Mathematics Grade 10

DEPED COPYGLOSSARY OF TERMS AND LIST OF THEOREMS USED IN THISMODULE:Degree of a Polynomial – the highest degree of a term in a polynomialFactor Theorem - the polynomial P(x) has x – r as a factor if and only if P(r) = 0.Mathematical Model – is a mathematical representation of some phenomena in real worldPolynomial – an algebraic expression of the form anxn + an – 1xn – 1 + an – 2xn – 2 + … + a1x + a0, where an  0, and a0, a1, a2, …, an are real numbersRational Root Theorem - Let an – 1xn – 1 + an – 2xn – 2 + … + a1x + a0 = 0 be a polynomial equation of degree n. If p , in lowest terms, is a rational q root of the equation, then p is a factor of a0 and q is a factor of an.Remainder Theorem - If the polynomial P(x) is divided by (x – r), the remainder R is a constant and is equal to P(r).Synthetic Division – a short method of dividing polynomial expressions using only the coefficient of the terms 80 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYREFERENCES AND WEBSITE LINKS USED IN THIS MODULE: Acelajado M. J., Beronque, Y. B. & Co, F. F. (2005) Algebra Concepts and Processes. (3rd Ed.), Mandaluyong City: National Book Store. Coronel, I. C., Coronel, A. C. & Munsayac, J. M. (2013) Growing Up with Math. Quezon City: FNB Educational, Inc. Coronel, I. C., Villano, L. M., Manalastas, P. R., Marasigan, J. A. & Coronel, A. C. (2004) Advanced Algebra Trigonometry and Statistic. Quezon City: Bookmark Inc. Coronel, I. C., Villano, L. M., Manalastas, P.R., Marasigan, J.A. & Coronel, A.C. (1992) Mathematics and integrated Approach (SEDP). Philippines: Bookmar Inc. De Leon, C. M. & Bernabe, J. G. (2002). Elementary Algebra Textbook for First Year. (Pilot Edition) Quezon City: JTW Publishing Co., Dilao, S. J., Orines, F. B. & Bernabe, J. C. (2003). Advanced Algebra, Trigonometry and Statistics. Philippines: JTW Publishing Co. Green, J. W., Ulep, S. A., Gallos, F. L., & Umipig, D. F. (n. d.) Teaching Mathematics IV Volume I. Philippines – Australia Science and Mathematics Educational Project. Hadlay, W. S., Pfluger, J. & Coratto, M. (2006) Algebra 1 Student Text. USA: Carnegie Learning Pittsburgh. Larson, R. & Hostetler, R.P. (2013). Algebra and Trigonometry. (8th Ed.) Cergage Learning Asia Ple Ltd. Orines, F. B., Esparrago, M. S. & Reyes, N. V. (2008) Advanced Algebra, Trigonometry and Statistics. Quezon City: Phoenix Publishing House. Oronce, O. A. & Mendoza, M. O. (2003) Exploring Mathematics Advanced Algebra and trigonometry. (1st Ed). Manila: Rex Printing, Company, Inc. Quimpo, N.F. (2005) A Course in Freshman Algebra. Mathematics Department Loyola Schools. Ateneo de Manila University. Villanueva, T. T. & Vanzandt, G. E.(2009) Hands-on Minds-on Activities in Mathematics IV. Advanced Algebra, Trigonometry and Statistics. Philippines: St. Jude Thaddeus Publication. Website Links http://www.mathsisfun.com/algebra/polynomials-division-long.html http://www.youtube.com/watch?v=qd-T-dTtnX4 http://www.purplemath.com/modules/polydiv2.htm https://www.brightstorm.com/math/algebra-2/factoring/rational-roots-theorem/ http://www.youtube,com/watch?v=RXKfaQemtii 81 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

