DWCRTTPORS *Algebra, *Tnstr:Uction, Mathematical Concepts, *Mathematicq, Mathematics Education, *Secondary School Mathematics

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AUTHOR nixon, Lyle J. TTmLE An Analysis of Certain Aspects of Approximation Theory as Related to Mathematical Instruction in. Algebra. Final Report. TmsTTTUmTON Kansas state Univ., Manhattan. MOONS AGENCY Office of Education (DHEW) , Washington, D.C. Bureau of Research. W17,ATT NO RP-8-F-03n PUP DAV' Aug 6() (1PANm OFG-6-9-008030-0035(097) NOTr 99D.

"UPS DRICP ;?)RS price MP-s0.50 HC-4;3.05 DWCRTTPORS *Algebra, *Tnstr:uction, Mathematical Concepts, *Mathematicq, Mathematics Education, *Secondary School mathematics

AsSTRACT This study is conc9rned with certain aspects of approximation theory which can he introduced into '.he mathematics curriculum at the secondary school level. The investigation examines existing literature in mathematics which relates to this subject in an effort to determine what is available in the way of mathematical concepts pertinent to this study. As a result of the literature review the material collected has been arranged in a structured mathematical form and existing mathematical theory has been extended to make the material useful to instructional problems in high school algebra. The results of the study are found in the expository material which comprises the ma-ior portion of this report. This material contains ideas of approximation theory which relate to elementary algebra. Also, included is a collection of references for this material. The report concludes that certain techniques of approximating a root of a polynomial by finding roots of a derived polynomial can be presented in a manner which is suitable for courses in elementary algebra. (DI') EDUCATION A WELFARE U.S. DEPARTMENT Of HEALTH, OFFICE OF EDUCATION

REPRODUCED EXACTLY AS RECEIVEDFROM THE THIS DOCUMENT HAS BEEN POINTS OF VIEW OR OPINIONS PERSON OR ORGANIZATIONORIGINATING IT, REPRESENT OFFICIAL OFFICE 0EDUCATION STATED DO NOT NECESSARILY POSITION OR POLICY.

FINAL REPORT Project No. 8-F-030 Grant No. 0EG-6-9-008030-0035(057)

AN ANALYSIS OF CERTAIN ASPECTS OFAPPROXIMATION THEORY AS RELATED TO MATHEMATICAL INSTRUCTION IN ALGEBRA

Lyle J. Dixon Kansas State University Manhattan, Kansas 66502

August 31, 1969

U. S. DEPARTMENT OF HEALTH, EDUCATION, AND WELFARE

Office of Education Bureau of Research

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FINAL REPORT

Project No. 8-F-030

Grant No. OEG-6-9-008030-0035(057)

AN ANALYSIS

OF CERTAIN ASPECTS OF APPROXIMATION THEORY

AS RELATED TO MATHEMATICAL INSTRUCTION IN ALGEBRA

Lyle J. Dixon

Kansas State University

Manhattan, Kansas 66502

August 31, 1969

The research reported herein was performed pursuant to a grant withthe Office of Education, U.S. Department of Health, Education, andWelfare. Contractors undertaking such projects under Government sponsorship are encouraged to express freely their professional judgment in the conduct of the project. Points of view or opinions stated do not, therefore, necessarily represent official Office of Education position or policy.

U.S. DEPARTMENT OF

HEALTH, EDUCATION, AND WELFARE

Officc of Education

Bureau of Research ,miArrmvrr,u,,s

TABLE OF CONTENTS

Title Page 1

Table of Contents 2

Summary 3

I. Problem Under Consideration 5

II. Methodology for the Investigation 6

III. Findings and Results 8

IV. Conclusions and Recommendations 9

Appendix A . Expository Presentation of Approximation 11 Theory as Related to Secondary School Algebra.

Appendix B - References 5?

Appendix C Bibliography 59

2 SUMMARY

mathematical principlethat certain solutions It is a well-known equation equation may beapproximated by considering an to an algebraic certain terms of the derived from the originalequation by "ignoring" introduced by this sortof procedure equation. Estimates of the errors At present,mathematical are given inadvanced courses inmathematics. include such principles. instruction at the secondarylevel does not have need This is true eventhough other areasof science instruction for the basic conceptsinvolved in thisprinciple. investigation is: Is there a The fundamentalquestion for this mathematical knowledgewhich covers thisapproxima- systematic body of in some con- tion technique and isit possible topresent this knowledge be established in venient form whereby a programof instruction can level?The basic courses inelementary algebra atthe secondary school task is to answerthis question. first necessary tomake a In order to answerthis question it was existing literature inmathematics rather extensiveexamination of the the way of mathematicalcon- to find out what wasalready available in of educationallitera- cepts pertinent to thestudy. A search was made these approxi- ture to determineif an attempt hadbeen made to relate the secondaryschool level. mation techniques toelementary algebra at of mathematical The search of theliterature revealed some sources literature on information but no reference wasfound in the educational concepts were classifiedby the subject of thisstudy. The mathematical put in a convenientstruc- areas and thosepertinent to the study were tured mathematical formfor future use. about the level ofmathemati- One significantobservation was made Almost all were at anadvanced cal concepts found inthe literature. to the study werefound in re- level and those whichmight be pertinent The trend in mathe- search journals in theperiod from 1900 to1930. included the areaof this spe- matical research inrecent years has not of the ideaspresented in the cific study. As a result of this, many developed and/orextended expository material inAppendix A had to be as originalresearch. concepts could beused in The criteria used todetermine if the to be included inthe elementary algebra andif these concepts were approximation expository material werethe following: Is it pertinent which is readableby the"average"high theory? Is it good mathematics in the teacher? Is it related tohigh school algebra school mathematics Is strictly algebraicand/or geometric concepts? sense it depends upon that a high in the expositorysection at a level the material presented what a high school school teacher canunderstand? Is it adaptable to student knows aboutmathematics? material in The results of thestudy are found inthe expository find the pertinentideas of approxi- Appendix A. In this material one can It is clear that the mation theory asrelated to elementaryalgebra. answered and in theaffirmative. basic question ofthis study can be be presented with somerigor Certain aspects ofapproximation theory can contribution to and/or in an intuitive mannerto make an additional at the secondaryschool level.This contribution mathematics education mathematical aspects. One, it broadensthe range of has two significant Secondly, it concepts which areavailable at a givenexperience level. which the appliedsciences provides the kind ofmathematical experience to be relevant tothe first courses can use andprovides it early enough in these sciencesat the secondaryschool level. additional insight into A by-product of thestudy has been the investigator duringthe course of mathematics obtainedby the principle way to measurethis, but it the investigation.There is no quantitative has been significant. should be made as theresult of The principlerecommendation which work to the last andultimate test the findings is thecontinuing of this and be used in an of its value. Some teaching unitsshould be prepared teachability of the conceptsout- experimental situationto determine the material. The expositorymaterial is essentially lined in the expository material at background for theunits and should notbe used as teaching for prospective the secondary schoollevel. It would be appropriate mathematics teachers. I. PROBLEM UNDER CONSIDERATION

Certain equations which arisefrom physical problems proveto be difficult to solve because of thenumerical computations whichhave to 6olution of the original be completed. Sometimes an approximation for a equation can be obtained byignoring a given term of theequation and solving the "reduced" equation. For example:

An equation such as

\IT n3.1 (1) 7 iloo can betransformed into the equation

d 6.2d d2 9.61 (2) 7 110o 1,210,000 =

A solution to (2) isapproximated by solving theequation

6.2d 9.61 . (3) 7 if55 =

Equation (3) is obtainedfrom (2) b-- "ignoring" ordeleting the term in reasonably d2. The solution of(3) is approximately 111 and this comes close to satisfying(2) and, hence, satisfying(1).

One might be tempted toconclude that this is an"accident" of the equation involved but thisis not the case. Physicists use the proce- known to use a dure just described quitefrequently and chemists are similar procedure in thesolution of some of theirequations.Moreover, there appears to be a desire onthe part of teachers inthese sciences that some instruction takeplace in the mathematicscurriculum on precisely these procedures.

The fundamental questionwhich should be answeredis: which covers Is there a systematicbody of mathematical knowledge such examples as justdescribed (and consequentlyothers) and is it form whereby a possible to present thisknowledge in some convenient program ofinstruction can be establishedin courses in elementary algebra at the secondarylevel? the first half of The last half of the questiondepends upon answers to readily lend them- the question and severalother factors which do not teachers, selves to research andanalysis. These other factors, such as However, curriculum, etc., are notnecessarily included inthis study. if the a significantportion of the fundamentalquestion can be answered specific objectives ofinvestigation are:

5 1(a) to determine if any significant body ofknowledge exists in approxi- mation theory which could be made available for secondary mathe- matics instruction, and

1(b) if such information is available, to systemize this information in order that secondary teachers of science and mathematics might have a significant and structured elementary mathematical model to use in relating mathematics and science,

2. to extend this information in whatever manner necessary to make the mathematical model more convenient for use at the secondary level and

3. to present the various aspects of the findings of the first three objectives in an expository form which would be readable by the "average" secondary mathematics teacher.

One rather clear limiting factor imposed within the objectives just noted is the relevancy of the material to be investigated to the secondary school level. This means, in essence, that no mathematical concepts should enter into the results of the investigation which do not have a basis within high school Algebra or Geometry. Any approximation te,lh- niques which depend upon the Calculus are automatically excluded fromthe investigation. For a similar reason infinite series are also excluded even if there might be a partialjustification for including this ap- proach on the grounds that one or two high school textbooks include some discussion of the subject. What should be clear is that the results of the investigation depend upon the "usual" (and perhapstraditional) con- tent of the subjects of Algebra and Geometry. This limitation has par- ticularly significant implications for educational applications of the results of this study.

II. METHOD3LOGY FOR THE INVESTIGATION

The procedures followed in this study were essentially the same as those of any scholarly investigation. These procedures are given here by phases with appropriate comments on each phase.

Phase 1. An extensive examination of existing literaturerelated to the general area of approximation theory wasmade. This was done in order to determine what is already knownin the area and what might be appro- priately related to this investigation. Library resources, such as the Educational Index, were examined for referencesto the topic. Books in the areas of numerical techniques,engineering applications, computer sciences and any area which might make useof approximation techniques were examined for suitablematerials and for references to other possible sources. The mathematical journals, both pureand applied, were examined for papers and references on approximations. Members of the mathematical community were consulted for sources of informationand the free ex- change of ideas with their members has been mosthelpful during the in- vestigation.

6 Phase 2. After the material and sources of literature had been collected, it was necessary to examine each reference foundto determine if it was related to the problem and if it might beuseable material. This was clearly a judgment decision determined by thepreviously men- tioned specific objectives of the study. It was not always clear, at that point, that some source was necessary for future use. This was minimized by the establishmentof certain criteria for selection of materials. These criteria were established before the investigation actually began and are enumerated here. In many instances during the search of the literature a source was rejected at that pointbecause it obviously failed to meet these criteria.

The criteria used ad(the following questions:

(a) Is it pertinent? Does it really belong in approximation theory? (Some material was related, but quite secondary in nature.)

(b) Is it good sound mathematics? Is it readable by the "average" high school mathematics teacher? (Much periodical literature in mathe- matics is presented so abstractly only another researchmathema- tician can read it.)

(c) is it related to high school Algebra? Does it depend upon strictly algebraic and/or geometric concepts(deducible from these subjects at the high school level)? (Some methods depend upon Calculus and were not specifically suited for thisproject.)

(d) Is this material at a level that a. high school teacher canunde - stand? (This means making a decision on what most teachers know and what their training has been.It is clear from various studies that this is not as high as desired and for this study to have any impact or application it must produce something which most teachers can study and, hence,learn.)

