EFFICIENCY LEVEL IN SOLVING POLYNOMIAL EQUATIONS AND THEIR PERFORMANCE IN MATHEMATICS OF GRADE 9 STUDENTS

A Thesis Presented to the Faculty of the Teacher Education Program
Ramon Magsaysay Memorial Colleges
General Santos City

In Partial Fulfillment of the Requirement for the Degree
Bachelor of Secondary Education
Major in Mathematics

Armando V. Delino Jr.
October 2015
TABLE OF CONTENTS

Contents Page
Title Page i
Table of contents ii CHAPTER I THE PROBLEM AND ITS SETTING 1 Introduction 1 Theoretical Framework Conceptual Framework Statement of the problem Hypothesis Significance of the study Scope of the study Definition of terms

CHAPTER II REVIEW OF RELATED LITERATURE
CHAPTER III METHODOLOGY Research Design Research Locale Sampling Technique Research Instrument Statistical Treatment

CHAPTER 1
PROBLEM AND ITS SETTING

Introduction In Mathematics, a polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials appear in a wide variety of areas of Mathematics and Science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the Sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social Science; they are used in calculus and numerical to approximate other functions. In advanced Mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry. (https://en.wikipedia.org/wiki/Polynomial). Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms” that can be combined using addition, subtraction, multiplication and division except not division by a variable (so something like 2/x is right out). (http://www.mathsisfun.com/algebra/polynomials.html). This modern notation was more or less developed by Ren´e Descartes (1596-1650) in his book La G´eom´etrie [Geometry] (1637). Before the 17th century, mathematicians usually did not use any particular notation. Before Descartes, Fran¸cois Vi`ete (1540-1603) has already developed the basic idea of introducing arbitrary parameters into an equation and to distinguish these from the equation’s variables. Furthermore, most of the students spend a lot of time in solving polynomial equation because it has 2 or more variables and coefficients that are involve. For the entire approach, the researcher would like to determine the efficiency level in solving polynomial equations and their performance in Mathematics of Grade 9 students.
Theoretical Framework This study is anchored on Piaget’s theory of cognitive development that reality is a dynamic system of continuous change and, as such, is defined in reference to the two conditions that define dynamic systems. Specifically, he argued that reality involves transformations and states. Transformations refer to all manners of changes that a thing or person can undergo. States refer to the conditions or the appearances in which things or persons can be found between transformations. For example, there might be changes in shape or form (for instance, liquids are reshaped as they are transferred from one vessel to another, and similarly humans change in their characteristics as they grow older), in size (for example, a series of coins on a table might be placed close to each other or far apart), or in placement or location in space and time (e.g., various objects or persons might be found at one place at one time and at a different place at another time). Thus, Piaget argued, if human intelligence is to be adaptive, it must have functions to represent both the transformational and the static aspects of reality. He proposed that operative intelligence is responsible for the representation and manipulation of the dynamic or transformational aspects of reality, and that figurative intelligence is responsible for the representation of the static aspects of reality.
Operative intelligence is the active aspect of intelligence. It involves all actions, overt or covert, undertaken in order to follow, recover, or anticipate the transformations of the objects or persons of interest. Figurative intelligence is the more or less static aspect of intelligence, involving all means of representation used to retain in mind the states (i.e., successive forms, shapes, or locations) that intervene between transformations. That is, it involves perception, imitation, mental imagery, drawing, and language. Therefore, the figurative aspects of intelligence derive their meaning from the operative aspects of intelligence, because states cannot exist independently of the transformations that interconnect them. Piaget stated that the figurative or the representational aspects of intelligence are subservient to its operative and dynamic aspects, and therefore, that understanding essentially derives from the operative aspect of intelligence. At any time, operative intelligence frames how the world is understood and it changes if understanding is not successful. Piaget stated that this process of understanding and change involves two basic functions: assimilation and accommodation. https://en.wikipedia.org/wiki/Piaget%27s_theory_of_cognitive_development
Conceptual Framework
The conceptual framework of the study depicts that the efficiency level in solving polynomial equations serves as Independent variable. Any factors related to this will affect their performance in Mathematics. The dependent variable that is being observed and will be affected by the independent variable is the performance in Mathematics.