VISIT DEPED TAMBAYANhttp://richardrrr.blogspot.com/1. Center of top breaking headlines and current events related to Department of Education.2. Offers free K-12 Materials you can use and share 10DEPED COPY Mathematics Teacher’s Guide Unit 2 This book was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at [email protected]. We value your feedback and recommendations. Department of Education Republic of the Philippines All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Mathematics – Grade 10Teacher’s GuideFirst Edition 2015 Republic Act 8293, section 176 states that: No copyright shall subsist in any workof the Government of the Philippines. However, prior approval of the government agency oroffice 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 ofroyalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names,trademarks, etc.) included in this book are owned by their respective copyright holders.DepEd is represented by the Filipinas Copyright Licensing Society (FILCOLS), Inc. in seekingpermission to use these materials from their respective copyright owners. . All means havebeen exhausted in seeking permission to use these materials. The publisher and authors donot represent nor claim ownership over them. Only institutions and companies which have entered an agreement with FILCOLSand only within the agreed framework may copy this Teacher’s Guide. Those who have notentered in an agreement with FILCOLS must, if they wish to copy, contact the publishers andauthors directly. Authors and publishers may email or contact FILCOLS at [email protected] or(02) 439-2204, respectively.Published by the Department of EducationSecretary: Br. Armin A. Luistro FSCUndersecretary: Dina S. Ocampo, PhDDEPED COPY Development Team of the Teacher’s GuideConsultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, andRosemarievic Villena-Diaz, PhDAuthors: Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D.Cruz, Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B.Orines, Rowena S. Perez, and Concepcion S. TernidaEditor: Maxima J. Acelajado, PhDReviewers: Carlene P. Arceo, PhD, Rene R. Belecina, PhD, Dolores P.Borja, Maylani L. Galicia, Ma. Corazon P. Loja, Jones A. Tudlong, PhD, andReymond Anthony M. QuanIllustrator: Cyrell T. NavarroLayout Artists: Aro R. Rara, Jose Quirovin Mabuti, and Ronwaldo Victor Ma.A. PagulayanManagement and Specialists: Jocelyn DR Andaya, Jose D. Tuguinayo Jr.,Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel Jr.Printed in the Philippines by REX Book StoreDepartment of Education-Instructional Materials Council Secretariat (DepEd-IMCS)Office Address: 5th Floor Mabini Building, DepEd Complex Meralco Avenue, Pasig City Philippines 1600Telefax: (02) 634-1054, 634-1072E-mail Address: [email protected] All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Introduction This Teacher’s Guide has been prepared to provide teachers of Grade 10 Mathematics with guidelines on how to effectively use the Learner’s Material to ensure that learners will attain the expected content and performance standards. This book consists of four units subdivided into modules which are further subdivided into lessons. Each module contains the content and performance standards and the learning competencies that must be attained and developed by the learners which they could manifest through their products and performances. The special features of this Teacher’s Guide are: A. Learning Outcomes. Each module contains the content and performance standards and the products and/ or performances expected from the learners as a manifestation of their understanding. B. Planning for Assessment. The assessment map indicates the type of assessment and categorized the objectives to be assessed into knowledge, process/skills, understanding, and performance C. Planning for Teaching-Learning. Each lesson has Learning Goals and Targets, a Pre-Assessment, Activities with answers, What to Know, What to Reflect on and Understand, What to Transfer, and Summary / Synthesis / Generalization. D. Summative Test. After each module, answers to the summative test are provided to help the teachers evaluate how much the learners have learned. E. Glossary of Terms. Important terms in the module are defined or clearly described. F. References and Other Materials. This provides the teachers with the list of reference materials used, both print and digital. We hope that this Teacher’s Guide will provide the teachers with the necessary guide and information to be able to teach the lessons in a more creative, engaging, interactive, and effective manner. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Table of Contents Curriculum Guide: Mathematics Grade 10 Unit 2 Module 3: Polynomial Functions................................................................ 82 Learning Outcomes ..............................................................................................82 Planning for Assessment......................................................................................83 Planning for Teaching-Learning ...........................................................................86 Pre-Assessment ...................................................................................................87 Learning Goals and Targets .................................................................................87 Activity 1 ....................................................................................................88 Activity 2 ....................................................................................................89 Activity 3 ....................................................................................................90 Activity 4 ....................................................................................................90 Activity 5 ....................................................................................................91 Activity 6 ....................................................................................................91 Activity 7 ....................................................................................................92 Activity 8 ....................................................................................................94 Activity 9 ....................................................................................................99 Activity 10 ................................................................................................100 Activity 11 ................................................................................................101 Activity 12 ................................................................................................102 Activity 13 ................................................................................................106 Activity 14 ................................................................................................107 Summary/Synthesis/Generalization ...................................................................108 Summative Test .......................................................................................................109 Glossary of Terms...................................................................................................114 References Used in This Module ........................................................................115 Module 4: Circles ........................................................................................... 116 Learning Outcomes ............................................................................................116 Planning for Assessment....................................................................................117 Planning for Teaching-Learning .........................................................................123 Pre-Assessment .................................................................................................125 Learning Goals and Targets ...............................................................................126 Lesson 1A: Chords, Arcs, and Central Angles................................................126 Activity 1 ..................................................................................................127 Activity 2 ..................................................................................................128 Activity 3 ..................................................................................................129 Activity 4 ..................................................................................................130 Activity 5 ..................................................................................................131 Activity 6 ..................................................................................................132 Activity 7 ..................................................................................................132 Activity 8 ..................................................................................................132 Activity 9 ..................................................................................................133 Activity 10 ................................................................................................136 Activity 11 ................................................................................................136 Activity 12 ................................................................................................137 Activity 13 ................................................................................................138 Summary/Synthesis/Generalization ...................................................................139 Lesson 1B: Arcs and Inscribed Angles.............................................................139 Activity 1 ..................................................................................................140 All rights reserved. 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DEPED COPY Activity 2 ..................................................................................................141 Activity 3 ..................................................................................................142 Activity 4 ..................................................................................................143 Activity 5 ..................................................................................................144 Activity 6 ..................................................................................................145 Activity 7 ..................................................................................................145 Activity 8 ..................................................................................................146 Activity 9 ..................................................................................................148 Activity 10 ................................................................................................151 Activity 11 ................................................................................................153 Activity 12 ................................................................................................154 Summary/Synthesis/Generalization ...................................................................154 Lesson 2A: Tangents and Secants of a Circle ................................................155 Activity 1 ..................................................................................................155 Activity 2 ..................................................................................................159 Activity 3 ..................................................................................................160 Activity 4 ..................................................................................................161 Activity 5 ..................................................................................................162 Activity 6 ..................................................................................................163 Activity 7 ..................................................................................................164 Activity 8 ..................................................................................................172 Summary/Synthesis/Generalization ...................................................................172 Lesson 2B: Tangent and Secant Segments .....................................................173 Activity 1 ..................................................................................................173 Activity 2 ..................................................................................................174 Activity 3 ..................................................................................................174 Activity 4 ..................................................................................................175 Activity 5 ..................................................................................................175 Activity 6 ..................................................................................................176 Activity 7 ..................................................................................................176 Activity 8 ..................................................................................................177 Activity 9 ..................................................................................................179 Activity 10 ................................................................................................180 Summary/Synthesis/Generalization ...................................................................180 Summative Test .......................................................................................................181 Glossary of Terms...................................................................................................189 List of Theorems and Postulates on Circles....................................................191 References and Website Links Used in This Module ....................................193 Module 5: Plane Coordinate Geometry .................................................. 198 Learning Outcomes ............................................................................................198 Planning for Assessment....................................................................................199 Planning for Teaching-Learning .........................................................................205 Pre-Assessment .................................................................................................207 Learning Goals and Targets ...............................................................................207 Lesson 1: The Distance Formula, the Midpoint Formula, and the Coordinate Proof....................................................................207 Activity 1 ..................................................................................................208 Activity 2 ..................................................................................................208 Activity 3 ..................................................................................................209 Activity 4 ..................................................................................................210 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Activity 5 ..................................................................................................212 Activity 6 ..................................................................................................212 Activity 7 ..................................................................................................213 Activity 8 ..................................................................................................215 Activity 9 ..................................................................................................216 Activity 10 ................................................................................................217 Activity 11 ................................................................................................220 Summary/Synthesis/Generalization ...................................................................221 Lesson 2: The Equation of a Circle ....................................................................221 Activity 1 ..................................................................................................221 Activity 2 ..................................................................................................222 Activity 3 ..................................................................................................223 Activity 4 ..................................................................................................225 Activity 5 ..................................................................................................226 Activity 6 ..................................................................................................227 Activity 7 ..................................................................................................227 Activity 8 ..................................................................................................228 Activity 9 ..................................................................................................228 Activity 10 ................................................................................................229 Summary/Synthesis/Generalization ...................................................................230 Summative Test .......................................................................................................231 Glossary of Terms...................................................................................................237 References and Website Links Used in This Module ....................................238 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Module 3: Polynomial FunctionsA. Learning Outcomes Content Standard: The learner demonstrates understanding of key concepts of polynomial functions. Performance Standard: The learner is able to conduct systematically in different fields a mathematical investigation involving polynomial functions. DEPED COPY Unpacking the Standards for UnderstandingSubject: Mathematics 10 Learning CompetenciesQuarter: Second Quarter 1. Illustrate polynomial functionsTOPIC: Polynomial 2. Graph polynomial functions Functions 3. Solve problems involvingLesson: polynomial functionsIllustrating PolynomialFunctions, Graphs of Essential EssentialPolynomial Functions and Understanding: Question:Solutions of ProblemsInvolving Polynomial Students will How do theFunctions understand that mathematical polynomial concepts help Writer: functions are solve real-life useful tools in problems that can Elino Sangalang Garcia solving real-life be represented problems and in as polynomial making decisions functions? given certain constraints. 82 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Transfer Goal: Students will be able to apply the key concepts of polynomial functions in finding solutions and making decisions for certain life problems.B. Planning for AssessmentProduct/PerformanceThe following are products and performances that students areexpected to come up with in this module.1. Write polynomial functions in standard form2. List all intercepts of polynomial functions written in both standard and factored forms3. Make a list of ordered pairs of points that satisfy a polynomial function4. Make a table of signs for polynomial functions5. Make a summary table of properties of the graph of polynomial functions (behavior, number of turning points, location relative to the x-axis)6. Formulate and solve real-life problems applying polynomial functions7. Sketch plans or designs of objects that illustrate polynomial functionsg. Create concrete objects as products of applying solutions to problems involving polynomial functions (e.g. rectangular open box, candle mold)DEPED COPY Assessment Map TYPE KNOWLEDGE PROCESS/ UNDERSTANDING PERFORMANCE Part I SKILLS Part I Pre- Part IIAssessment/ Part I Products and Diagnostic Illustrating Illustrating Graphing performances polynomial polynomial polynomial related to or functions functions functions involving (Recalling the (Recalling (Describing the quadratic definition of the definition properties of functions polynomial of polynomial graphs of (Solving area functions and functions and polynomial problems) the terms the terms functions) associated associated with it) with it) Solving problems involving Graphing polynomial polynomial functions functions (Describing the properties of graphs of polynomial functions) 83 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