(e) Is it adaptable to what a high school student knowsabout mathe- matics?This follows naturally from (d).

(f) If more than one technique is available to accomplish someapproxi- mation procedure, which one best lends itself toteaching and learning theory?

These criteria were adequate for the job intended,except the last one was never actually used. The reasons for this are given in the findings.

Phase 3. The next step was to order the material by areasof appli- cation. As it turned out this was essentially done concurrentlywith Phase 2, and frequently as early as Phase 1. For instance, to order or group desirable information on procedures toobtain approximations to roots of an algebraic polynomial into one areacalled for all material on. Horner's rule, Newton's method and others tobe classed together as a of certain general approximationconcept. In contrast, the approximation a solution of aquadratic equation by solving alinear equation obtained from the quadratic constitutes anentirely different order of approxima- somewhat arbitrary but, gene- tion concepts. The areas of grouping were rally speaking, weredetermined by what the investigatorsfound in the literature. in- Phase L. The next phase consistedof arranging the accumulated formation in some convenientlystructured mathematical form. This was essentially a matter ofdeveloping the mathematical aspectsof existing theory into a logicallycoherent system which can be usedby teachers of secondary school mathematics. In some instances this waseasily done due only as a result of to the circumstancesbut in others this was possible work done during Phase5.

Phase 5. Existing mathematical theory wasextended, wherever possi- ble and where needed, in a mannerto make the system of Phase4 more com- plete and useful inapplication to instructionalproblems in Algebra. While it Some very large gaps werenoted during Phase 4 in certain areas. was not anticipatedthat significant mathematicalresearch would be needed, it turned out that the literatureleft significant gaps in themathematics (which may be filled by information known orassumed by others and not available to thisinvestigation). These gaps were filled bymathematics generated by the principalinvestigator.

Some aspects of this phase began asearly as Phase 4 and extendedin- to Phase 6, with a greatdeal of this kind of activityin the last phase.

of Phase 6. The last step of theinvestigation was the preparation explicit mathema- an expositorypresentation with adequate reference to tical development which coveredthe structured system developedand ex- panded in Phase 5.The presentation was prepared in aform consistent with the objectives of thestudy and special emphasis wasplaced upon the level of abstractness ofthe material. The expository development is found in Appendix A of thisreport.

FINDINGS AND RESULTS

Since the major portion ofthis investigation is the preparationof some expositorymaterial, it would be proper toinclude that material at it this point. However, due to the structureof the expository material, will be found in AppendixA, along with the collection ofreferences for observations which are perti- this material. However, there are several nent to the investigation as awhole and these are made at thistime.

A review of theliterature reveals two rathersignificant facts. First, there is a sizeablebody of knowledge available invarious places about approximation theory. Nearly all of this is in advancedtexts, journal articles and technicalreports. All of these sources are available to mathematicianswho regularly require their useand are versed in tY. general area. A good deal of the recent informationis frequently linked to numericalanalysis and the computer. One also

8 finds the application sources are frequently the best sources of readable mathematics in this area.This suggests that most mathematics is so abstractly written and presented that only a relatively small seg- ment of the population has an opportunity to digest and use the informa- tion presented in such form.

The second significant fact obtained from the review of the litera- ture is that there is no source on approximation theory which presents in an elementary fashion those aspects of approximation theory which are used in moderate amounts by the various scientific disciplines found in secondary school courses of study. This was somewhat anticipated and was a motivation factor of this investigation. However, it was anticipated that some sources, maybe obscure, would be found, This was not the case. In fact, the search for this kind of material continues, even as the final report is being prepared.

Another related observation was the recognization that mathematics changes rather rapidly. Prior to 1935 a great deal of attention was de- voted to algebraic polynomials and their solutions. By 1940 there are almost no references in periodical literature to such matters and fewer references are to be found since then.The general area of mathematics which had its central theme in courses called Theory of Equations has essentially disappeared or been absorbed by other areas of mathematics. This shift in interest by mathematicians probably contributed to the shortage of useable material on certain topics in approximation theory and forced the creation of certain theorems to fill gaps left by this shift.Their theorems may not be as original as some but no references have been found for them. The theorems are identified in the expository material by a D in the theorem number, such as Theorem D2.

IV. CONCLUSIONS AND RECOMMENDATIONS

The conclusions of the investigation are to be found in the exposi- tory material as this was to be the major task of the study. As has been previously indicated the material is not as extensive as had been antici- pated and there might be an aspect which could have been included, but the expository material is representative and respectable mathematics.

There are some recommendations which should be made as a result of this study and all are within the capability of educational endeavors. First, and not necessarily dependent upon this study but obviously recognized in carrying out the investigation, is the need for expository materials in mathematics which translate some phases of the subject into a form which high school teachers can use. This project has attempted to do this for an area of mathematics (applied in nature) and there are many and perhaps more important areas which need the same treatment. For instance, the social sciences are becoming more mathematical as these areas develop tho science aspect of their endeavors. The present high school mathematieF, is almost totally mathematical or physical science orientated. If rahe other areas are important to our society, there should be some literature which describes these relationships and which suggests how it might be adapted to existing patterns of curricu-

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lum and instruction. Examples of problems from economics,psychology,and other disciplines are appropriately a partof mathematics instruction and expository material would seem to be the best avenueto get these ideas into the hands of teachers.

Second, a natural consequence of this study would bethe preparation of suitable material for lessons for high schoolstudents to use.This is essentially the adaption of this report to apractical situation. This set of lessons should be tested in aclassroom situation to deter- mine the success of the whole idea.Mathematically, there is no reason why it cannot be done. Perhaps there are non-mathematical reasons why approximation theory of any sort has no place in secondaryschool mathe- matics, but none are now known.

Finally, the investigation revealed the lack ofmaterial at any level on the general area of "theory ofequations." It would seem that in our rush into "modern mathematics" we mayhave left out significant educational concepts which ought to be restored. It is unlikely that the college and university mathematics courses willinclude these ideas and perhaps a major portion of it can go into high schoolmathematics. It is mathematically feasible and has already been done on anexperimental basis.

10 Appendix A

Expository Material on Approximation Theory

11 I. Introduction.

The general area of approximation theory is quite broad, too broad to be adequately covered in a publication of thissize. However, there are aspects of this general theory which are quitto pertinent to secondary school mathematics and on more than one occasion(for example, see Bowen DO) science teachers have suggested that someportions of this material might be quite appropriate for the curriculumin mathematics and at an early stage in the secondary schools.The question of what portions are available and appropriate is essentially anunanswered question as far as the literature is concerned. Moreover, ifthere are portions which are appropriate, can these portions be organized in a manner which makes good mathematical sense and which is strictly dependent upon a bare minimum of secondary school mathematics?

The purpose of this material isto provide some affirmative answers to these questions and to present some aspects of approximation theory which might be appropriate for secondary school mathematics.This we shall do in the following sections. We begin by examining some history of mathematics and some types o approximation. Following this we will present the mathematics of oneof these types by citing pertinent theo- rems and facts. Most of thetheorems are already in the literature some- place and they only need tobe assembled into a meaningful presentation. There are included certaintheorems which were not found but were needed to fill gaps in the area.

One very importantprinciple was kept in mind in the presentation of this material. Itshould be material which the "average" high school teacher can use withrelative ease and it should be ,based upon elementary enough mathematics so that it could be adapted to a classroom situation. The latter restrictLion essentially means a year of algebra and maybe a reasonably good ae.quaintance with graphing.

II. Historical Background on Solving Equations.

In this section we will give a very brief review of the historical development of the solving of equations and in particular, the solving of the quadratic and cubic equations.The purpose in reviewing the back- ground of these equations is to recognize that approximations played an important part in the early history of solving of equations, and to recog- nize that this process was a long, evolutionary one. We shall not be complete in our presentation, but will touch upon only the significant highlights of that history. This will be accomplished by first treating the quadratic, then the cubic and finally all other degrees of algebraic polynomials.

1. The Quadratic Equation. The earliest solutions of problems involving equations were obtained by trial.At least no written record has been found to indicate otherwise. By 1500 B.C. the Egyptians (as described in in the Ahmes Papyrus) were able to make a trial guess and adjust the re- sults of the trial guess in a manner which gave the correct solution of

12 the equation under consideration.This technique was called the "Rule of False Position" and the procedure was used by subsequent students of mathematics as late as the 1700's.

A similar technique called the "Rule of Double False Position" was used rather extensively for many centuries. This technique, as well as the other one, were subsequently replaced by our modern techniques but only after the algebraic symbolism was developed by Vieta and others.

A study of the history of early mathematics (before 1600) reveals that most solving of algebraic type equations was done by trial and error with the initial step being to make a guess orapproximation of the solution and make subsequent adjustments on the basis of the first ap- proximation. This procedure was applied to all degrees of polynomial equations which arose from the problems to be solved.

In the Berlin Papyrus (c. 2160-1700 B.C.) one finds the first known solution of a quadratic equation.The problem reduces to solving the equations

2 3 .tY2 ir 100 and Y x . x s"

The solution technique reduces to a simple case of the "Rule of False Position."

The Greeks were able to systematically solve quadratic equations by geometric means. Euclid (c. 300 B.C.) gave solutions and procedures to solve problems such as

xy = k and x - y = a and a(a - x) =x2.

The Hindus may have been able to solve quadratic equations by 500 B.C. although no record of the method of solution has been found. By 500 A.D. we do find a rule, relating to the sum of geometric series, which shows that the solution of the quadratic equation was known, but we have no rule for the solution of the equation itself. About 628 A.D. Brahmagupta gave a definite rule for solving a quadratic. Smith [19] gives the steps of solution for an example of a quadratic(x2- 10x =-9) which turns out to be, in modern terminology,

-9.1 +(-5)2-(-5) x = m 9 1

One might note the similarity of the solution to the quadratic formula except he found only one solution, i.e., the positive one. This was quite typical of all early attempts to solve equations in that only posi- solutions were acceptable.

13 Mahavira (c. 850) had a way of solving quadratics for a positive root, but did not write it out. However, an analysis of his solution leads one to believe he knew substantially the modern rule for finding a positive root of a quadratic.

The Hindu Rule was first given about 1025 and was widely used in India.

Al-Khowarismi (c. 825) gave two general methods of solving quadra- tics of the form

x2+ px q .

Negative roots were neglected. Omar Khayyam (c. 1100) also gave a rule for solving the equation

x2+ px = q .

The techniques of solution mentioned above are typical of ways of solving quadratics until about 1600.Solutions for other higher degree equations were essentially based upon an approximation and a subsequent adjustment, as in "Rule of False Position".

In 1631 Harriot gave the first important presentation of the solu- tion of a quadratic (and other equations) by factoring.

Vieta (c. 1600) replaced the geometric method of solving quadratics by an analytic method.The solution of the equation x2 + ax + b 0 was given as

1 1 x- -fa:Ey a - 4n .

The symbolism introduced by Vieta and others of his time made possi- ble such a representation and a closeness to the complete solution ofany quadratic. It is easy to see that the trial and error stage is being re- placed by a systematic and complete method of solving a quadratic.