Independent Variable Dependent Variable

Their performance in Mathematics

Efficiency Level in solving Polynomial equations

Figure1. Conceptual Framework of the Study

Statement of the Problem The researcher aimed to find out the Efficiency Level in Solving Polynomial Equation and their Performance in Mathematics of Grade 9 students in Ramon Magsaysay Memorial Colleges (Annex) for school year 2015-2016. Specifically, the researcher sought to seek answer on the following questions:
1. What is the efficiency level in solving polynomial equations of the Grade 9 students?
2. What is the level of their performance in Mathematics?
3. Is there a significant relationship between the efficiency level in solving polynomial equations and their performance in Mathematics of Grade 9 students?
Hypothesis
Null Hypothesis
There is no significant difference in solving polynomial equations and their performance in Mathematics of Grade 9 students of RMMC (annex).
Alternative Hypothesis
There is a significant difference in solving polynomial equations and their performance in Mathematics of Grade 9 students of RMMC (annex).

Significance of the Study The findings of this study will be of great help to members of persons that are engage to this research. Students. Students will be given a chance to develop and become a competent individual. Moreover, this study will aware them on how they can improve their skills in dealing with Mathematical problems especially in solving polynomial equations. Teachers. Teachers will be able to find out the gap between students as they perform algebraic tasks. Also, teachers will be able to use this as a method to easily asses the level of achievement of students and how responsive they are to the subject. Furthermore, teachers can develop their system of instructions, fitting to reinforce students according to their respective levels of achievement. School Administration. They will be informed about the performance of the students, and encourage to assist the education department specifically in math areas. Parents. The result of this study will lead to the awareness of their involvements in the education of their children wherein education needs cooperation between home and school. Community. The community will be aware of the school needs and their cooperation for the school and students development. Researchers. This study will give them better understanding and additional knowledge as they conduct other studies relevant to this study.
Scope and Delimitation of the study This study will focused only on the Efficiency Level in Solving Polynomial Equations and their Performance in Mathematics of Grade 9 students of RMMC (annex). Pioneer Ave. General Santos City.
Furthermore the study will involve a number of students that is randomly selected as respondents from Grade 9 students of RMMC (annex) as of 2015-2016.
Definition of Terms For better understanding of this study, the following terms used in study are here by defined base on the context of the study.
Efficiency Level: (Conceptual) A level of performance that describes a process that uses the lowest amount of inputs to create the greatest amount of outputs. (Operational)
Polynomial Equation
(Conceptual) stands in contrast to a polynomial identity like (x + y)(x − y) = x2 − y2, where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality. (Operational)

Performance in Mathematics (Conceptual) The accomplishment of a given task measured against preset known standards of accuracy, completeness, cost, and speed. (Operational)

CHAPTER 2
REVIEW OF RELATED LITERATURE

Hilbert stated that polynomials are functions that are not linear, and you can multiply them to get higher degrees. It was his great insight to consider “ideal’s,” linear combination of polynomials, with common zeros. In his theory of homology, Hilbert was very sophisticated way of trying to reduce a non-linear situation to a linear situation. Essentially he produces a complicated system of linear relations that encapsulates some of the information about non-linear objects, the polynomials.
In 1857 Bouniakowsky made a conjecture concerning prime values of polynomials that would, for instance, imply that P(x) = x 2 + 1 is prime for infinitely many integers x.
According to Dickson any finite set of linear polynomials with qj ≥ 1 and (qj, aj) = 1 for every j = 1...k. Suppose that no integer m > 1 divides P(x) P2(x)...Pk(x) for every x ∈ N. Then there are infinitely many natural numbers n for which all the numbers P1 (n)...Pk (n) are simultaneously primes.
Schinzel stated the following conjecture better known as “Schinzel’s hypothesis H” which is a wide generalisation of a Dickson’s conjecture. Any finite set of irreducible polynomials in one variable with positive leading coefficients. Suppose that no integer m > 1 divides P(x) P2(x)...Pk(x) for every x ∈ N. Then there are infinitely many natural numbers n for which all the numbers P1 (n)...Pk (n) are simultaneously primes.
Bateman and Horn states any finite set of polynomials in one variable with positive leading coefficients, and of degree h1...hk respectively. Let each of these polynomials is irreducible over the field of rational numbers and no two of them differ by a constant factor.
Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss, doubted several of his claims, especially given the difficulty of some of the problems and the limited mathematical methods available to Fermat. His famous Last Theorem was first discovered by his son in the margin on his father's copy of an edition of Diophantus, and included the statement that the margin was too small to include the proof. He had not bothered to inform even Marin Mersenne of it. It was not proved until 1994 by Sir Andrew Wiles, using techniques unavailable to Fermat.
Although he carefully studied, and drew inspiration from Diophantus, Fermat began a different tradition. Diophantus was content to find a single solution to his equations, even if it were an undesired fractional one. Fermat was interested only in integer solutions to his Diophantine equations, and he looked for all possible general solutions. He often proved that certain equations had no solution, which usually baffled his contemporaries.
John Friedlander and Henryk Iwaniec states that general quadratic irreducible polynomial in two variables is known [Iw] to represent the expected order of primes (these are not characterized in any simple fashion). Polynomials in one variable are naturally more difficult and only the case of linear polynomials is settled, due to Dirichlet.
Roger Heat-Brown proved that there are infinitely many prime numbers of the form x3 + 2y3. In collaboration with S. J. Patterson in 1978 he proved the Kummer conjecture on cubic Gauss sums in its equidistribution form. He has applied Burgess's method on character sums to the ranks of elliptic curves in families. He proved that every non-singular cubic form over the rational numbers in at least ten variables represents 0. Heath-Brown also showed that Linnik's constant is less than or equal to 5.5. More recently, Heath-Brown is known for his pioneering work on the so-called determinant method. Using this method he was able to prove a conjecture of Serre in the four variable cases in 2002. This particular conjecture of Serre was later dubbed the ``dimension growth conjecture" and this was almost completely solved by various works of Browning, Heath-Brown, and Salberger by 2009.
According to Lynne Angela Teachey (2003) on her study about, Investigations in Conceptual Understanding of Polynomial Functions and the Impact of Mathematical Belief Systems on Achievement in an Accelerated Summer Program for Gifted Students; students who were able to depend on conceptual knowledge and make clear connections between the definition of factor and various polynomial operations and between equations; were also able to control their processes. A result which is steadily parallel to the study of Hiebert and Lefevre's (1986) belief that, "conceptual knowledge contributes to students’ abilities to monitor their progress". However, the students still showcased 2 conceptual deficiencies; the difficulties with the function concepts of independent and dependent variables and the misunderstandings about polynomials and symmetry. The conceptual difficulties with the independent and dependent variables were consistent with Sierpinska’s (1992) findings about difficulties that students encounter with functions. The first conceptual difficulty related to function variables was the tendency to view the variables as entities separate from the measures that they represented. The second conceptual difficulty with functions was the students’ lack of understanding of the asymmetry between the independent and dependent variables. Hence, the study is presenting a parallel idea with the current topic, which is, the importance of the students’ level of understanding the concepts of polynomials and its relevance to students' performance in solving mathematical equations.
( http://repository.lib.ncsu.edu/ir/bitstream/1840.16/4836/1/etd.pdf)