TYPE KNOWLEDGE PROCESS/ UNDERSTANDING PERFORMANCEFormative Quiz 1: SKILLS Quiz 3: Quiz 2: Illustrating Graphing Graphing polynomial polynomial polynomial functions functions functions (Writing (Finding the (Preparing table polynomial intercepts of of signs) functions in polynomial (Describing the standard form functions) behavior of the and in (Finding graph using the factored form) additional Leading points on the Coefficient Test) Quiz 4: graph of a polynomial Quiz 6: function) Quiz 5:DEPED COPYSummative Graphing Graphing Solving problems Part IIAssessment polynomial polynomial involving functions functions polynomial Products and functions performances (Identifying (Sketching (Solving real-life related to or the number of the graph of problems that involving turning points polynomial apply polynomial polynomial and the functions functions) functions behavior of using all (Solving the graph properties) Part I problems based on Graphing related to multiplicity of Solving polynomial volume of an zeros) problems functions open involving (Describing the rectangular box) (Sketching the polynomial properties of the graph of functions graph of polynomial polynomial functions Part I functions) using all Solving problems properties) Illustrating involving Part I polynomial polynomial functions functions Illustrating (Recalling polynomial the definition functions of polynomial (Recalling the functions and definition of the terms polynomial associated functions and with it) the terms associated Graphing with it) polynomial functions 84 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