2. The Cubic Equation. The earliest known (c. 350 B.C.) cubic equation was of the form x3 = h although same Babylonian tablets give tables of cubes about two thousand years earlier.The problem of the duplication of the cube (said to have been known by Hipprocrates (c. 160 B.C.)) de- pends upon the finding of two mean proportionals between two given lines. This means to find x and y in the equations

a x X y b

Archimedes referred to a problem of cutting a sphere bya plane so that the two segments shall have a given ratio. The problem reduces to the proportion

14 2 c x c x2

Eutorius (c. 560) tellshow and this produces thecubic x3 +c2b=cx2. Archimedes solved the problemby finding the intersectionof two conics. Diophantus apparently solvedthe equationx3 +x =4x2 + 4, possibly by noting thatx(x2 + 1) = 4(x2 + 1). 860), Tabit ibn Several Arabic mathematicians,notably Almahani (c. (c. 1000) dis- Qorra (c. 870), Abu Jafar al-Khayin(c. 960) and Alhazen cussed the cubic and atleast one found solutionsto certain equations in a manner similarto Archimedes. forms of the cubic that Omar Khayyam(c. 1100) specificed thirteen in general theory but a had positive roots. This is a distinct advance However, he did expand long way from completesolutions for any cubic. be solved by the the number of cubics ofcertain forms which could that "intersection of conics"techniques. Most Arab writers believed to solve. the cubic equation,in general, was impossible

From about 1200 to1500 several writers in westerncivilization although several mentioned the cubic but wereunable to solve all cubics, special cubics were solvedby special techniques,such as factoring.

The real interest in thecubic lies in the work ofCardan and Tartaglia.. History doesnot clearly describewho should receive the controversy make credit and the storiessurrounding the Cardan-Tartaglia solution for the cubic interesting reading. It is clear that the that a method x3 + ax2 = c was obtained by1535 and very shortly after Cardan in his Ars Magna of solution for thecubic x3 + bx = c was found. (1545) showed how to transformthe first form into thesecond and suc- x3 + ax2 + bx + c = 0 into a re- ceeded in transformingthe general cubic to solve and, duced equation typexi + px = q. This he already knew how positive therefore, every cubic wassolvable. Cardan's examples showed and negative roots, thelatter being included forthe first time.

Vieta (1615) subsequentlyfound the transformationwhich reduced the general cubic to theform y3 + 3by = ?c and,by a second transforma- b2 and the latter tion or substitution,arrived at a formz0+2cz3= was solved as aquadratic inz3. equation and the Again history reveals thestruggle to solve an known existence of subsequent success some twothousand years after the of the the equation. The cubic equation canbe solved although some frequently approxi- solutions involve complicatedradicals and these are to a physical mated for ease ofcomputation. The application of a cubic to have a suita- problem almost necessitatesthe approximation in order ble answer to usein a physical sense.

15 that any conside- 3. The Quartic Equation. One should probably suspect ration of a quartic equationwould be minor during thetime the cubic was One can find only slight unsolved. History reveals this to be the case. The first serious mentionof the Quartic prior to thetime of Cardan. consideration of the Quartic camein 1540, almost immediately onthe heels of the solution of thecubic.

A young Italian student by the nameof Ferrari solved theproblem Cardan did a by reducing the equation to anequation involving a cubic. great deal to clarify the processand spread the knowledge ofthe method in his book Ars Magna. were Not all solutions werenecessarily included as negative ones frequently ignored. Vieta(c. 1590) and Descartes (1637) improvedthe system of solution and themodern form is due to Simpson(1745).

from Ferrarils 4. The Quintic Equation. Euler found a method different for reducing the solution of thegeneral quartic equation tothat of a cubic equation. He attempted, to apply thistechnique to the Quintic in hopes of reducing the solutionof, it to the solution of thequartic and thereby solve the quintic. However, he failed as did othernotables in the world of mathematics. These failures prompted Ruffini(1803, 1805) to try and show that the quinticcould not be solved by such means.

Abel and Falois eventually resolvedthe question. Galois (1846) essentially answered the totalquestion in his posthumouslypublished works. Abel had earlier shown(1824) that the roots of the general quintic cannot be expressed interms of its coefficients by meansof radicals. This we are able to do for thequadratic, cubic and quartic. Intuitively it would seem possibleto be able to do the samefor the quintic, but it is not. degree equations 5. Solutions of other equations.The question of higher does not seem to haveinterested mathematicians ofwestern civilization until quite late. The examination of the equationsof higher order takes equation all you two major directions. The first is to find out from the (We often do this can about theroots without actually solvingfor them. for the quadratic by lookingat the value of itsdiscriminant.) Cardan may have knownabout the Rule of Signs which tellshow many positive formulation of the roots were possible. Harriot (1621) may have made the Rule oftSigns and Descartes(1637) certainly gave such a Rule.

The second direction for thesolution of equation of higherorder was to approximatesolutions and by some process ofiteration subse- quently obtain better approximations.The Chinese scholars of the13th and 14th centuries were outstandingin this area and this seemsto be China's particular contributionto mathematics. In 1247 Chin Kiu- shao's writings reflect a high degree ofperfection in this aspect ofequation solving and equivalent form ofHomer's method (1819) is given.

Fibonacci (1225) attempted someimprovements as did Vieta(1600).

16 Newton (1669) simplified Vieta'swork and perfected an approximation technique. Horner (1819) further simplifiedNewton's work.

It is interesting to note thatNewton's method was replaced by Horner's method (as can be observedin any of the Theory ofEquations books) because of, the ease of hand computation. However, with the ad- vent of computers, some of theearlier methods are better onesto use because they adapt easier to machinelanguage. Progress does not always make old things obsolete.

6. The Fundamental Theorem ofAlgebra. Perhaps the most interesting problem related to the 'solution ofequations was the apparent disregard of negative solutions long aftertechniques were available to find them. It is not too difficult tounderstandthe ignoring of complexsolutions, as concepts ofcomplex numbers had not been fullydeveloped. However, there came a time(c. 1608) when mathematicians beganto assert that an equation of nth degree has exactly nroots. It was restated at various times without proof and Gauss(1799) gave the first rigorous demonstra- tion of the theorem.

The theorem, together with thetechniques of the solving of linear, quadratic, cubic and quartic, completesthe major theory of equation solving. For the quintic and others of higherorder, the theory is not quite as complete, but complete asmathematically possible.

We have just sketched 7. Review and Summary of HistoricalBackground. the basic highlights of the solutionof certain types of equations. It should be noted that all early attemptsat any particular type of equa- tion were essentially based uponapproximations (and, in some instances, adjustments on these approximations). After rather lengthy periods of time, solutions were found for just one,then another and then another type of equation. Only after the symbolism becamefully developed does one find significantcontributions in this area.

For those equationswhich no techniques ofsolution were found, mathematicians seemed tobe able to approximateroots of equations with some degree of accuracy. Subsequently, certaintechniques were deve- loped to improve a givenapproximation, usuallyby an iteration process.

In recent times the users ofmathematics, i.e., physicists, chemists, etc., have begun to solvecertain equations by deleting some term of the equation and solvingthe resulting equation. This is fre- quently done even when formulas areavailable to solve the starting equation. This is essentially an approximatingtechnique and tends to indicate a cyclic treatment of thesubject as far as history is con- cerned. It also suggests that a secondstage should follow at some fu- ture time and a general theory befounded to justify the proceduresand approximations now being used. In what follows we partiallyprovide an indication of what this second stageshould contain.

3.7 :no.

III. Types of Approximations.

When one begins to examine approximation theory it becomes clear that there are different types of approximations which may arise from or be related to different circumstances. In this section we will examine a few different types which will serve to illustrate different aspects of approximation theory. Some are elementary and have already been seen even by the beginning high school student.

1. Approximating a number by another number. In the elementary school arithmetic one finds the establishment of a decimal representation of the rational (and sometimes irrational) numbers by giving decimal equivalents to certain fractions. Thus

1 .5

1 = .25

= 0.25

The list is usually limited to the more common (in usage) fractions. However, we may see statements like this:

7= .333

2 -5= .666 or, in some instances, 22 T or = 3.14 .

As every good student of mathematics knows, these last statements are not actually correct. The fraction1 is not .333, and n is neither 22nor 3.14. 7 7 1 What should be properly written is a statement that is approximately 3 22 .333 or that n is approximately or 3.14. This is sometimes written with a variety of symbols for the equality but a ". or me distinguishes these examples from the first ones. Thus we may write

1 .333

22 and n. --- or n 3.14 . 7

18 What we really mean in theseinstances is that .333 is adecimal 1 whose value is reasonably closeto the value of3and that 3.14 is reasonably close to value of v. This is usually done incomputational situations either because wedo not know the exactvalue of some number (v for instance) or because we arewilling to settle for an answer area reasonably close to somenumber. For example the circumference or of a circle with radius of 10 canbe precisely given by theformula

C 2vr

and A = vr2

which in this case become

C = 20v

and A = 100v .

If, because the problemdepends upon physical data orbecause we want to approximation for it know approximately whatC and A are,, we may use an and obtain

C cts 20(3.14) or C ti 62.80

and A pe 100(3.14) or A Av., 3114 and Thus 62.80and 314 become approximationsfor the exact values 20v 100v.

When public officials say160 million Americans watched therecent guess) walk on the moon they areapproximating (actually an estimate or counts the the actual number whodid watch. When an engineering student number of steps between twopoints and multiplies this by3 he determines approximately the number of feetbetween the two points. These are but two instances in whichthe actual number involvedis not known but is approximated by a number, thecloseness of the approximationbeing deter- mined by the actual amountof information exactlyknown. Each step of in length and the com- the engineering studentis not exactly three feet puted distance varies as thelength of the step.Sometimes the exact value of a number couldbe stated, but we are satisfiedwith something inch but most which approximates it. It may have rained .475 of an people would probably sayit rained one-half of an inch,being satisfied with the message one-halfconveyed in the conversation.

In all these instances we see anactual value of a numberbeing re- placed by a number which isapproximately correct. For the fraction 1 we might use .3 or.33 or .3333, depending onprecisely how close we 3 1 wish.to approximate . 3 The concept of finding a 2. Approximating a root of an equation.

19 number which approximates another number is also found in other circum- stances. If the second number is the root of an equation, i.e., a number which satisfies some mathematical expression, it is frequently possible to use the equation or parts of it to determine an approximation for one of its roots. This is similar to finding an approximation for, say 1but the significant difference is that we have an equation and the 3 equation becomes the source of approximations whereas one must guess or develop a source of approximations forIre . There is no mathematical equation which would determine how many people watched the walk on the moon. In the case of the equation, we have a basis for making an approxi- mation. In section II we observed how early attempts to solve equations began with guesses and the guess was refined by making use of the equa- tion. In the case of the Rule of False Position a refinement produced the correct number and not just an approximation. The equation itself was used to determine what the second approximation should be. In the case of Hornerts Method (which we will describe shortly) the first ap- proximation for the root of an equation, together with the equation, prl duce a second and better approximation. There are several techniques which are similar and the computer makes it possible to repeat the pro- cess as long as one wishes and one can refine the sequence of approxima- tions to obtain as close an approximation as one wishes.

We give a description of Horner's Method as a device to approximate a root of algebraic equation*. First, a real root of the algebraic equa- tion P(x) = aoxil + axn-1 + a2xn-2 + + an = 0 must be isolated, that 1 is, it must be determined that a root lies between a andb, a < b. This can be done by several techniques, but the most elementary way is to find (guess at) two values a and bsuch that P(a) is opposite in sign from P(b). If this be true thenthere exists c , where a < c < b such that P(c) = 0. Hence, c is a root of P(x) = 0.

Second, the equation

(1) P(x) = aoxP + aixn'l + + an = 0 is now transformed by replacing x by x - a, and this diminishes the roots by a . We now have a new equation.

( 2 ) P1(x) = ao21 + blxP-1+b2xP'2+ + bn = 0 with a root of this equation between 0 and (b - a). We now determine two more values c and d such that 0 < c < d< (b - a) and P (c) is oppo- site in sign from P1(d). Now repeat the transformation again. Each new

*We assume for all discussions that the algebraic equations are poly- nomial equations with real coefficients.

