special case quartic.pdf

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NEGROS ORIENTAL STATE UNIVERSITY

COLLEGE OF EDUCATION

THE FORMULA OF FINDING THE ROOTS

OF THE QUARTIC EQUATION

A RESEARCH

Presented to

Mr. Arnel M. Yurfo, MS MATH

Negros Oriental State University

Bayawan-Sta. Catalina Campus

Bayawan City, Philippines

In Partial Fulfillment

of the Requirements for the Subject

SPEC MATH 5 (FUNDAMENTALS OF MATHEMATICS)

Mike Loel P. Balbon

Lee Marie M. Marfiel

Elmo E. Labrador

Rommel A. Binalayo

Jonas S. Alcano

Elmer M. Baluran

Noel Q. Papasin

Sandro M. Herrero

March 2014

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ACKNOWLEDGEMENT

The researchers would like to express their profound gratitude to the following people for

their invaluable contribution that helped in the completion of this study:

To their families. They are the greatest team, friends and companies who welcomed and

surrounded them with incredible love, prayer and support;

To their friends and classmates who have stood with them and their families without question

and in absolute loyalty;

To Mr. Arnel M. Yurfo, an indescribable mentor who spent countless hours encouraging and

helping them with this research paper. His examples and perpetual optimism defied circumstances

and discouragement and gave them confidence that they can do this.

To www.google.com for such a wonderful web browser.

To those who have loved, prayed and financially supported them through these years, who have

gone with them in every trial, who have fought for them in prayers and in strength.

Above all, the LORD ALMIGHTY, for His Divine Love and Mercy that has kept the researchers

in good health and determination to accomplish their study.

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DEDICATION

To all the people along the roads of the world who have been kind to us and our families. To

those who have fed us, given us place to sleep, rest, and loved a stranger.

To Mr. Arnel M. Yurfo, our exemplar, whose encouragement and motivation bolster us to pursue

our dreams.

Lastly, to all the NUMERICAN SOCIETY members of NORSU-BSC, whose enthusiasm and

youthfulness serves as our morphine and antidote to stress, discouragement and pain all

throughout.

- The FAO Group

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ABSTRACT

The main purpose of this study is to define and find the roots of a quartic

equation with specified coefficients

In the course of history, there are five algorithms already established to solve the general

quartic equation. But in order to create a formula in solving special cases such as

, the researchers were able to formulate the FAO theorem.

The solutions presented in this paper are newly introduced by the researchers based on the

solutions derived by some mathematicians. The researchers come up with the following results:

a) ; b) ; c) ; d)

Moreover, no one can guarantee computational success to any particular algorithms.

Excessive complications may be adverse especially polynomials in the degrees higher than 3.

However, the theorem has been proven and examples are also presented to support the results.

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TABLE OF CONTENTS

TITLE PAGE..i

ACKNOWLEDGEMENT..ii

DEDICATION...iii

ABSTRACT...iv

TABLE OF CONTENTS....v

LIST OF NOTATIONS..vi

CHAPTER 1. The Problem and its Scope.....1

Introduction.....1

Statement of the Problem....3

Significance of the Study.....3

Objectives........4

Scope and Limitation...4

Review on Related Literature..5

CHAPTER 2. Methodology and Basic Concept........ 9

Research Methodology.,...9

The general quartic formula.....9

CHAPTER 3. Results And Discussions............12

FAO Theorem.........12

Proving FAO Theorem...... 14

Examples... 17

CHAPTER 4. Conclusion and Recommendations...22

REFERENCES...23

CURRICULUM VITAE24

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LIST OF NOTATIONS

Notations

Name

Page

= Equal sign iv,1,2,4,5,7,8,9,10-21

Not equal iv,2,4,9

_ Subtraction, negative iv,1,2,5,7,8,10-21

+ Addition, positive iv,1,2,4,5,8-21

Square root 1,2,7,11,13,16,18,20

Element of 2,4,9

R Real Numbers 2,4,9

Greater than 9,14

imaginary 18,19,20,21

Plus minus 2,7

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CHAPTER 1

THE PROBLEM AND ITS SCOPE

Introduction

Equation is a statement of an equality between two expressions. It is used in almost all

branches of pure and applied mathematics and in the physical, biological, and social sciences.