Chapter 3
METHODOLOGY
This chapter will show the models and design of the study. It detailed how to conduct a study in particular places from the respondents to get more information’s and ideas to them.
Research Design

Research Locale This study will be conducted at Ramon Magsaysay Memorial Colleges (Annex). A private and non-sectarian institution built in 1960 and presently serves as the good education that produces competent and quality of graduate.
The first and only institution/programs to be accredited in region 12, Such distinction was given by the Philippine Association of Colleges and Universities Commission on Accreditation (PACUCOA).
Its Business Education/Commerce, Liberal Arts and Criminology programs have achieved level 2 accreditation while its Elementary and Secondary Education courses have reached the candidate level . The level 1 status programs are accountancy, high school, Master of Arts, Information Technology, Computer Science, Office Administration. In the years to come , Ramon Magsaysay Memorial Colleges is bent on receiving accreditation for its other programs as well. RMMc likewise others three TESDA accredited programs namely Medical Transportation, Contact Center Services, and Ladderized BS Information Technology.
Research Instrument In getting the data needed for the result, the researcher formulated a questionnaire that intends to measure the Efficiency level in Solving Polynomial Equations in Mathematics of Grade 9 Students of RMMc.

Respondents and Sampling Used The respondents of this study are 20 Grade 9 students enrolled in the said subject during the school year 2015-2016 at Ramon Magsaysay Memorial Colleges, General Santos City.
Data Gathering Procedure
The researcher observed the following procedures in data gathering :
Asking permission to conduct. The researcher sent a letter to the college president through the dean of education program to conduct a study.
Identification of the respondents. Upon the approval of the letter request, the researcher went to the area and conduct the interview to the Grade 9 students of RMMC.
Distribution of Questionnaire. The researcher will personally collect the questionnaire right after the respondents finished answering the questions.
Collection and Tallying of Data. The data from the retrieve questionnaire were thencollated and tallied and submitted for statistical treatmet.
Statistical Treatment of Data.

CURRICULUM VITAE
Armando V. Delino
Malandag, Malungon, Saranggani Province

Personal Data Nickname: Yosh/Yox
Birth Date: March 14 1987
Status: Married
Religion: Catholic

Educational Background
Course:
Bachelor of Secondary Education Major in Mathematics
College:
Ramon Magsaysay Memorial Colleges
Secondary:
Davao City National High School
Elementary:
Davao Holytrinity Academy