20 step of the sequence, 0 < o < x

Newton's Method (1676) is a similarprocedure and it has an advan- tage over Horner's Method in thatit can also be applied totrigono- metric, or logarithmic, or othersimple functions. Horner's method is strictly reserved for algebraic polynomialswith real coefficients.

The technique of successiveapproximations where each new approxi- mation is a refinement of thepreceding approximation is an excellent technique to apply to the task offinding a root of an equation. Ber- noulli's method of approximating thelargest root of the equation

X3 111 ax2 + bx + c

with real coefficients is one suchexample. The method uses a set of recursion formula such that successivevalues for a variableAn(a,b,c) when compared with the preceding valueAn_1(a,b0c) gave anapproxima- tion for the absolute value of thelargest root of the cubic, ifthat root were real.

T. A. Pierce f13] gave a recursivetechnique for finding the least root of a cubic by developingBernoulli-type formulas for approximating the absolute value of that root,if it is real. The formulas are strict- ly algebraic in nature, butrather complicated to deduce. Moreover, other techniques seem to be moreelementary and cover more typesof equations than just the cubics.

Other techniques employing the methodof successive approximations can be found which usethe equation and variousderivatives of the poly- nomial function. These techniques employ mathematicsbeyond the secon- dary school level and, therefore, arenot appropriate for thisinvesti- gation. As an example of this type involvingthe derivative, see Ford C71.

Other techniques were developed in thetwenties and thirties for solving certain polynomials. For example, Kennedyro showed an alge- braic treatment for polynomial equationsof a certain kind via a loga- rithmic process to find approximationsfor the roots of those certain polynomials. Running rio gave techniques forfinding real roots of cubics and quartics from graphs ofcertain straight lines whichdepended upon the discriminantof the equation. Grant 18] gave still more infor- mation on how to solve quarticequations by graphical means.

This brief summary of some pertinent papers onroots of polynomials shows the search for varioustechniques to simplify or avoidlaborious

21 calculations necessary in findingapproximations to roots of an equation by methods such as Hornerls Method. An entire period of mathematical work has been devoted to this kind of research. In some ways, the compu- ter has made some of this work obsolete, whilemaking other aspects of it even more valuable.

3. Approximating a function by another function. One of the most widely used approximation procedures inanalysis and applied mathematics is the procedure of approximating some functionby another function. This technique is particularly useful if theoriginal function is compli- cated and the approximating function is simpler.We are not mathemati- cally able to find the values of certain functions,but we are able to use these functions because we catiapproximate them by ones which we can evaluate.

A classic example of this sort of a procedureis given by a very im- portant theorem from analysis. The first Weierstrass ApproximationTheo- rem [11 asserts that

if f(x) is a function which is continuousin the finite interval Ca0b3 then for every e > 0there exists a polynomialPn(x) of degree n n(e) such that the in- equality

If(x) Pn(x)I se

holds throughout the intervalra,b3.

The theorem is really very powerful for two reasons. First, any continuous functions on a closed interval can beapproximated by a poly- nomial of degreen , where the degree n depends only upon how close one wishes to approximate thefunction.When we say any continuous func- tion, this includes an extremely large collectionof functions which are riot polynomials. It would include, except for certaininstances or for certain levels of definition, the logarithmic,trigonometric, hyperbolic and magyothers. 1 The function f(x) = on[01.9] can be approximated by a polynomial in 1 -x The functions which can be approximated seemendless, except they are all continuous on [alb].

Second, these functions are approximated by asingle kind of func- tion, namely the polynomial. The polynomial function is also a con- tinuous function and, perhaps is the most usedclass of continuous func- tions. It is clearly simple, and it is the first kindof function dis- cussed in secondary school mathematics. It is a well-defined classof functions with many facts already established andwell-known. There is no general class of continuousfunctions which is simpler than thepoly- nomial class, and at the same time, has so muchknown about it.

Another theorem which illustrates this type ofapproximation is

22 WeierstrassIs Second Theorem [1] and it says

If F(t) is a continuous function of period2u, then for every e > 0 there exists a trigonometric sum

Sn(t)811 ao + (ak cos kt + bk sin kt) k -i

where n = n(e), such that

1F(t) Sn(t)1 e

for allt .

This theorem is not as general as the firsttheorem but illustrates a similar kind of procedure. Here periodic functions are approximated by a function of sines and cosines. The two observations about the first Weierstrass Approximation Theorem would also seem tobe correspondingly appropriate for the Second WeierstrassApproximation Theorem.

Both theorems are existence theorems in the sensethat an nth degree polynomial exists which approximate anf(x) on [a,b]. It may be very difficult to actually find such a polynomial.However, it turns out that Bernstein [3] provided a subclass of polynomials which woulddo the job. The Bernstein theorem states that

If f(x) is continuous on [0,11, its Bernstein polynomials Bn, where

n

Bn(x) <11ni )(1 xko. 20n-k(n) k kt(n - kg 0 k

converge uniformly to it on[0,1] as n 4 0.

Thus we have a specific collection of polynomialswhich are known and which can be used to approximate any continuous function on[0,1].

it. Approximating Roots of One Function by Roots of AnotherRelated Function. The last type of approximation technique tobe mentioned is not obvious and, as we shall see later, has its difficultiesand limitations. We shall describe the technique by using theexample men- tioned in section I. SuppoL,, we drop a stone into a well and measure the time from releasing the stone until the soundof the stone hitting the water reaches our ears.Suppose this time is 3.1 seconds. Then the time for the stone to fall can be written asti and the time for 1 the sound to return to the ear as t where 1100 ft. per sec. is 2 1100

23 Thus the total time elapsedis the assumed velocityof sound in air.

d 3.1 (1) tl t2 TAT 117%

If we want to know the depthof the well, we must solvethis equation for technique of squaring bothsides to eliminate d . To do so, we use the the square root and weobtain

d2 d 6.2d 9.61 (2) 16 1100 1,210,00

This is a quadratic ind and can be solved by thequadratic for- So we look mula but the computationsinvolved are going to be lengthy. because for a way out. Ige observe that the wellcould not be very deep therefore, d/1100 is rela- the two times, t1 and t2have to be small and, tively small.As a consequenced2/1,210,000 would be very small relative consider only to the other terms of(2). If we ignore this term, or

d 6.2d (3) ilTX5 '9.61

Incidentally, the we find a newequation which can be solvedfor d . than solution of (3) ford certainly requiresless computational work solving (2) for d

Solving (3), we find d = 141. Checking this in(2) one can verify that it approximately satisfies(2) and therefore, can be verified as satisfying (1). So a solution to our originalproblem has been obtained.

The basic technique used inthe solving of thisproblem is an ap- proximation. We found equation(2) which needed to be solved. We ob- The tained equation(3) from equation (2) by deleting acertain term. solution of equation(3) is an approximation for a solutionof equation by the (2). Thus we are actually approximatingthe roots of one function in some way roots of another relatedfunction. The related function must depend upon the originalfunction, or in the case ofpolynomial equations, depend upon the coefficients ofthe polynomial.

It would appear, at firstglance, that this techniquemight be ques- blemish or two, but itworks in a tionable. In fact, it is not above a surprisingly large number of cases. We shall explore this kindof an approximation in some depth inlater sections. It is this procedure which the science teacherswould like to see developedin our courses in algebra. four types of approximations 5. Comments. If one looks at these one observesthat they all involve finding onenumber which approximates

214. another number. The basic difference is where does oneget the informa- tion to make an approximation. In the last three types described there is given something about a desired number, i.e., rootof an equation, etc., and the equation has been the basic source ofinformation for the approximations. In sections 2 and L. we did quite different things with the same information, thus providing different approximation procedures.

IV. Some Theorems Related to A22roximations.

1. Introduction. In this section we shall cite some theorems which have some bearing upon roots of polynomial equations. These are not to be considered all that could be cited here, but are representative of those already available.A few will be essential for later sections and many illustrate related kinds of information.Some are too advanced to be considered at the secondary level, but those which can be proven by elementary algebraic means are proven.The proofs are included merely to illustrate their adaptability to secondary school mathematics.

2. Locations of Roots of P(x) = 0. A short reference to the loca- tion of roots of a polynomial equation was given in the historical back- ground. We would like to expand on this here by citing several different types of locating theorems and concepts. At least one is elementary enough to be intuitively justified at the secondary level. In all the examples selected one determines something about the roots byexamining the coefficients of the polynomial or by examining certaincomputations entirely dependent upon these coefficients.

Mathematically, we are justified in doing this because thecoeffi- cients of any polynomial equation can be expressed in terms of theroots of the equation. For example, a quadratic equation with roots a and b can be written as

(x a)(x-b) = 0

Multiplying and collecting terms we have

x2-(a + b)x + ab = 0 .

Thus any quadratic of the formx2- px + q =0 has its coefficients uniquely determined by the roots of that quadratic. All quadratic equa- tions may be transformed to this form by dividing by the coefficient of x2, if it is not already 1.

A similar argument for a cubic equation produces a form

x3-(a + b + c)x2 + (oh + ac + be )x - abc = 0

wherea, band c are the roots of the cubic. Likewise the nth de- gree polynomial equation has as itscorresponding form

xn-(al + a2 + a3 + an)xn'l+ al a2 a3 an = 0

25 where the signs depend uponn being odd or even.

Thus, every coefficient of any polynomial P(x) = 0 is a function of the roots of that equation and the coefficients should determine some- thing about the location of the roots of P(x) = 0. In what follows the equation P(x) = 0 shall be understood to be

(1) P(x) = xn + aixn -1 + a2xn -2 + an = 0 unless otherwise stated with ai being real numbers.

We take as a first example of location of roots the RouthHurwitz criterion. The Routh-Hurwitz criterion provides an algebraic manipula. tion of the coefficients of a polynomial to find out if the roots are to the right or left of the y-axis. Given the polynomial equated to zero

P(x) = xn axn-1 axn-2 + an = 0 1 2 form the Hurwitz determinants D1, D2,...pn given by

a1 a3 as a2k1

1 a2 O a2k-2

al a3 a Dk 0 2k-3

k = 1,2, ...,n 0 1 a2 a2k-14.

000 000 000 000

0 0 0 0 where the coefficients with indexes larger thann or with negative in- dexes are replaced by zeros.

A necessary and sufficient condition that there be no zeros of P(x) = 0 to the right of the y-axis is that all the Dic's bepositive° For each determinant with a negative sign there is a root to the rjjht of the y-axis. Further, if Dn is the only one which is negative, there is a single root to the right of the y-axis, and this root must lie on the x-axis. If Dn is zero, the single root is at the origin. It is clear from this statement, which is given without proof that one can determine something about the character of the roots of ?(x) = 0 without being able to actually find the roots.

Van Vleck [151 proved a theorem which further aids in the location of the roots of P(x) = 0 if the theorem is applied along with the Routh- Hurwitz criterion. The theorem is as follows: If ci is real and the termsof the sequence

coCl c2 c

cl c2 crol co cl c2 Cl ... Co Clc2 c3 ... c c1 ) . 1 2 c2 c3c4 ...

cncn+1 c2n4.2

are positive,all the roots of the equation

2 211_ co + cix + c2x "' c2nx are imaginary,and all but one of the rootsof the equation

co + cx + c2x2 + + c x211+1 - 0 1 2n+1

are imaginary.

