − 4𝑥 − 12? a. 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 2)(𝑥 + 3) b. 𝑓(𝑥) = (𝑥 + 2)(𝑥 + 2)(𝑥 + 3) c. 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 2)(𝑥 + 3) d. 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 2)(𝑥 − 3) 34
- 43. Answer Key What I Know (Pre-test) on page 2 1. B 6. B 11. B 2. D 7. C 12. C 3. B 8. D 13. A 4. D 9. B 14. A 5. A 10. C 15. B What’s New on page 7 Expression Polynomial or Not Reason/s 1. Polynomial 2. Not The variable of one term is inside the radical sign. 3. Polynomial 4. Not The exponent of the variable is not a whole number. 5. Not The variables appear in the denominator. 6. Polynomial 7. Not The exponent of the variable is not a whole number. 8. Polynomial 9. Not It has a negative exponent. 10. Polynomial What’s more (A) on page 9 Polynomial Function or Not Reason Degree Leading Coefficient Consta nt Term 1. Polynomial 0 None 0 2. Not The variable of one term is inside the radical sign. 3. Polynomial 1 -1 0 4. Not The exponent of the variable is not a whole number. 5. Not The variable appears in the denominator. 6. Polynomial 1 √5 0 7. Polynomial 2 1 0 8. Not It has a negative exponent. 9. Polynomial 3 1 1 10. Polynomial 2 -4 1 11. Polynomial 0 None 2020 12. Not The variable of one term is inside the radical sign. 13. Polynomial 1 3 √4 0 14. Polynomial 1 1 2 15. Polynomial 1 3 0 Lesson 1: What I Know on page 5 1. D 6. A 11. A 2. C 7. C 12. A 3. B 8. C 13. C 4. B 9. C 14. D 5. B 10. B 15. A 35
- 44. What’s More (B) on page 10 Polynomial Function or Not Degree of Polynomial Type of Polynomial Leading Term Constant Term 1. Polynomial 1 Linear x 0 2. Polynomial 2 Quadratic 4x 2 0 3. Not 4. Polynomial 2 Quadratic x 2 12 5. Polynomial 2 Quadratic -5x 2 10 6. Polynomial 0 Zero None 1 2 7. Not 8. Polynomial 2 Quadratic 1 2 x 2 −3 9. Polynomial 5 Quintic 𝑥 5 2 10. Not 11. Not 12. Polynomial 1 Linear 5 8 x 0 13. Polynomial 1 Linear 2𝑥 3 2 3 14. Polynomial 2 Quadratic 𝑥 2 0 15. Not What I Have Learned on page 10 1. polynomial function 2. nonnegative 3. degree 4. cubic 5. x 3 6. constant term 7. leading coefficient 8. 3 Polynomial Function or Not Reason Degree Leading Term, Coefficient Constant Term 9. Polynomial 0 None 20 10. Not The variable of one term is inside the radical sign. 11. Polynomial 1 −1991𝑥, −1991 None 12. Not The exponent of the variable is not a whole number. 13. Not The variables appear in the denominator. 14. Polynomial 1 √4𝑥 or 2𝑥, √4 or 2 None 15. Polynomial 2 𝑥 2 , 1 20 What I Can Do on page 11 Answers may vary. Additional Activity on page 14 Answers may vary. Assessment on page 12 1. A 6. C 11. B 2. C 7. D 12. B 3. B 8. B 13. B 4. B 9. D 14. C 5. A 10. A 15. A 36
- 45. Lesson 2 What I Know on page 16 1. C 6. D 11. A 2. B 7. C 12. B 3. B 8. C 13. A 4. D 9. B 14. B 5. A 10. C 15. B What’s New on page 21 Polynomial Function Term with highest exponent Term/s with lower exponents in descending order Constant term 1. 𝑥 4 −4𝑥 2 −45 2. 3𝑥 3 6𝑥 2 ; 4𝑥 0 3. 5𝑥 4 𝑥 3 ; −2𝑥 −5 4. −11𝑥 4 9𝑥 2 2 5. 2𝑥 3 −8𝑥 2 ; 6𝑥 0 What’s more (A) on page 26 Polynomial Function Term with highest exponent Term/s with lower exponents in descending order Constant term Standard form 1. 4𝑥 4 8𝑥 4 𝑓 ( 𝑥 ) = 4𝑥 4 + 8𝑥 + 4 2. 𝑥 2 0 −4 𝑓 ( 𝑥 ) = 𝑥 2 − 4 3. 𝑥 3 2𝑥 1 𝑦 = 1 + 2𝑥 + 𝑥 3 4. 5𝑥 10 5𝑥 5 −5 𝑦 = 5𝑥 10 + 5𝑥 5 − 5 5. −9𝑥 5 𝑥 2 6 𝑓 ( 𝑥 ) = 𝑥 2 − 9𝑥 5 + 6 What’s more (B) on page 26 1. 𝑦 = 343𝑥 3 + 27 Find 𝑎 and 𝑏 (𝑎 is the cube root of the first term) (𝑏 is the cube root of the second term) 𝑎 = 7𝑥 𝑏 = 3 Substitute the values of 𝑎 and 𝑏 in (𝑎 + 𝑏)(𝑎 2 − 𝑎𝑏 + 𝑏 2 ) 𝑦 = ( 7𝑥 + 3 ) [(7𝑥) 2 − 2(7𝑥)(3) + ( 3 ) 2 ] So the factored form is 𝒚 = ( 𝟕𝒙 + 𝟑 ) (𝟒𝟗𝒙 𝟐 − 𝟐𝟏𝒙 + 𝟗) 2. 𝑦 = 27𝑥 3 − 8 Find 𝑎 and 𝑏 (𝑎 is the cube root of the first term) (𝑏 is the cube root of the second term) 𝑎 = 3𝑥 𝑏 = 2 Substitute the values of 𝑎 and 𝑏 in (𝑎 − 𝑏)(𝑎 2 + 𝑎𝑏 + 𝑏 2 ) 𝑦 = ( 3𝑥 − 2 ) [(3𝑥) 2 + (3𝑥)(2) + ( 2 ) 2 ] So the factored form is 𝒚 = ( 𝟑𝒙 − 𝟐 ) (𝟗𝒙 𝟐 + 𝟔𝒙 + 𝟒) 37