TYPE KNOWLEDGE PROCESS/ UNDERSTANDING PERFORMANCE SKILLS Self-Assessment (Describing the (optional) properties of the graphs of polynomial functions) Journal Writing: Expressing understanding of polynomial functions, graphing polynomial functions, and solving problems involving polynomial functionsAssessment Matrix (Summative Test)DEPED COPY Levels of What will I assess? How will I How Will I Score?Assessment assess?Knowledge 15% The learner Paper and 1 point for every demonstrates Pencil Test correct response understanding of key concepts of Part I items 1, 2, polynomial functions. and 3Process/Skills Illustrate polynomial Part I items 4, 5, 1 point for every 25% functions. 6, 7, and 8 correct responseUnderstanding Graph polynomial 30% functions Part I items 9, 1 point for every 10, 11, 12, 13, correct response Solve problems involving polynomial and 14 functions The learner is able to Part II Rubric for the Solution conduct systematically (6 points) to the Problem a mathematical Criteria: investigation involving  Use of polynomial polynomial functions in different fields. function as model  Use of appropriate Product/ Solve problems involving polynomial mathematicalPerformance functions. concept 30%  Correctness of the final answer Rubric for the Output (Open Box) Criteria:  Accuracy of measurement (Dimensions)  Durability and Attributes 85 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYC. Planning for Teaching-Learning Introduction This module is a one-lesson module. It covers key concepts of polynomial functions. It is composed of fourteen (14) activities, three (3) of which are for illustration of polynomial functions, nine (9) are for graphing polynomial functions, and two (2) are for solving real-life problems involving polynomial functions. The lesson as incorporated in the activities is designed for the students to: 1. define polynomial functions and the terms associated with it; 2. write polynomial functions in standard and factored form; 3. write polynomial functions in standard form given real numbers as coefficients and exponents; 4. recall and apply the different theorems in factoring polynomials to determine the x-intercepts; 5. determine more ordered pairs that satisfy a polynomial function; 6. investigate and analyze the properties of the graphs of polynomial functions (like end behaviors, behaviors relative to the x-axis, number of turning points, etc.); and 7. solve real-life problems (like area and volume, deforestation, revenue-advertising expense situations, etc.) that apply polynomial functions. One of the essential targets of this module is for the students to manually sketch the graph of polynomial functions which later on can be verified and validated with some graphing utilities like Grapes, GeoGebra, or even Geometer’s Sketchpad. In dealing with each activity of this lesson, the students are given the opportunity to use their prior knowledge and required skills in previous tasks. They are also given varied activities to process the knowledge and skills learned and further deepen and transfer their understanding of the different lessons. Lastly, 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 (for examples, slow/fast learners, and localized situations/examples). 86 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY As an introduction to the main lesson, show the students the picture mosaic below, then ask them the question that follows: In this mosaic picture, can you see some mathematical representations? Give some. Motivate the students to find out the answers and to determine the essential applications of polynomial functions through this module. Objectives: After the learners have gone through this module, they are expected to: 1. illustrate polynomial functions; 2. graph polynomial functions; and 3. solve problems involving polynomial functions. PRE-ASSESSMENT: Check students’ prior knowledge, skills, and understanding of mathematics concepts related to polynomial functions. Assessing these will facilitate your teaching and the students’ understanding of the lessons in this module. LEARNING GOALS AND TARGETS: Students are expected to demonstrate understanding of key concepts of polynomial functions, formulate real-life problems involving these concepts, and solve these using a variety of strategies. They are also expected to investigate mathematical relationships in various situations involving polynomial functions. 87 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Answer KeyPart I: Part II.1. B 8. B (Use the rubric to rate students’ work/output)2. C 9. A Solution to the problem3. A 10. A Since P  2l  2w , then 36  2l  2w or 18  l  w , and4. D 11. D w  18  l .5. A 12. D The lot area can be expressed as A(l )  l(18  l ) or6. D 13. A A(l )  18l  l 2 .7. C 14. A A(l )  (l 2  18l ) A(l )  (l 2  18l  81)  81DEPED COPY A(l )  (l  9)2  81, in vertex form. Therefore, l  9 meters and w  18  l  18  9  9 meters, yielding the maximum area of 81 square meters.What to KNOW The students need first to recall the concept of polynomialexpressions. These will lead them to define and illustrate mathematicallythe polynomial functions.Activity 1: Which is which?Answer Key 1. polynomial 2. not polynomial because the variable of one term is inside the radical sign 3. polynomial 4. not polynomial because the exponents of the variable are not whole numbers 5. not polynomial because the variables are in the denominator 6. polynomial 7. not polynomial because the exponent of one variable is not a whole number 8. polynomial 9. not polynomial because the exponent of one variable is negative 10. polynomial 88 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Let this activity be the starting point of defining a polynomialfunction as follows:A polynomial function is a function of the formP(x)  an x n  an1x n1  an2 x n2  ...  a1x  a0 , an  0,where n is a nonnegative integer, a0, a1, ..., an are real numbers calledcoefficients, an xn is the leading term, an is the leading coefficient,and a0 is the constant term.DEPED COPYOther notations: f (x)  an x n  an1x n1  an2x n2  ...  a1x  a0 or y  an x n  an1x n1  an2x n2  ...  a1x  a0 ,Activity 2: Fix and Move Them, Then Fill Me UpAnswer Key Polynomial Function Polynomial Function in Degree Leading Constant Standard Form Coefficient Term1. f (x)  2 11x  2x22. f (x)  2x3  5 15x f (x)  2x 2 11x  2 2 2 2 3 5 33 f (x)  2x3  15x  5 3 2 33. y  x(x2  5) 33 3 04. y  x(x  3)(x  3)5. y  (x  4)(x  1)(x 1)2 y  x3  5x 1 0 y  x3  9x 3 -1 4 y  x4  3x3  5x2  3x  4 4 1 89 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 3: Be a Polynomial Function ArchitectAnswer Key 4. f (x)  7 x3  1 x2  2x 1. f (x)  2x 3  7 x 2  1 x 46 46 2. f (x)  2x 3  1 x 2  7 x 5. f (x)   1 x3  7 x2  2x 64 64 3. f (x)  7 x3  2x2  1 x 46 6. f (x)   1 x3  2x2  7 x 64DEPED COPY The answers above are expected to be given by the students. Inaddition, instruct them to classify each polynomial according to thedegree. Also, let them identify the leading coefficient and the constantterm.What to PROCESS In this section, the students need to revisit the lessons and theirknowledge on evaluating polynomials, factoring polynomials, solvingpolynomial equations, and graphing by point-plotting.Activity 4: Do you miss me? Here I Am AgainAnswer Key 6. y  x(x  3)(x  4) 7. y  (x  2)(x  2)(x 2  4) 1. (x 1)x  3(x  2) 2. x  3x  2(x  3)(x  3) 8. y  2(x 1)(x  1)(x  1)(x  3) 9. y  x(x 1)(x 1)(x  3)(x  3) 3. (2x - 3)x -1(x - 3) 10. y  (2x  3)(x 1)(x  2)(x  3) 4. (x  2)(x  2)(x  3) 5. (2x  3)(x 1)(x  2)(x  3) The preceding task is very important for the students because ithas something to do with the x-intercepts of a graph. These are the x-values when y = 0, and, thus the point(s) where the graph intersects thex-axis can be determined. 90 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 5: Seize Me and Intercept MeAnswer Key -4, 3 -3, 3 1. x-intercepts: 0, 1, -3 2. x-intercepts: 2, -1, -3 3. x-intercepts: 1, -2 4. x-intercepts: 2, 1, -1, 5. x-intercepts: 0,Activity 6: Give Me More CompanionsDEPED COPYAnswer Key 1. x-intercepts: -4, -2, 1, 3 y-intercept: 24 x -5 -3 0 24 y 144 -24 24 -24 144ordered pairs: (-5,144), (-4,0), (-3, -24), (-2,0), (0,24), (1,0), (2-24), (3,0), (4,144)2. x-intercepts: -5,  3 , 2, 4 2y-intercept: -90 x -6 -4 -0.5 3 5 y -720 240 -101.2 72 -390ordered pairs: (-6, -720), (-5, 0), (-4, 240), (  3 , 0), (-0.5, 101.2), 2 (2, 0), (3, 72), (4, 0), (5, -390)3. x-intercepts: -6, 0, 4 y-intercept: 0 3 x -7 -3 1 2 y 175 -117 7 -32 4ordered pairs: (-7,175), (-6,0), (-3,-117), (0,0), (1,7), ( 3 ,0), 91 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