Descartes' Rule of Sign is asimilar kind of criterion fordeter- mining the character of theroots of P(x) = 0. The rule states that the number of positive real rootsof P(x) = 0 is equal to thenumber of changes of sign in thecoefficients (when taken fromxn down to x°) di- The minished by 2k, where kis an integer with minimumvalue of zero. rule simply describes themaximum possible number of realpositive roots to and not actually how manyexists. Descartes' Rule of Sign applied P(-x) = 0 gives the maximum possiblenumber of real positive rootsof P(-x) = 0 and hence, the maximum possiblenumber of real negative roots of P(x) = 0.

These criteria are helpful indetermining some informationabout P(x) = 0 but in no way aid in solvingP(x) = 0 or in approximating any roots of P(x) = 0.

Consider the example

(2) x3 + 6x2 + llx + 6 0 ,

the Hurwitz determinants maybe computed'and they give

D= 6 D= 60 D= 360 1 2 3

27 and the Routh-Hurwitz criterion applied to this casereveals no roots to the right of the y-axis.

The Descartes Rule of Signs applied to this example indicates no changes in sign so there are zero positive real roots. If we replace x by -x in the example, we obtain

(3) -x3+6x2 llx + 6 = 0 and the Descartes Rule of Signs shows 3 sign changes so equation(3) has at most 3 positive roots or possibly only 1.Hence, the original equa- tion (2) has 3 or 1 negative roots. The Routh-Hurwitz criterion applied to (3) indicates 3 roots to the right of the y-axis. Hence, (2) has 3 real negative roots. The Descartes Rule of Signs gave 3 as a maximum num- ber but we could not know that 3 was actually the right number. In this sense, the Routh-Hurwitz criterion isbetter than the Descartes Rule of Signs, but it is also more complicated to apply (in the computational sense).

A second, (and frequently useful ) piece of information about the character of the roots of P(x) = 0 can be found if one candetermine the limits or bounds upon the roots themselves. By this, we mean, is it possible to say that all roots of P(x) = 0 are numerically smaller than some M, and M being a function of thecoefficients of P(x)?A great deal of attention was directed to this kind of a mathematical problem during the early 1900's and much of it is quite appropriate to this discussion. We cite several of these to illustrate this technique.

If we refer to a theorem from complex variables we note that the region in which all roots of P(x) = 0 lie is a circle with center atthe origin and a radius of

(4) 1 + laid max.

max means to use the absolute value of the largestcoefficient The lak of P(x). For example all roots of (2) would be within 12 of the origin. This is true because the roots are -1, -2, and -3. For this example the radius is too large for it to be really effective or to be used as an ap- proximation for the largest root.However, it is very easy to compute.

There are other estimates and approximations for the location of roots of P(x) = 0.We give three such expressions to illustrate further this kind of investigation into bounds on the roots.

The expression in (4) can be replaced by other, yet similar, expres- sions giving the values of the radius of a circle containing all the roots of P(x) = 0. Walsh 116]gave the following limits on roots.

1. All roots of x2 + aix = 0 lie in or on the circle with center at origin and a radius oftall.

28 2. All roots of x2 + a1x + a2 = 0 liein or (Center is on a circle withradius of 1a 1 +v47.7 1 21' always at the origin.)

All roots ofx3+ ax2+ a2x + a3 = 0 lie in 3. 1 or on a circle withradius lall +Fia2 + 671 3

For the example (2) this radius would be

161 + + or approximately

6 + 3.3166 + 1.8171 = 11.1337 which is some better, but not much. This sequence of values for the radius finally can be expressed as

All roots of P(x) =xn +axn'l+ + an = 0 4. . 1 lie in or on a circle at the originand with a radius of

Tall "' FIC11

Carmichael and Mason r53 proved that allroots of P(x) = 0 are in absolute value less than or equal to

1/1 'a112 la212 lan12

For example (2) this would be091 which is actually larger in value than the value of 12 which we obtainedearlier.

Williams [18] later showed that the absolute valueof any root of P(x) = 0 was less than

+ la an A. 1 112 1a2 a1l2 an-112 1an12

Again for our example this would beVIM, or better than the other esti- mates.

From the literature on this subject oneis notable to determine, except for special equationsP(x) = 0, when these values stand a chance of being a reasonably good approximationfor the largest root ofP(x) = 0. There are several sources which giveexpressions for the least root of P(x) = 0 and we cite a few atthis time.

29 Landau [10] was able to show that every equation of the formaxn + x + 1 = 0 has a root, the absolute value of which isnot greater than 2, and that every equation of the form axn + bxm + x + 1 = 0 has a root, the absolute value of which is not greater than 8. These two observations were to establish a chain of discoveries and refinementswhich are excel- lent illustrations of the creative process in mathematics.

Allardice [21 was able to generalize these results to obtain the following theorems:

Theorem. The equation axn + x + 1 = 0 has a root, the absolute value of which is not greater than n/n-1, which is the value of equal roots.

Theorem. The equation axn + bxm + x + 1 always has a root whose n m absolute value is not greater than (n-1)(m-1)

Theorem. The equation axn + bxm + cx + + aix + ao = 0 has a root whose absolute value is not greater than

ao J2 al n-1 m-1 .k -1 regardless of the other coefficients of the equation.

Applying Allardice's approximation to example (2) we find this value to be18 with 1-11 less than this, a surprisingly good approximation. T Landau's condition would have been a root whose absolute value wasless than 8 and not a good approximation.

About the same time Fejer L61 also generalized Landau's results by a very elegant theorem. We state the theorem for algebraic polynomials with integral exponents although the theorem is more general than this case.

Let

ao + alx +a2x2+ + anxn = 0 be an equation of n+1 terms, ao )i 0, al 0. Let pbe the root of this equation of least absolute value. Then

a() p s n al

30 ao The value n -- gives both Landau's value for axn + x + 1 and al Fejgr's value for the same equations. For our example (2) we see

6 I 18 P 3 TTI

Carmichael and Mason rsi, Montel[113 and others were able to pro- vide a slightly generalized statement on the last root. It is easy to intuitively verify these results for the quadratic equation and should make interesting enrichment material for some secondary schoolstudents.

3. Some Approximation Theorems.

In this section we will note some theorems which have previously been proven and are related to our discussion. One should note that there are various and diverse techniques for isolated instances of root finding. In the following theorems we shall be concerned with the nth degree polynomial equation of the form

(1) P(x) = xn + alxn'l + a2xn°12 + + an = 0

If P(x) = 0 has a numerically large real root it is possible to find an approximation for this root by solving the equation

x - al = 0

The reasoning for this is that the coefficient al in equation(1) is the sum of all roots ofP(x) = 0. If al is this numerically large real root, then

al = + a2 +a3 + . + an where a2,m3,...lanare the other roots of P(x) = 0, al is approximately al if al is large ini comparison to cf2 + a3 + + an. However, this approximation idea has limited application because al must be so large with respect to other roots.

There is, however, a technique for finding this large root and it is stated in the following theorem.

Theorem l.*If al is anumerically large root of P(x) = 0, where

*The Theorem is said to be well-known by Oldenburger[123 and we give a similar proof.

31 7 ,51fw,..si., V77. "raTI

P(x) = xn + alxn'l + a2xn-2 + + an

Then m1 approximately satisfies the equation

x2 +aix + a2 = 0 .

Proof: Let abe the sum of the other roots em ft of (1). It 1 2,3"n followsthat al = -(mi + al). Let (1.2 be the sum of the products of a2/a3"..lan in pairs (that is, (72 = m2m3 + met + m3mt + ) . From generalknowledge about theory of equations the coe. °ficient a2 can be writtenas

a2 = (1,17,2 mlm3 0.1% apn + a2a3 + a2a4 + a3a4 + an-104,n.

Hence,

a2 = al (a2 + a3 + a14 + + an) + a2m3 + a2a4 + a3m4 + or

a2 = + (12 alai

Since mial is large compared to a2 (since 02 does not contain a term with al in it), then we can say a2 is approximately The equation x2+ apt + a2 = 0 can now be written as

x2- + al)x + alai = 0 and ft, is a root of this equation.

There are equations for which x2 + alx + a2 = 0 will not give a good approximation for in in which case we increase the degree of the equa- tion by one to obtain

x3+ a1x2+ a2x + a3 = 0

A similar argument can be given to show that ml, if large, satisfies this equation. One must also assume ft, is the only large root, as one tacitly assumed in Theorem 1.

In fact, one may generalize this to produce the following theorem:

32 Theorem 2. If al is a numerically large root ofP(x) = 0, then al ap- proximately satisfies the equation

xn -1 axn '2 + axn -3 + 0 . 1 2 + an -1

The proof is similar to the proof of Theorem 1.

A word of caution is necessary here. By a numerically large root, we mean either positive ornegative. The proof essentially rests upon absolute values and this is clear if we goback to equation (2) of section IV, 2.The equation x3 +6x2+ 11x + 6 0 has as its largest root 1-31. If x2 + 6x + 11 - 0 is used as anapproximating equation, the roots of it are

-6 4:16a x 2

approximation The absolute value of x is "6 '"/"T III 17 a reasonably good 2 ° for 1-31 considering that1-31 is not relatively large with respect to the other roots. If one takes the equation

x3- 9x2- 12x +20 =0

which has roots of 10, 1 and -2, theapproximating equation becomes

x2- 9x -12 0

and its largest root is 9 1579or approximately10.18, which is a 2

very good approximationfor the largest roots which was 10.

We mentioned previously in section III theWeierstrass Approxima- tion Theorem where a function isapproximated by a polynomial under a certain set of conditions. We restate the theorem for immediate reference.

Weierstrass Approximation Theorem: If f(x) is a continuous func- tion on the closed interval[alb], then for e > 0 there exists an n =n(c) and a polynomial Pn(x) of degreensuch that

(2) 11(x)-Pn(x)1 < e for a s x s b.

The polynomial Pn(x) and nare not unique. That is, one may find a

33 second polynomial of degree mwhich would satisfy (2) for the same e. Thus, polynomial approximationsto f(x) exist which have apredetermined accuracy onra,b1. The theorem says the polynomialsexist, but does not give any clue as to how to findthem.

One class of functionsf(x) which are continuous is the class of polynomials. Thus P(x), a polynomial ofdegree m is a continuous there- function. In fact, it is continuousfor every real number x, and fore, would be continuous for anyclosed interval Ca,b). The Weierstrass Approximation Theorem applied to thiscontinuous function Pm(x) over some interval ralbl states there exists apolynomial of degree n which is within a certain els distanceof Pm(x). It is clear that one of thepoly- nomials which approximatesPm(x) is the polynomial Pm(x) itself. However, this is not necessarily the only one as we observed in thepreceding paragraph. Immediately, two questions which areof importance to this study need to be answered. First, are there polynomialsof degree n where n s m, which approximatePm(x) to a desired accuracy?' Second, if there is a polynomialPn(x), n s m which approximatesPm(x) on some in- terval, do the roots ofPn(x) approximate the roots ofPm(x)?

What we really would like to find is apolynomial Pn(x), n< m, which approximatesPm(x) and where the coefficients ofPn(x) are obtained in some fashion from thecoefficients of Pm(x). If the roots of Pn(x) could approximate the roots ofPm(x) whereever possible, then we could solve for the roots ofPn(x) 0 and obtain approximationsfor the roots of Pm(x) 0.

In what follows we shallestablish some conditions under whichthe answers to the twoquestions just raised are affirmative,and, in par- ticular, which allow us to dowhat we expressed in the lastparagraph as a desire tobe able to do.The following theorems, someof which were not found in the literature,will outline some of theseconditions. We shall be much more specific in thesection on quadratics.