A polynomial equation has the form;

in which the coefficients , ... an are constants, the leading coefficient is not equal to zero,

and n is a positive integer. The greatest exponent n is the degree of the equation. Equations of the

first, second, third, fourth, and fifth degrees are often called, respectively, linear, quadratic, cubic,

biquadratic or quartic, and quintic equations.

Other important types of equations are algebraic, as in = = 7; trigonometric, as

in sin ; logarithmic, as in ; and exponential, as

in

. Diophantine equations are equations in one or more unknowns, usually with

integral coefficients, for which integral solutions are sought.

A root is a value of x that when plugged into the polynomial equation yields ;

a polynomial equation is solved when all the roots of the equation have been found.

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We are all familiar with a linear equation or an equation of the first degree and has only

one root. The single root of the linear equation is is

Likewise, the quadratic, or second-degree, equation ax2 + bx + c = 0 has two roots, given by the

formula

Processes on solving the two equations above can be used with complex numbers and

trigonometry to derive the formula in solving cubic and quartic equations.

Quartic equations often referred to as quartics, in the form

where a is nonzero. However, they are the highest degree

polynomials which can be solved analytically, by radicals, with no repetitive techniques.

According to history there were five and only five algorithms so far has been discovered

on solving the general quartic equation. However, there are some special cases which solutions

are yet to be discovered. Still emphasis has been placed on finding the real roots of the quartic

equation in the form .

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Statement of the Problem

The main purpose of this study is to define and find the roots of special case quartic

equation with specified coefficients

Significance of the Study

The most evident role of this study is to have a more profound and correct understanding

on quartic equation. The general quartic equation has been subject to great controversy on who

was able to derive the formula first. However, its sense and significance remains limited and

obscure.

Finding the complete algorithm for the roots of the said equation, this research will give

enough background information to make the key ideas on solving accessible to non-specialists

and even to mathematically oriented readers who are not professional mathematicians.

One of the applications of this study is in the field of computer science. Quartic often arise

in computer graphics. It is a proven fact that in this field proper algorithm is important to make

successful computer programs starting from the basic to the most complicated ones. The results of

this study will help computer geniuses in creating programs with minimal errors resulting to more

efficient and effective programs that will foster more learning and eventually will aide in the total

development.

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Moreover, quartic often appears in problems involving optimization. Optimization, also

known as mathematical programming is a collection of mathematical principles and methods used

for solving quantitative problems in many disciplines, including physics, biology,

engineering, economics, and business. The subject grew from a realization that quantitative

problems in manifestly different disciplines have important mathematical elements in common.

Because of this commonality, many problems can be formulated and solved by using the

unified set of ideas and methods presented by this paper that make up the field of optimization.

Furthermore, the results of this research may be used as basis for more related studies and

may encourage other researchers to ponder more on this topic.

Objectives

After this Mathematical research, the researchers are expected to find out the special

formula of finding the roots of quartic equation , where

, and a, b, c, d, e .

Scope and Limitation

This study focuses only in creating the formula of a special case quartic equation

with specified coefficient Thus,

it also covers some related studies of the said problem so that we could come up to the exact

formula in finding its roots.

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Review on Related Literature

One of the landmarks in the history of mathematics is the proof of the nonexistence of

solutions based solely on radicals and basic arithmetic operations namely addition, subtraction,

multiplication, and division for solutions of general algebraic equations especially of degrees

higher than four. Equations in general has been the most interesting part of the history of

mathematics.

Prior to the 17th century, the theory of equations was handicapped by the failure of

mathematicians to recognize negative or complex numbers as roots of equations. Only ancient

Indian mathematicians, such as Brahmagupta, recognized negative roots, and outside of India and

China negative coefficients of the polynomials were not recognized. Instead of one type of

quadratic equation, as given above, there would be six different types, depending on which

coefficients were negative.

In 1629 the French mathematician Albert Girard recognized both negative and complex

roots of equations and so was able to complete the partial insight of Franois Vite into the

relation between the roots of an algebraic equation and its coefficients. Vite discovered that if a

and b are the roots of , then and

More generally, Vite showed that if the coefficient of the first term of the equation

is unity, then the coefficient of the second term with its sign changed equals the sum

of all the roots; the coefficient of the third term equals the sum of all the products formed from the

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multiplication of two of the roots at a time; and the coefficient of the fourth term with its sign

changed equals the sum of all the products formed from three of the roots at a time. If the degree

of the equation is even, the last coefficient equals the product of all the roots; if odd, the

coefficient with its sign changed equals the product of all the roots. Vite also contributed

important numerical methods for approximating the roots of equations.