(2,-32) 4. x-intercepts: -3, -1, 0, 1, 3 y-intercept: 0 x -4 -2 -0.5 0.5 2 4 y 1680 -60 1.64 1.64 -60 1680 ordered pairs: (-4,1680), (-3, 0), (-2, -60), (-1, 0), (-0.5, 1.64), (0, 0), (0.5, 1.64), (1, 0), (2, -60), (3, 0), (4, 1680)Activity 7: What is the destiny of my behavior?DEPED COPYAnswer KeyValue Value Relation of y-value to Location of the Point of x of y 0: (x,y): above the x- -5 144 y  0, y  0, or y  0 ? axis, on the x-axis, or below the x-axis? -4 0 y 0 above the x-axis -3 -24 y=0 on the x - axis -2 0 below the x-axis 0 24 y 0 on the x - axis 1 0 y=0 above the x-axis 2 -24 y 0 on the x - axis 3 0 below the x-axis 4 144 y=0 on the x - axis y 0 above the x-axis y=0 y 0Answers to the Questions: 1. (-4,0), (-2,0), (1,0), and (3,0) 2. The graph is above the x-axis. 3. The graph is below the x-axis. 4. The graph is above the x-axis. 5. The graph is below the x-axis. 6. The graph is above the x-axis. Show the students how to prepare a simpler but similar table, thetable of signs. 92 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Example: The roots of the polynomial function y  (x  4)(x  2)(x 1)(x  3)are x = -4, - 2, 1, and 3 . These are the only values of x where the graphwill cross the x-axis. These roots partition the number line into intervals.Test values are then chosen from within each interval. Test Value x  4 Intervals 1 x  3 x 3 x4 -5  4  x  2  2  x  1 2 4 x2 – + + x 1 – -3 0 + + x 3 – ++ + + – –+ – +y  (x  4)(x  2)(x 1)(x  3) + –– – + ––position of the curve above –+ aboverelative to the x-axisDEPED COPY below above below Give emphasis that at this level, though, we cannot yet determinethe turning points of the graph. We can only be certain that the graph iscorrect with respect to intervals where the graph is above, below, or onthe x-axis as shown on the next page. 93 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 8: Sign on and Sketch MeAnswer Key1. y  (2x  3)(x 1)(x  4)(a)  3 , 1, 4 2(b) x   3 ,  3  x  1, 1 x  4, x4 2 2DEPED COPY(c) Intervals x  3 3  x 1 1 x  4 x4 2 2 5 Test Value -2 02 + - + 2x  3 - ++ + x 1 - + – -+ x4 abovey  (2x  3)(x 1)(x  4) below -- position of the curve +– relative to the x-axis above below(d) 2. y  x3  2x 2 11x 12 or y  (x  3)(x 1)(x  4) (a) -3, 1, 4 (b) x  3 ,  3  x  1, 1 x  4 , x  4 94 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