Let P(x) = xn +alx11-1 + a2xn"2 + + an = 0 with singleroots. Let Q(x) be composed of all termsof P(x) exceptthat one term, say b.x11"j0lsjsn-1,isdifferentfromthecorrespondingtermweiajx n by an arbitrarily small amountless than e over the interval[0,13. Let 0 be a lower bound and 1be an upper bound on the rootsof P(x) - 0. Then we have of Theorem D3. If x2 is a root ofQ(x) = 0, then it is an approximation a root xl ofP(x) = 0.

Since P(x) is continuous andQ(x) is arbitrarily close toP(x) over We know that an interval[0,11 the Weierstrass Approximation Theoremholds.

1P(x)-Q(x), = Ibixn-j - < e for some fixed j 1 s j s n-1.

Suppose x2 is a root ofQ(x) = 0 and xl is the corresponding root of P(x) - 0. We know that

IP(x1) P(x2) (4(x2)1 s Mx].) Cl(x1)1 IP(x2) Q(x2)I

< c + e

But Q(x2) 0 and P(xl) 0 so

1P(x2)-Q(x1)1 < 2e

This can be written as

layTi <

Since x2'i and are positive, and we lose no generality assuming aj >bi, then we have

laix3-3-aixT.31slaix3-J- < 2e

Therefore

lail - xi'3l < or

<2 IxTmj 'ail but

41.-j -1) x'i-xy. = (x -2. xl + 2 xl)(x3 -1+

We have

1x2 1x2-j -xil -1+x2-j-2 xi + + < 2 !ail or 2e

I IM1 where

xr1..j.11 14-j-1 xl +

35 Since 'ail and IMI are finite, we have

1x2 xil < el

Thus x2 and xi are arbitrarily close, dependingupon the closeness of P(x) and Q(x). Hence, a root of Q(x) = 0 is anapproximation to a root of P(x) = 0.

This theorem is intuitively obvious if oneconstructs a graph of P(x) and looks at where Q(x) mustgo.

Q(x)

)44

Since Q(x) must lie between P(x) + e and P(x)-e it must also cross the x -axis in some small interval about xl. Hencean approximation for xl is obtained at x 2.

It is also intuitively obvious that not all roots of Pn(x) = 0 can be roots of any polynomial of degree n-1.The following theorems show this to be true and under what circumstances.

Lemma Dl. If bi,b2,b3,...,bn are the roots of the nth degree polynomial equation

P(x) = xn + aixn'l + a2xn'2 + + an = 0 and if an / 0, then bi / 0, i = 112,...,n.

36 # O. For Proof: Let bi be a root ofP(x) = O. Since an # 0, then bi if bi = 0, then (b )n (b)n-1 + anII0 1 1 -r and every term containing a bi is also zeroand this would mean an would also have to be zero for bi to be a rootof P(x) = O. Since an # 0, then bi # O.

Theorem Db. Let 131013201)30 ,bn be the roots of the nth degreepolyno- mial equation n-2 xnml+ a2x + + an = 0 P(x) = xn + a1

Further, let an be non-zero. Then no root of P(x) = 0 can be a rootof Q(x) = 0, where Q(x) is obtained from P(x) by deleting thexn term.

Proof: Assume bi also is a root of Q(x) = O. Then Q(bi) 111 0 and we already knowP(bi) = 0; but P(x) = xn + Q(x) for all x and therefore,

P(bi) (bi)n + Q(bi) or

0 = (bi)n + 0

The only way this last equation can holdis for bi = O. But by the lemma bi # 0, thereforeobi cannot be a rootof Q(x) O.

Theorem D5. Let bi, (i = 1,...,n) be the rootsof P(x) = O. Let Q(x)

P(x)-aixn'i, j= aj # 0, an # O.Then no root of P(x) = 0 can be a root of Q(x) = O.

Proof: Since an # 0, then bi # 0, Assume bi is a root of Q(x) = O. Then we have Q(bi) = P(bi) ai(bi)nmior 0 so 0 -aj(bi)n'i. Since Q(x) = O. a.,i0and(bl.)21-i #0dthen bi cannot be a root of

In both of these theorems the condition an# 0 holds and the theo- rems hold for large collectionsof polynomial equations, but not all. For example, if P(x) =x4-x3+ x2 - x = 0, a root of thisequation satisfies the equation

Q(x) = -x3 + x2 x = 0

or

37 Q(x) = x4 -x3- x= 0 .

It is clear that the root which does this is the zeroroot. We will now examine the cases in which an = 0 and see if we canenlarge the collec- tion of polynomial equations which satisfy TheoremsD4 and D5. If an 0 0 then the polynomial equation P(x) = 0 can be written inthe following manner:

P(x) = xk R(x) where k = 1,2, .0

If an is the only zero constant of P(x) then k = 1. R(x) is a polynomial of degree n-k with the constant an_k being nonzero.This expression for P(x) is obtained for any P(x) with an = 0 by factoring the highest power of x from P(x) and R(x) is the remaining factor. One may deduce two interesting relationships between the roots ofP(x) = 0 and R(x) = O.

Theorem D6. If an = 0, b1 m 0 and P(bi) = 0, thenR(bi) 'I O.

Proof: Let P(x) = xk R(x)

or

P(x) = xk (xn 'k + aixn 'k '1 + + an -k)

where

an_k yi 0

(If an_k were equal to zero, an additional factor of x could be removed from R(x) and some other nonzero coefficient would bethe constant term of a new R(x).) Then consider

n-k-1 R(bi) = brk + a(b-) 1 an-k

= 0 we have Since b1

R(b1) = R(0) = On'k +al(0)//44-1+ + an-k

and

R(0) = an_k

Fencelthe zero root of. P(x) = 0 is not a root ofR(x) = O.

Theorem D7. If an = 0, and bi is a nonzero root ofP(x) = 0, then R(bi) = O.

38 Proof: Let P(x) = xk R(x). If bi # 0 and is a root of P(x) = 00 we have

P(bi) = 0 (bi)k R(bi)

Since bi # 0,(bi)k #0 and consequently, R(bi) must be zero. Hence bi is a root of R(x) = O.These two theorems together say that the nonzero roots of P(x) - 0 (where an 0) are the nonzero roots of R(x) = 0, and any zero roots of P(x) = 0 are not roots ofR(x) = O. Lemma 1 further says that R(x) so 0 has no zero roots. We shall now reconsider theorems D14 and D5 in a broader sense.

Theorem D8. Let b1lb2,...lbn be roots of the nth degree polynomial equa- tion

P(x) = xP + alx114 + a2x114 + + an In 0

Then no nonzero root of P(x) = 0 can be a root of Q(x) n 0 where Q(x) is obtained from P(x) by deleting the xn term.

Proof: If an # 0, then D4 holds. If an = 0, then P(x) 0 takes on the form

P(x) = xkR(x) =xk(xp-k + an -k) = 0

Then Q(x) = P(x) - xn. If bi # 0, then

Q(bi) P(bi)- bi .

Since, by hypotheses P(bi) = 0 and (bi)n.yi 0, we have

Q(bi) = (bi)n # 0

and bi is not a root of Q(x) O.

Theorem D9. 'Let bi, i = 112,...0 be' the roots of P(x) =0(where P(x) is the polynomial of Theorem 108). Let Q(x) = P(x) - ajO-J where al # O. Then no nonzero root of P(x) = 0 can be a root of Q(x) O.

Proof: If an ,0, then Theorem D5 holds. If an - 0, then we can write P(x) = 0 as xk R(x) 0 as we did in Theorem D8. Then Q(x),

=xk R(x)-ajxn-j. If bi # 0, then we have

Q(bi) = 0i)k R(x) ai(bi)nmi

but, by Theorem D7, if P(bi) = 0, P(x) =xk R(x), and bi # 0, then R(bi) = 0, and

39 Q(bi) =01 0 -aj(bOnnsi or

Q(bi) =-aj(bi)11-i #0 .

Hence bl cannot be a root ofQ(x) al 0. The sequence of theorems have shown that a nonzero root of apolynomial equationP(x) = 0 cannot be a root of a polynomial equationobtained by deleting some nonzeroterm of the original polynomial equation.

If we examine Theorems D3 andD9 together, we may observe one very interesting possibility.Theorem D9 says no nonzero rootof P(x) = 0 canbearootofQ(x)=P0 ajx se 0. Theorem D3 says if P(x) and Q(x) are alike except for some term aj.xrilej and these terms are arbi- trarily close, then a root ofQ(x) = 0 is an approximation to a root of F(x) = 0. The roots cannot be the samebut can be close to being the same. Theorem D3 can be used on theP(x) and Q(x) of Theorem D9 if aj is rela- tively small with respect to theother coefficients andQ(x) then can be takentohavenoterri ajx in it. In this case we have the following theorem.

Theorem D10. If P(x) = 0 has a termaixn-i with airelatively small with respect to the other coefficientsof P(x), thenthe polynomial Q(x) P(x) = 0. =P(x)- = 0 has a rootwhich approximates a root of

the Weier- IfajxIlmj is relatively small on some interval then, by strass Approximation Theorem,Q(x) is an approximation for.P(x) over this interval. Hence, by Theorem,D3Q(x) = 0 has a root which is an approxi- mation for a root ofP(x) = 0, and the theorem is proved.

In the section on thequadratic equation we shall seethe implica- tions of this theorem inapplication.

V. Some Approximation Techniquesfor Quadratic Equations.

general theorems on 1. Introduction. We have already presented some approximations of polynomials of nthdegree. Now we will direct our attention to the quadratic equationand the approximating equationsfor the quadratic.The quadratic is especiallyimportant because it is the most elementary polynomial, besidesthe linear one, which is found inthe algebra taught at the secondaryschool level. Moreover, we can be more specific for this polynomial inseveral ways and this makes theapproxi- mations more valuable.

We will not consider the case where weapproximate the roots of a quadratic by the roots of anotherquadratic. Since we would have to solve the second quadratic anyway, theeffort involved might just as well be directed toward solving thefirst quadratic. It is also true that, for elementary problems, this is theleast interesting case. We will consider approximating the roots of aquadratic by deleting a term of the quadratic and solving the resultingequation for its roots. This will be done by considering the variouspossible types of roots of the quadratic equation and the various terms which canbe deleted for each type of root considered. We will be concerned with the deletionof only one term from thequadratic, and there will be a total of six casesto he considered.

2. The Quadratic Equation with Real ,Roots.Write the quadratic equation P(x) = 0 in the form

P(x) = x2-(a+b)x + ab = 0

wherea and bare the real roots ofP(x) = 0 and a # b. We now con- struct Q(x) = 0 from P(x) 0 by deleting a term of P(x) = 0.

Case A. The first term we shall delete isx2. Let

Q(x) = P(x) - x2 = 0 .

Then

Q(x) = -(a+b)x + ab = 0 .