In 1635 French philosopher and mathematician Ren Descartes published a text on the

theory of equations, including his rule of signs for the number of positive and negative roots of an

equation. A few decades later, English mathematician and physicist Isaac Newton gave an

iterative method of finding roots of equations, it is known today as the Newton-Raphson method.

At the end of the 18th century, German mathematician Carl Friedrich Gauss proved that

every polynomial equation has at least one root. The question remained, however, whether it is

possible to express that root by an algebraic formula involving the coefficients of the equation, as

had been done for degrees one to four. A major step toward answering this question was the idea

of French mathematician and astronomer Joseph Lagrange of permuting the roots of an equation

to study its solutions. This fruitful idea led, in the work of Italian mathematician Paolo Ruffini,

Norwegian mathematician Niels Abel, and French mathematician variste Galois, to a complete

theory of polynomials that among other things, showed that a polynomial could be solved by

means of a general algebraic formula only if the polynomial has a degree less than five.

However, in the course of history, algebraic equation in the fourth degree has been said to

be the highest degree polynomial that can be solve logically by means of radicals; the solution for

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quadratic equation uses a single square root; the cubic equation solution uses a square root inside

a cube root; although quartic equations uses the same means but more complicated than the cubic.

The emphasis of this paper has been placed on finding the real roots of the quartic equation

A concrete solution on quartic was discovered during work on a problem proposed to

Cardan in 1540. After an unsuccessful attempt at solving this equation, Cardan turned it over to

his follower Ludovico Ferrari (15221565). Ferrari, using the rules for solving the cubic or the so

called resolvent cubic, eventually succeeded where his master had failed. At least, Cardan had the

pleasure of incorporating the result in the Ars Magna, with due credit given to Ferrari.

The central role of the resolvent cubic in the solution of the quartic was appreciated by

Leonard Euler (17071783). Eulers quartic solution first appeared as a brief section (S 5) in a

paper on roots of equations [1, 2], and was later expanded into a chapter entitled Of a new method

of resolving equations of the fourth degree (S 773783) in his Elements of algebra [3, 4].

Eulers quartic solution was an important advance, in which he showed that each of the

roots of a reduced quartic can be represented as the sum of three square roots, say

where the are the roots of a resolvent cubic.

As the cubic formula is significantly more complex than the quadratic formula, the quartic

formula is significantly more complex than the cubic formula. Wikipedia's article on quartic

equations has a lengthy process by which to get the solutions, but does not give an explicit

formula.

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In fact, there are five algorithms being formulated for solving general quartic equations.

However, none of these five would completely suit the needs of stable computations.

The five algorithms are listed below:

1. Descartes-Euler-Cardanos one

(

) (

)

2. Ferrari-Langranges

3. Neumarks

4. Christianson- Browns

(

)

5. Yacoub-Fraidenraich-Browns

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CHAPTER 2

METHODOLOGY AND BASIC CONCEPT

Research Methodology

This section presents the steps the researcher had done to formulate the results of the

study. First, they studied the formulas in solving the roots of quadratic and cubic equations. Then,

solutions to quadratic and cubic equations are used in solving the general quartic equations. The

researchers then have come up to the following steps in solving the quartic equation in the form

with specified coefficients .

2.1.1 The General Quartic Formula

1. The general quartic formula:

If the value of a > 1, lets first divide all coefficients by a so that will be equal to 1.

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Now, lets find the values of f, g and h

2. Then, plug these values to the following cubic equation:

3. Now, lets determine the values of p, q, and r. In getting the values of this

variables, lets consider three cases.

Case 1. If the roots of the resolvent cubic are non-negative, then the two nonzero roots

will be used to determine the values of p and q.

Case 2. If the roots of the resolvent cubic are two or three imaginary numbers, then select

the two imaginary numbers to be used to determine the values of p and q.

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Case 3. If two or three negative numbers exist in the roots of the resolvent cubic, then you

will select the two integers to be used to determine the values of p and q.

Now, substitute these selected roots to the following formulas,

4. The four roots are:

CHAPTER 3

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RESULTS AND DISCUSSIONS

This chapter presents the main result if this study.