(c) Test Value x  3 Intervals x4 3  x 1 1 x  4 5 x 3 -4 + - 02 + x 1 - ++ + x4 - -+ -- -y  (x  3)(x 1)(x  4) + -+position of the curve above below above belowrelative to the x-axisDEPED COPYNote: Observe that there is one more factor, -1, that affects the final sign of y. For example, under x  3 , the sign of y is positive because -(-)(-)(-) = + .(d)3. y  x4  26x2  25 or y  (x  5)(x  1)(x 1)(x  5)(a) -5, -1, 1, 5(b) x  5 ,  5  x  1, 1 x  1, 1 x  5 , x  5(c) Intervals x  5  5  x  1 1 x  1 1 x  5 x  5 Test Value -6 -2 0 26 x5 - + + ++ x 1 - - + ++ x 1 - - - ++ x 5 - - - -+ + – + –+y  x 4  26x2  25 above below above below aboveposition of thecurve relative tothe x-axis 95 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

(d)DEPED COPY4. y  x 4  5x3  3x 2 13x 10ory  (x  5)(x  2)(x 1)2(a) -5, -2, 1  2  x  1, x 1(b) x  5 ,  5  x  2,(c)Test Value x  5 Intervals x 1  5  x  2  2  x  1 2 x 5 -6 + - -3 0 + x2 - ++ -+ + (x 1)2 + ++y  (x  5)(x  2)(x 1)2 - + --position of the curve below above below belowrelative to the x-axis Note: Observe that there is one more factor, -1, that affects the final sign of y. For example, under x  5 , the sign of y is negative because -(-)(-)(+) = - . .(d) 96 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

5. y  x 2(x  3)(x 1)4 (x 1)3 (a) -3, -1, 0, 1 (b) x  3 ,  3  x  1, 1 x  0 , 0  x  1, x  1 (c) Test Value x  3 Intervals 0 x 1 x 1 x2 -4 0.5 2  3  x  1 1 x  0 x3 + + + (x 1)4 - -2 -0.5 + + + + + (x 1)3 ++ -y  x2(x  3)(x 1)4 (x 1)3 ++ + ++ position of the curve relative to the x-axis aboveDEPED COPY -- -+ –– –+ below below below above (d) Broken parts of the graph indicate that somewhere below, they are connected. The graph goes downward from (-1,0) and at a certain point, it turns upward to (-3,0).Answers to the Questions:1. For y  (2x  3)(x 1)(x  4)a. Since there is no other x-intercept to the left of  3 , then the 2 graph falls to the left continuously without end.b. (i)  3  x  1 and x4 (ii) x  3 and 1 x  4 2 2c. Since there is no other x-intercept to the right of 4, then the graph rises to the right continuously without end.d. leading term: 2x3e. leading coefficient: 2, degree: 3 97 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