We wish to know how the root of Q(x) = 0 is relatedto either root of P(x) = 0. Solve Q(x) = 0 for x .Since

Q(x) = -(a+b)x + ab = 0

we have ab x a+b

Note that the expression ---ab is a symmetric oneina andb, so a+b whatever we might say relative to a could be said relative tob . b Since a # b, let a be greater thanb . If ais large relative to

the expression22Lbecomesrelatively close to b i.e., ::+b is an ap-

proximation for b if ais large relative tob . But this means that ab 7743 approximates aroot of P(x) = 0 under this set ofconditions.The

41 approximation one has canbe deter- related question ofjust how good an mined by looking atthe followingtable.

ofAL Table I - Values a+b

a = 20b a = 100b a a n 2b a = 4b a = 8b b b b b b b 20 100 ab 5b2 14 8 2-1b im. b a+b

ab expression gg takes onvalues which get As one observesin the table the (relative tob). In fact, closer and closer tobas abecomes larger ab increases withoutbound. Moreover, from the limit of a.--ftisi b, as a particular case, theapproximation ob- the conditionsdescribed for this tained is alwaysfor the smallest ofthe real roots. direction, i.e., is sufficientlylarge in the negative Ifa Examine the table la' > b, the same sort ofapproximation is obtained. below. ab Table II - MoreValues ofa+b

40100b a = -2b -4b 40.8b =10b

b= b b b b b 100b ab . 8b 10b DT. = +2b 7 -9 99

b ifla' is greater than b .The Thusab is an approximation for a+b depends only onthe' closeness of theapproximation, for a givenb , value of a. Tables I and II andputs this .n If one takes thevalues found in hyperbola for agraph, prO- graphical form oneobtains the equilateral Table III is onesuch graph for a vided one fixes bat sine value. made about fixed b .The graph reflectsthe two observations ab provided lal isrelatively large. being an approximationfor b , a+b abdoes not approximateb in the central Or should also note that a+b--- In the event ais close region of the graphwhere a is close to -b. small and thecoefficient of x to -b in value weobserve (a+b) is very terms of the quadratic. This gives is small withrespect to the other

14.2 Table III - Graph ofjab , for fixed b . a+b

Values of a.

6 ab us f ctvas to when we eon fiepend upon37-4.13 being a epod approximation

['or roet P(x) 0. We )ormlize this in the followingtheoreei.

Theorem D11. If P(x) = x2 (011))x 4- ab= 0, with a,b real roots of P(x)= U. a X 0, b 0 and lei > b, then the equation

Q(x) = P(x)-x2= -(a+b)x + ab = 0 ab has a ro)t, x =17.6., which is an approximation for the smallest root of P(x) = 0, provided the coefficient of x2is relatively small in compari- son to t.te coefficient of xand the constant term ab.

Comment 1. Note that if either root iszero, we have a special quadra- tic which need not be solvedby approximation.

ab Comment 2.Note that the keyas to when one can depend upon to be an a+b approximation of a root of P(x)= 0 is the coefficients of the polynomial P(x). the have written P(x)- 0 in the form x2-(a+b)x + ab= 0 so as to reduce the number of unknowncoefficients. If the coefficient of x2,

i.e., 1, is small relative to 1-(a+b)!and !abl then It. providesan ap- a+b proximation to a root of P(x)= 0.

Comment 3. The theorem does notsay so, but since lal > b, the rootap- proximated is, numerically, thesmallest.

Comment 4. The error of approximationis likewise not given in the theo- rem, but it is easily determined bycomputing the following: ab Error = b- a+b However, this requiresa knowledge of whatb is and if we must know b in order to compute theerror then we already know a root of P(x)= 0 and all this approximation businessis needless. However, the Tables I and II suggest a means todetermine relativeerror and Table III gives enough of a graphic picture toapproximate the relativeerror. Error and rela- tive error are strictly functionsof two variables, a and b and, therefore, for any fixedequation P(x) = 0 the amount oferror is pre- viously fixed by the roots ofthe equation.

Comment 5. This theorem justifies theprocedure used in the introduction to solve a certain well problem and putsthe procedure on a sound mathe- matical basis.

Case B. Consider how Q(x)= P(x) (a+b)x= 0; that is, take P(x) = 0 and delete thex term. We are still assuming a and bare real roots of P(x) and a X b. Then

Q(x) = x2 + ab= 0

104 and

x2= -ab for x2 of -ab to have real roots,either a orb must be negative. Assume a is. Then bis positive and -ab ispositive. Hence x =41/75,173 and the latter is a real number. But v75b can be av approxi- mation for bonly when a = -b. In either case the expression x =V-ab can approximate aroot of P(x) se 0 only if ais approximately equal to -b. When this happens the coefficientof x i.e., -(a+b) becomes very from P(x) = 0 small and approximately zero. Thus the term being deleted has a relatively small coefficientin comparison to the othercoeffi- cients in the equation.

For example, the equation

x2+ .01x =(12)(11.99) al 0 has a root which can beapproximated by a root ofx2-(12)(11.99) = 0.

i.e., x =407211.99)

In fact, we have approximationsfor both roots. These observations can be stated in the followingtheorem.

Theorem D12. Let P(x) =x2-(a+b)x + ab = 0, with aand breal, a yi 0, b 0. Let1-(a+b)1 be relatively small as compared to 1 and lab!. Then, both roots of P(x) = 0 areapproximated by the roots of Q(x) = x2 + ab = 0.

Comment 1. If -(a+b) is approximately zero,thenais approximately ab the negative, ofb . Thus aand bmust be real roots, otherwise would be complex or, numericallysmaller than a+b and neither areto be allowed in the theorem.

Case C. Now we consider the case whereQ(x) = P(x) - ab = 0 or Q(x) = x2-(a+b)x = 0. We still assume a and bare real roots of P(x) = 0, a i b. Solving Q(x) =x2-(a+b)x = 0, we obtain

x 0 and x = (a+b) .

If ab is to be relatively smallwith respect to 1 and1-(a+b)11 then either ais close to zero or b is close to zero. If ais approxi- mately zero thenI-(a+b)1 is approximately b . In this case the two roots of Q(x) = 0 give approximationsfor the two roots ofP(x) = 0. Similarly, ifbis approximately zero,1-(a+b)1 is approximately a and again we have two approximationsfor the roots of P(x) = 0.

We have, therefore, the following theorem.

Theorem D13. Let P(x) =x2-(a+b)x + ab = 0, with a and breal,

145 small as compared to 1 and 0, b 0, a > b. Let labi be relatively a approximated by the rootsof 1-(a+b)1.Then the two rootsof P(x) = 0 are Q(x) = x2 (a+b)x = 0. We consider now the caseswhere the 3. The Quadratic withComplex Roots. roots are complex.Suppose

x2- aix + a2 =0

Let a+bi and a-bi bethese roots. rye know complex has complex roots. Thus the quad- roots of. P(x) = 0 occur inpairs and are conjugatepaira3 ratic becomes

(a2+b2)= 0 (1) x2-(2a)x +

Now consider the variousequations obtained from(1) by the deletion of a term. Then Gone A. We begin by consideringthe deletion ofx2.

Q(x) = -tax +(a2+b2)= 0 .

Solving for x we have a2+b2 x 2 a

or a b2 x + 75".

value If a is relatively large andbis relatively small, thR for x becomes approximatelya/2. If bis not small, thenlace/2a and x may be much lar- becomes a significant partof the value for x to a then the root of ger thana/2. But if b is small in comparison P(x) = 0.This Q(x) = 0 enables us to find anapproximation to a root of from Q(x) = 0 and seems to be acontradiction for xis a real number longer a contra- the roots ofP(x) are complex. This contradiction is no The roots of diction if one locates thesevalues in the complexplane. P(x) = 0 and Q(x) = 0 are put onthe graph as follows:

coif: 4"1111111119,1.....r o

146 Since bis relatively small the roots of P(x) = 0 lie close to the x-axis. The root o' Q(x) - 0 is a/2 and it does not approximate either root of P(x) = 0, but a does: Thus, for this case, Q(x) = 0 gives a value which can be multiplied by 2 to get an approximation of either root of P(x) = 0. Therefore, we may state the following theorem.

Theorem DI4. If P(x) = x2- + a9 ms 0 has complex roots a + bi, and a is relatively large andhis relatively small, then the root of Q(x) = -a lx + a2 - 0 (or Q(x) P(x)-x2Is 0), when doubled, becomes an approximation for the roots of P(x)= 0.

Case B. Instead of deleting x2 from P(x) = 0, consider the Q(x) obtained by deleting thex term. We have

Q(x) = x2 + (a2+b2) = 0

Solving for xwe obtain

x = 4:4a4b2) or

x = iv4T;i

If a is relatively small, then the value of (a2+b2) is approximately b2,ar1 thusxis approximately + ib. Hence, ifais small, the solutions of. Q(x) = 0 are approximations to the solutions of P(x) = 0. Again this bewlmes obvious if we locate these valueson the complex plane.

I a+ b

I/1 1111

-

Thus we have the following theorem.

147 Theorem D15. If P(x) = six + a2 = 0 has complex roots a bi, and a is relatively small and bis relatively large, then the roots of Q(x) - x2 + (a2+b2)- 0 (where a2 =a2+b2),then the roots of Q(x) 0 are approximations for the roots of P(x)m O.

Case C. If we delete the constant termwe know that Q(x) 0 is defined to be P(x)- a2 = 0 or

Q(x) m x2- 2ax = 0 .

The roots of Q(x) 0 are x = 0 and x = 2a. If the roots of P(x) m 0 are a 4: bi, then the only way for the roots of (x) to approximate the roots of P(x) = 0 are for the roots to be close tozero or close to a . In the first instance, a andbwould both be close to zero andso would a 4: bi. Thus both roots of (4) are approximations to theroots of P(x) = O. In the second instance, ifbis relatively small then 1/2 of the root x m 2a will give an approximation toa * bi.

From this analysis we can state the followingtheorems:

Theorem D16(a). If P(x) x2- alx + a2 - 0 has complex roots a 4: bi and a is relatively large and .bis relatively small, then the nonzero root of Q(x) x2 sax = 0 multiplied by 1/2 is an approximation for either root of P(x) = O.

Comment: The roots of P(x) - 0 lie close to the x-axison a graph and thus a is an approximation for either root. Notice the similarity of this case to Case A considered under the sectionon complex roots.

We also have one other theoremwe may state at this time.

Theorem D16(b).. If a is also relatively small in Theorem D16(a), then both roots of Q(x) = 0 are approximations to theroots of P(x) = O.

This must be true for if botha and bare very small, then a t bi both lie close to the origin. Hence either 0 or 2a will be approxima- tions for a bi.

It is interesUng to observe that in allcases of complex roots the amount of error of an approximation toa root of P(x) = 0 obtained from a Q(x) = 0 depends almost entirelyupon just one part of.the complex root, that is, either the real partor the imaginary part. The graphs asso- ciated with the various theorems of this section showthis rather clearly.