The solution to the quartic equation is derived from the General

quartic formula by eliminating all b and c. After numerous attempts, the researchers come up with

the following formula and solutions.

3.1 FAO Theorem

If , divide all coefficients by so that the value of ,

There exist g and h:

Then,

1. Plug these values to the following cubic equation:

2. Now, lets determine the values of p, q, and r. In getting the values of this variables,

lets consider three cases.

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Case 1. If the roots of the resolvent cubic are non-negative, then the two nonzero roots

will be used to determine the values of p and q.

Case 2. If the roots of the resolvent cubic are two or three imaginary numbers, then select

the two imaginary numbers to be used to determine the values of p and q.

Case 3. If two or three negative numbers exist in the roots of the resolvent cubic, then you

will select the two integers to be used to determine the values of p and q.

Now, substitute these selected roots to the following formulas


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3.2 Proving FAO Theorem

In proving this formula the researchers used the general quartic formula. Since, the

researchers assume that b = 0 and c = 0, the researchers can conclude that if they remove b and c

from the general quartic formula, then they could come up with the right formula that satisfies the

restriction.

Proof.

1. The general quartic formula:

If the value of a > 1, lets first divide all coefficients by a so that . Now, lets

proceed in finding the values of f, g and h:

since b = 0 and c = 0, then we can conclude that

Likewise, the value of g is

eliminating all b and c will result to:

Consequently, the value of h is

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Since b and c = 0, then researchers can eliminate all the b and c from the formula and gives

2. The cubic equation:

Since f = 0,

This then gives us,

3. Determine the values of p, q, r and s.

Case 1. If the roots of the resolvent cubic are non-negative real numbers, then the two

nonzero roots will be used to determine the values of p and q.

Case 2. If the roots of the resolvent cubic are two or three imaginary numbers, then

select the two imaginary numbers to be used to determine the values of p and q.

Case 3. If two or three negative numbers exist in the roots of the resolvent cubic, then

you will select the two negative numbers to be used to determine the values of p and q.

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Since b = 0, then this will give us


Since s = 0, we can remove this variable from the formulas. This then gives us:

Examples

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Solution:

Lets determine first the values of g and h.

Then, lets plug this values to the cubic equation:

Solving the roots by cubic equation calculator gives:

Now, lets determine the values of p, q and r. Since there are negative roots which fall in

our 3rd

case. Therefore we are going to use and to determine the values of p and q.

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Then, the four roots are,

-

+

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Solution.

Lets divide first the numerical coefficients of the equation by 3 so that a is equal to 1.

This then gives us

Now, lets find the values of g and h,

Then, plug these values to the cubic equation:

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Solving by cubic equation calculator gives:

Next, lets determine the values of p, q and r.

= 0.117873888i

Then, the four roots are:

0.8446456519

0.8446456519

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0.8446456519

0.8446456519

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CHAPTER 4

CONCLUSION AND RECOMMENDATIONS

The general quartic equation is said to be the highest degree polynomials which can be

solved logically by means of radicals and complex numbers. However, the process we have to

undergo is very extensive that nobody could guarantee a computational success. Excessive

complications may be adverse.

We may guess that the five previously published methods have been elaborated so as to

attach some algebraic quality to the resolvents coefficients, but their robustness in actual

computations should be a subject of further analysis. The same principle is applied to the solution

derived by the researchers.

Furthermore, there are several cases about quartic equations that are left insoluble. The

researchers thereby recommend for a more comprehensive study regarding this topic.

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REFERENCES

Herbison-Evans, D. (2005). Solving Quartics and Cubics for Graphics. Retrieved

January 15, 2014 from http://linus.it.uts.edu.au/~don/pubs/solving.html.

Strong, T. (1859). Elementary and Higher Algebra. Pratt and Oakley.

Neumark, S. (1965). Solution of Cubic and Quartic Equations, Pergamon Press, Oxford.

Brown, K.S. (1967). Reducing Quartics to Cubics. Retrieved January 15, 2014 from

http://mathpages.com/home/kmath296.htm.

Smith, J. T. (2013).Cubic and Quartic Formulas.

Burton, D. M. (2011). The History of Mathematics: An Introduction, seventh ed., McGraw-

Hill

Companies, Inc., New York.

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