2. For y  x3  2x 2  11x 12 or y  (x  3)(x 1)(x  4)a. Since there is no other x-intercept to the left of -3, then thegraph rises to the left continuously without end.b. (i) x  3 and 1 x  4 (ii)  3  x  1 and x  4c. Since there is no other x-intercept to the right of 4, then thegraph falls to the right continuously without end.d. leading term:  x3e. leading coefficient: -1, degree: 33. For y  x 4  26x2  25 or y  (x  5)(x 1)(x 1)(x  5) a. Since there is no other x-intercept to the left of -5, then the graph rises to the left continuously without end. b. (i) x  5 and 1 x  1 (ii)  5  x  1 and 1 x  5 c. Since there is no other x-intercept to the right of 5, then the graph rises to the right continuously without end. d. leading term: x 4 e. leading coefficient: 1, degree: 4DEPED COPY4. For y  x 4  5x3  3x 2 13x 10 or y  (x  5)(x  2)(x 1)2a. Since there is no other x-intercept to the left of -5, then thegraph falls to the left continuously without end.b. (i)  5  x  2 (ii) x  5 ,  2  x  1 and x  1c. Since there is no other x-intercept to the right of 1, then thegraph falls to the right continuously without end.d. leading term:  x4e. leading coefficient: -1, degree: 45. For y  x2(x  3)(x 1)4(x 1)3 a. Since there is no other x-intercept to the left of -3, then the graph rises to the left continuously without end. b. (i) x  3 and x  1 (ii)  3  x  1, 1  x  0, and 0  x  1 c. Since there is no other x-intercept to the right of 1, then the graph rises to the right continuously without end. d. leading term: x10 e. leading coefficient: 1, degree: 10 Let the students reflect on these questions: Do the leadingcoefficient and degree of the polynomial affect the behavior of itsgraph? Encourage them to do an investigation as they perform the nextactivity. 98 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 9: Follow My Path!Answer KeyCase 1: b. odd degree c. falling to the left a. positive b. odd degree rising to the right b. even degreeCase 2: b. even degree c. rising to the left a. negative falling to the rightCase 3: c. rising to the left a. positive rising to the rightCase 4: c. falling to the left a. negative falling to the rightDEPED COPYSummary table:Sample Polynomial Function Leading Degree: Behavior of Possible Coefficient: Even the Graph: Sketch Rising or n0 or Odd Falling or Left- Right- n0 hand hand1. y  2x3  7x2  7x  12 n0 odd falling rising2. y  x5  3x4  x3  7x2  4 n0 odd rising falling3. y  x4  7x2  6x n0 even rising rising4. y  x4  2x3  13x2 14x  24 n0 even falling falling 99 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Synthesis: (The Leading Coefficient Test) 1. If the degree of the polynomial is odd and the leading coefficient is positive, then the graph falls to the left and rises to the right. 2. If the degree of the polynomial is odd and the leading coefficient is negative, then the graph rises to the left and falls to the right. 3. If the degree of the polynomial is even and the leading coefficient is positive, then the graph rises to the right and also rises to the left. 4. If the degree of the polynomial is even and the leading coefficient is negative, then the graph falls to the left and also falls to the right. DEPED COPY You should also consider another helpful strategy to determinewhether the graph crosses or is tangent to the x-axis at each x-intercept.This strategy involves the concept of multiplicity of a root of apolynomial function, the one generalized in the next activity.Activity 10: How should I pass through?Answer KeyRoot or Multiplicity Characteristic Behavior of Graph Relative Zero of to x-axis at this Root: 2 -2 3 Multiplicity: Crosses or is Tangent to -1 4 Odd or even 1 1 tangent to x-axis 2 even crosses the x-axis tangent to x-axis odd crosses the x-axis even oddAnswer to the Questions: a. The graph is tangent to the x-axis. b. The graph crosses the x-axis. The next activity considers the number of turning points of thegraph of a polynomial function. The turning points of a graph occurwhen the function changes from decreasing to increasing or fromincreasing to decreasing values. 100 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 11: Count Me In Sketch Degree NumberAnswer Key of Polynomial Turning Function Points1. y  x4DEPED COPY x 4 12. y  x4  2x2 15 y 4 33. y  x5 0 x y x 54. y  x5  x3  2x 1 5 2 x 101 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

y5. y  x5  5x3  4x 5 4 xAnswers to the Questions: a. Quartic functions: have an odd number of turning points; at most 3 turning points Quintic functions: have an even number of turning points; at most 4 turning points b. No. It is not possible. c. The number of turning points is at most (n – 1).Important: The graph of a polynomial function is continuous, smooth, and has rounded turns.DEPED COPYWhat to REFLECT on and UNDERSTANDActivity 12: It’s Your Turn, Show MeAnswer Key  2x4 1. y  (x  3)(x 1)2(2x  5) rises to the left, falls to the right a. leading term: 5 b. end behaviors: -3, -1, 2 c. x-intercepts: 5 (-3,0), (-1,0), ( 2 ,0) points on x-axis: d. multiplicity of roots: -3 has multiplicity 1, -1 has multiplicity 2, e. y-intercept: 5 point on y-axis: 2 has multiplicity 1 f. no. of turning points: 15 (0,15) 1 or 3 102 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

g. expected graph:DEPED COPYNote: At this stage, we cannot determine the exact values of all the turning points of the graph. We need calculus for this. For now, we just need to ensure that the graph's end behaviors and intercepts are correctly graphed.2. y  (x2  5)(x  1)2(x  2)3 x7 falls to the left, rises to the right a. leading term:  5 , 1, 5 , 2 b. end behaviors: c. x-intercepts: (  5 ,0), (1,0), ( 5 ,0), (2,0) points on the x-axis:d. multiplicity of roots:  5 has multiplicity 1, 1 has multiplicity 2, 5 has multiplicity 1, 2has multiplicity 3e. y-intercept:point on y-axis: 40 (0, 40)f. no. of turning points: 2 or 4 or 6g. expected graph: Note: Broken parts of the graph indicate that somewhere above, they are connected. The graph goes upward from (1, 0) and at a certain point, it turns downward to (  5 , 0). 103 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