- 46. 38 3. 𝑦 = 𝑥 3 + 3𝑥 2 − 4𝑥 − 12 Group the terms (𝑎𝑥 3 + 𝑏𝑥 2 ) + (𝑐𝑥 + 𝑑) 𝑦 = (𝑥 3 + 3𝑥 2 ) + (−4𝑥 − 12) Factor 𝑥 2 out of the first group of terms. Factor the constants out of both groups. 𝑦 = 𝑥 2 ( 𝑥 + 3 ) + (−4𝑥 − 12) 𝑦 = 𝑥 2 ( 𝑥 + 3 ) − 4(𝑥 + 3) Add the two terms by adding the coefficients 𝑦 = (𝑥 2 − 4)(𝑥 + 3) This can be further factored as a difference of squares 𝑦 = (𝑥 + 2)(𝑥 − 2)(𝑥 + 3) So the factored form is 𝒚 = (𝒙 + 𝟐)(𝒙 − 𝟐)(𝒙 + 𝟑) 4. 𝑦 = 𝑥 4 − 5𝑥 2 + 4 Treat 𝑎 4 as (𝑎 2 ) 2 (𝑥 2 ) 2 − 𝑏(𝑥 2 ) + 𝑐 𝑦 = (𝑥 2 ) 2 − 5(𝑥 2 ) + 4 Let 𝑥 2 = 𝑥, thus, it shows a quadratic trinomial: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Factor it: 𝑦 = 𝑥 2 − 5𝑥 + 4 𝑦 = (𝑥 − 4)(𝑥 − 1) Put it back. (Substitute 𝑥 = 𝑥 2 ) 𝑦 = (𝑥 2 − 4)(𝑥 2 − 1) This can be further factored as a difference of Two squares 𝑦 = (𝑥 + 2)(𝑥 − 2)(𝑥 + 1)(𝑥 − 1) So the factored form is 𝒚 = (𝒙 + 𝟐)(𝒙 − 𝟐)(𝒙 + 𝟏)(𝒙 − 𝟏) What’s more (C) on page 27 What I Have Learned on page 28 1. B 6. F, H 2. E 7. M, J 3. D 8. I, O 4. C 9. K, N 5. A 10. G, L What I Can Do on page28 1. 𝐿 ( 𝑡 ) = −0.01𝑡 3 + 0.4𝑡 2 + 0.3𝑡 + 10 2. 𝑓 ( 𝑟 ) = 3𝑟 2 − 3𝑟 + 1 Assessment on page 29 1. C 6. A 11. C 2. B 7. B 12. B 3. B 8. C 13. A 4. D 9. B 14. B 5. A 10. A 15. A Additional Activity on page 31 Answers may vary. 1. 𝑦 = 𝑥 4 − 512𝑥 = 𝑥(𝑥 3 − 512) = 𝑥(𝑥 − 8)(𝑥 2 + 8𝑥 + 64) 2. 𝑦 = 9𝑥 3 − 36𝑥 2 + 4𝑥 − 16 = (9𝑥 3 − 36𝑥 2 ) + (4𝑥 − 16) = 9𝑥 2 ( 𝑥 − 4 ) + 4 ( 𝑥 − 4 ) = (9𝑥 2 + 4) ( 𝑥 − 4 ) Assessment (Post-Test) on page 33 1. C 6. D 11. B 2. D 7. D 12. C 3. C 8. A 13. B 4. D 9. B 14. A 5. A 10. C 15. A
- 47. References • Admin, Unknown. “Polynomial Functions- Definition, Formula, Types and Graph With Examples.” BYJUS. BYJU'S, January 7, 2020. https://byjus.com/maths/polynomial-functions/. • Admin, Unknown. “Polynomial Functions- Definition, Formula, Types and Graph With Examples.” BYJUS. BYJU'S, January 7, 2020. https://byjus.com/maths/polynomial-functions/. • Gloag, Andrew, Melissa Kramer, and Anne Gloag. “Polynomials in Standard Form.” CK. CK-12 Foundation, November 20, 2019. https://www.ck12.org/c/algebra/polynomials-in-standard- form/lesson/Polynomials-in-Standard-Form-BSC-ALG/. • “Polynomial.” Merriam-Webster. Merriam-Webster. Accessed June 23, 2020. https://www.merriam-webster.com/dictionary/polynomial. • “Writing Polynomials in Standard Form.” Math. Accessed June 23, 2020. https://www.softschools.com/math/algebra/topics/writing_polynomials_i n_standard_form/. • SparkNotes. SparkNotes. Accessed June 23, 2020. https://www.sparknotes.com/math/algebra2/factoring/section2/. • SparkNotes. SparkNotes. Accessed June 23, 2020. https://www.sparknotes.com/math/algebra2/factoring/section3/. 39
- 48. For inquiries and feedback, please write or call: Department of Education –Learning Resources Management and Development Center (LRMDC) DepEd Division of Bukidnon Fortich St. Sumpong, Malaybalay City, Bukidnon Telefax: ((08822)855-0048 E-mail Address: bukidnon@deped.gov.ph