3. y  x 3  2x 2  2x  4 or in factored form y  (x 2  2)(x  2)a. leading term:  x3b. end behaviors: rises to the left, falls to the rightc. x-intercept: 2 (2, 0) point on x-axis:d. multiplicity of root: -2 has multiplicity 1e. y-intercept: 4 (0, 4) point on y-axis:f. no. of turning points: 0 or 2g. expected graph:DEPED COPYNote: The graph seems to be flat near x = 1. However, at this stage, we cannot determine whether there are any “flat” parts in the graph. We need calculus for this. For now, we just need to ensure that the graph's end behaviors and intercepts are correctly graphed.4. y  x2(x2  7)(2x  3) 2x 5 a. leading term: b. end behaviors: falls to the left, rises to the right c. x-intercepts:  7,  3 , 0, 7 points on the x-axis: 2 d. multiplicity of roots: ( 7 , 0), (  3 , 0), (0, 0), ( 7 , 0) 2  7 has multiplicity 1, 3 has 2e. y-intercept: multiplicity 1, 0 has multiplicity 2, 7 has multiplicity 1 0point on the y-axis: (0, 0)f. no. of turning points: 2 or 4 104 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

g. expected graph:DEPED COPY5. y  2x4  3x3 18x2  6x  28 or in factored formy  (x 2  2)(2x  7)(x  2)a. leading term: 2x 4b. end behaviors:c. x-intercepts: rises to the left, rises to the right points on x-axis: 7 -2,  2 , 2 , 2 (-2, 0), (  2 , 0), ( 2 , 0), 7 ( 2 , 0)d. multiplicity of roots: -2 has multiplicity 1,  2 has1 multiplicity 1, 7 2 has multiplicity 1, 2 has multiplicitye. y-intercept: 28 point on y-axis: (0, 28) 1 or 3f. no. of turning points:g. expected graph: 105 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 13: Investigate Deeper and Decide WiselyAnswers to the Questions 1. a. 50% b. The value given by the table is 23.7%. The polynomial gives a value of 26.3%. The given polynomial is the cubic polynomial that best fits the data. We expect it to give a good approximation of the forest cover but it may not necessarily produce the exact values. c. The domain of the function is [0,98]. Since year 2100 corresponds to x = 200, we cannot use the function to predict forest cover during this year. Moreover, if x = 200, the polynomial predicts a forest cover of 59.46%. This is very unrealistic unless major actions are done to reverse the trend. You can find other data that can be modelled by a polynomial. Use the regression tool in MS Excel or GeoGebra to determine the best fit polynomial for the data. 2. The figure below can help solve the problem.DEPED COPY x 24x x x18 18 - 2xx x x x 24 - 2x 106 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Solution: Let x 18 – 2x 24 – 2x be the height of the box be the width of the box be the length of the box Working Equation: lwh  V (24  2x)(18  2x)x  V(x) (24  2x)(18  2x)x  560 4x3  84x2  432x  560 4x3  84x2  432x  560  0 x3  21x2 108x 140  0 (x  2)(x  5)(x 14)  0 To meet the requirements, the height of the box is either2 inches or 5 inches. Both will result in the volume of 560 cubicinches. In this problem, it is impossible to produce a box if theheight is 14 inches, so x = 14 is not a solution.DEPED COPY Encourage the students to write their insights. Let them show theirappreciation of polynomial functions. The following questions might behelpful for them: Were you surprised that polynomial functions havereal and practical uses? What mathematical concepts do you need tosolve these kinds of problems?What to TRANSFERThe goal of this section is to check if the students can apply polynomialfunctions to real-life problems and produce a concrete object thatsatisfies the conditions given in the problem.Activity 14: Make Me Useful, Then Produce SomethingAnswers to the QuestionsSolution:Let x be the side of the square base of the pyramid. So, B  x2 area of the base (B): h x2 height of the pyramid (h):Working Equation: V  1 Bh 3 107 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

V (x)  1 x 2 (x  2) 325  1 x2(x  2)  x3  2x2  75  0 375  x3  2x2 (x  5)(x2  3x 15)  0 The only real solution to the equation is 5. So, the side ofthe square base is 5 inches long and the height of the pyramid is3 inches.Students’ outputs may vary depending on the materials used and in theway they consider the criteria.DEPED COPYSummary/Synthesis/Generalization:This lesson was about polynomial functions. You learned how to:  illustrate and describe polynomial functions;  show the graph of polynomial functions using the following properties: - the intercepts (x-intercept and y-intercept); - the behavior of the graph using the Leading Coefficient Test, table of signs, turning points, and multiplicity of zeros; and  solve real-life problems that can be modelled with polynomial functions. 108 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

SUMMATIVE TESTPart IChoose the letter that best answers each question.1. Which of the following could be the value of n in the equation f(x) = xn if f is a polynomial function?A. – 2 1 C. 4DEPED COPYB. 0 D. 32. Which of the following is NOT a polynomial function? A. f (x)   C. f (x)  x  5x3 B. f(x)   2 x3 1 1 3 D. f (x)  x 5  2x23. What is the leading coefficient of the polynomial function f (x)  x  2x3  4 ? A. – 4 C. 1 B. – 2 D. 34. How should the polynomial function f (x)  1 x  x 2  11x 4  2x3 be 2 written in standard form? A. f (x)  11x 4  2x3  1 x  x 2 2 B. f (x)  x 2  1 x  2x3  11x 4 2 C. f (x)  11x 4  2x3  x 2  1 x 2 D. f (x)  1 x  x 2  2x 3  11x 4 25. Which polynomial function in factored form represents the given graph? y A. y  (2x  3)(x  1)2 B. y  (2x  3)(x  1)2 C. y  (2x  3)2(x  1) D. y  (2x  3)2(x  1) 109 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.