4. Comments and Observations. In every case considered wewere able to deduce conditions under which Q(x)= 0 produced roots which were approxi- mations to P(x) = O. In each case the condition containedsome stipula- tion upon the roots in order to insure that approximationswere possible. If we do not know what the rootsare, then we cannot insure whether the

14.8 and pro- We could make someassumptions however theorem applies ornot. necessarily safe in of the assumptions. This i.3 not ceed on the basis observation about the There is, however,an the game ofmathematics. exclusive incharacter. six cases we haveconsidered. They are mutually just certainconditions and no This means we can useeach theorem under others. 7x + 9 0. Looking over the Take an example. Suppose P(x) =x2- which couldpossibly apply(without) 6 theorems we concludethe only ones Dll and D14. These two arethe only knowing the rootsof P(x) = 0 are of x2 is thenumerically smallestcoeffi- ones inwhich the coefficient + 9 in 0to approximate a cient. In both theorems weused Q(x) = -7x to the smallestroot root of P(x) alO. Thus x 1111;is an approximation P(x) = 0 has real orcomplex roots: of P(x) = 0regardless of whether (7 +b2= 9 musthold If P(x) = 0 hascomplex roots, weknow that must be relatively But for TheoremD114 to apply b and b = J9-(2)2 . true in this case. Hence, D14 small with respectto a and this is not is out, leavingonly D11. -(a+b) -7 and one If P(x) = 0 hasreal roots(and it has) then 9 40 9 other must be +7 - or + --, and we root is approximately so the theorem D11. observe that we werejustified in using selection, check thecoef- wished to insurethe right fl If one really the the discriminant. Not only does ficients ofP(x) = 0 and compute but the size P(x) tell if the roots arereal or complex, discriminant of roots negative determineswhen the complex of the discriminantwhen it is fl imaginary axis. may lieclose to the real or really need toknow the roots to It turns out thenthat we do not because thecoefficients choose the appropriateapproximation theorem This is due tothe fact thatthe of P(x) = 0 canreallytell us this. functions of theroots and, hence, coefficients ofP(x) =0 are specific but know how toread them. tell us as much astheroots do if we about quadraticshas to do One last observationwhich can be made differ by a smallamount. with two quadraticfunctions which aand b are rootsof P(x) - 0. Let P(x) =x2 (a+b)x + ab, where Q(x) is a quadraticwhich differsfrom P(x) Let Q(x) 141P(x) - el where Approximation Theoremand Theorem D10 by exactly e. From the Weierstrass approximate the rootsof P(x) = 0. In this we know theroots of Q(x) a 0 case we have

Q(x) = x2 -(a+b)x + ab - e = 0

1i9 Solving by quadratic formula we have

(a+b) (a +b)2- x = 2

(a+b) 4:V/(a-b)2+ICE 2

(a+b) ffVf(a-b)2+ V51 2

(a+b) + (a-b) (1) x + = a + 2 and

(2) x (a+b) (a-b) - = b - 2

Thus the roots of Q(x) = 0 differ from the roots ofP(x) = 0 by at most When Q(x) differs from P(x) by exactly e. A similar statement can be made regarding a pair ofcubic functions which differ by exactly

e , except the roots of one cannot differ from the roots of theother by more than K where K is a specific function of the coefficients of the cubic P(x). It is probably true that a corresponding statement can be made for the nth degree polynomial.

VI. Approximations and Other Equations.

On the basis of what we have already seen, one would hope that the concepts of sections IV and V could be extended to other polynomial equa- tions of higher degree. The problem of such an extension is that we can- not always solve an nth degree equation and, therefore, don't know if approximations to the roots of one equation can be actually computed from a derived equation. The Weierstrass Approximation Theorem and Theorem D10 state that approximations do exist but finding them is another matter. Even the cubic equation presents some serious problems as far as applying the concepts of section V. We shall examine this one very briefly.

Suppose P(x) = x3 + a1x2 + a2x + a3 = 0. If we delete certain terms one by one we have thefollowing derived equations:

Ql(x) = alx2 + a2x + a3 = 0

Q2(x) x3 a2x a3

So Q3(x) mix3+alx2 + a3 0

Q4(x) = x3 +alx2+ a2x = 0

would have Equations Q2(x) = 0 andQ3(x) = 0 are both cubics which solutions for to be solved toobtainsolutions which approximated precisely P(x) = O. There is really noadvantage in solving these as exerted as would havebeen the same kind of,effort would have to be One necessary to solvethe original equation,which is also a cubic. obtain the actual might just as well solveP(x) = 0 in these cases and roots. root of P(x) = 0 isapproxi- In the case ofQh(x) we are assuming one obtained by solving aquad- mately zero and theother roots are readily applied, obtain solutions ratic equation. One could, if the theorems V but this would befinding of the quadratic viathe theorems of section and this could be approximations to approximationsto the actual roots unsatisfactory as a process. promise is Q1(x) = 0, The only derivedequation which shows any since it is a quadraticderived f:omP(x) = 0 on the condition that coefficient al, a2 and a3 areall relatively largewith respect to the (which are x3, which is 1. Without going into allthe possibilities of examples really beyond the levelintended in thisstudy) we will cite some to show how theapproximations turn out. Ql(x) = -9x2 - 12x Example 1. If P(x) =x3-9x2-12x + 20 = 0, then + 20 = 0 and wehave

-(-12) f1(-12)2-4(-9)(20)

or 12 x -18 d -41.39an Taking Vga. to be29.39 we find xto be approximately Thus the roots of 39 The three roots ofP(x) = 0 are 10, -2 and 1. 17.18 of P(x) = O. The Q1(x) give approximations tothe two smaller roots third root ofP(x) = 0 can be approximatedby the expression

20 (-41.39V17.39N -73--7V-18--) of the product ofthe three roots. This is true because20 is the negative 200 = 0 then Q1(x) =-10x2-6x Example 2. If P(x) =x3-10x2-6x + 200 = 0 and 6 +U3,6 x -20

The roots of Taking VST16 to be89.6 we find x to be-4.73 and 4.18. the real root P(x) = 0 are 7 i and 4 so oneroot of Q1(x) approximates of P(x) = 0. 01(x) = 13x2 + 32x Example 3. If P(x) =x3 + 13x2 + 32x + 20 = 0 then + 20 = 0. But -32 VT-376 x8` 26

If we consider and P(x) = 0 has onlyreal roots of -10, -2and -1.

lx1 = -32 26

we find

1.1 3'41

and this is numerically anapproximation for the smallestroot.

These examples are not, intended totake the place of the theory the which could be developed, butmerely serve to show that, in one case, concepts of section V on thequadratic are extendable, withmodifications and additional conditions, tothe cubic equation. Since the solutions of the general cubic equation arenot a part of the algebraicmaterials at the secondary school level, there seemsto be little need indeveloping the approximation theory for thispresentation.

It should be noted thatapproximations to solutions of annth degree equation can be obtained by methodssuch as Hornerls method orothers of a similar nature. These methods are adequatelydescribed in any of the books on theory of equations(for example see Dickson, Theory ofEquations).

VII. Implications for Secondary SchoolMathematics.

In this last section we wish torelate the previous sectionsto the business of teaching secondaryschool mathematics and indicate someim- plications for the future.We do this in the context ofanswering the original basic question whichmotivated this entire investigation.

We wished, if possible, to determine ifthere was a systematic body

S2 the technique ofapproximating a of mathematicalknowledge which covered of a derivedpolynomial and if it root to a polynomialby finding roots knowledge in someconvenient formwhereby were possibleto present this established in coursesin elementary a program ofinstruction could be We believe the answerto this algebra at the secondaryschool level. for both parts and weelaborate two part questionis in the affirmative by giving our reasonsfor this belief. Appendix, we note thefour types In reviewing sectionIII of this for which an extensiveamount of of approximationslisted include three structured mathematicsexists. The three are by another number 1. Approximating a number 2. Approximating a root of anequation by another function 3. Approximating a function old techniques,which we hope Of these three, thefirst two are quite material of sectionII amply indicates. The techniques the historical computations to such involved in these two rangefrom simple arithmetic formula or othersof a complicated computations asfinding Sterling's in mathematics. Approxi- similar nature fromthe theory of numbers area equation can be assimple as guessing or ascomplica- mating roots of an derivative of the recursive formulasdepending upon the ted as using well founded In either case,these procedures are polynomial equations. although they may and are an integralpart of themathematical structure In fact, the firsteduca- occur atvarious levels withinthe structure. stated at this point, tional implicationmight very well be and universitylevel mathe- In the processof modernizing college discontinued several coursesas a matics, we have,for the most part, required of thosewishing to be mathema- main portion ofthe mathematics Of particular impor- ticians (and I includeteachers in thisgroup). related of all such courseswas the tance to this studyand most closely Equations. This type of course courseoffered by the titleof Theory of catalog. There are probablygood has nearly disappearedfrom the college of such coursesapparently has reasons forthis. Much of the content notably the abstractalgebra. teen absorbed, intoother courses, most examines these other coursesone findsthe so-called However, if one they are too have frequentlybeen excluded because elementary concepts concepts in a College One may find someof these elementary elementary. students coming to Algebra course, butnot all. With more and more Algebra as a backgroundin mathematics,it seems college with College include, in a content of thesecondary school must that the algebraic equations. The significant way, thebasic concepts fromthe theory of This would meanthat key here is thephrase, "in asignificant way". of equations, such as techniques for findingapproximations to roots appropriate part of thecontent of secon- Horner's method,would be an this would be anessential dary school mathematics. It would seem that The mathematics isnot part of any acceleratedprogram at thatlevel. We need only findthe proper too difficult forthe secondaryschool. not only feasible,but prac- kind of presentationto make such a program accomplishment inmathema- tieal in implementation. In fact, the major tics education for the next generation might very well be thecreating of elegant but simple ways to present the more complex mathematicsat an s earlier stage of instruction than at present. This would be only one of many possible areas in which this couldbe done.

The third type of approximation referred to in section III,approxi- mating a function by another function, is also a part of anexisting mathematical structure. In fact, it is a foundation principle for the area in analysis which has made possiblemuch of our current mathematics, particularly the applied mathematics. Much of this mathematics depends upon the calculus and, therefore, isnot adaptable to the secondary school curriculum. Moreover, there appears to be no mathematical reason orneed to suggest the necessity for such an adaptation.

With respect to the fourth type of approximation given insection III, it is possible to make more positive statements ofimplication for secondary school mathematics.This type of approximation, the approxi- mating of roots of one function by roots of another relatedfunction, was examined in some detail with various relations spelled outin the theo- rems of sections IV and V. The following observations seem to be perti- nent to these sections.

1. There is a mathematical structure available for thistype of approximation technique. It is not as clearly delineated in the litera- ture as the other types and, for the purposes of thispresentation, seve- ral gaps were found. 2. It is possible to establish certain theorems, basedessentially upon concepts from the theory ofequations and the elementary algebra which show the connections necessary to fill the gaps noted. 3. The essential features of the mathematical structure arenoted in these two sections, first in a general way and then specificallyfor the quadratic equation. This expository presentation was not intended to be complete in every detail and perhaps there is a more elegantpresentation, but the features presented show that it is possible to put somesystema- tic order to this type of approximation. ?. Most of the concepts presented are dependent upon very elemen- tary mathematics of the secondary school level and, hence, areadaptable to this level. By implication, this means much of the material presented herein can be made a part of mathematics instruction at the secondary level. The material presented in this Appendix is intended for thehigh school teacher who will have to make the necessary adaptation tothe classroom situation. There does not appear to be any source available which has done this for the teacher. The fact that it appears to be feasible to adapt theseconcepts to secondary school mathematics and the fact that there appearsto be at least a suggested need for this material seems to imply thatsuch an adaptation should be attempted. The basic suggestion here is that a por- tion of time and effort be directed to making the high schoolstudentls mathematics more relevant to his science instruction. (The same impli- cation could be made relative to the social sciences, but from other facts not necessarily related to approximationtheory.)

54 theory should probably 6. The presentationof this approximation This is essentially apsychological he more intuitivethan rigorous. evidence from groupssuch as question or observationand we have ample Study Groupthat such an approachis feasible the School Mathematics concepts in mathematics. and that it actuallyworks in the classroom on of mathematics inapproxi- In summary, thereis a significant area to secondary mation theory whichis elementaryenough to be adaptable possible to make such anadap- school mathematics. It is mathematically adaptation could make acontri- tation, and, ifproperly presented, this instruction at thatlevel. bution to mathematicsand to science Appendix B: References

Frederick Ungar ri3 Achieser, N. I. Theory of Approximations, Publishing Co., New York,1956. notions of approxi- This book contains almostall of the fundamental and one mating one function byanother function. It is well written of the better sourcesfor this type ofapproximation theory. is Allardice, R. E. "On a limit of the rootsof an equation that [2] Ameri- independent of all but twocoefficients", Bulletin of the can MathematicalSociety, 13(1906-07), 443-447.

approximation des fonctionscontinues par [33 Bernstein, S. N., "Sur 1' Sciences, (Paris), des polynomes", ComptesRendus de l'Academie des 152(1911)

ro Bowen, John J., "Mathematicsand the teaching ofscience", The Mathematics Teacher,LIX(1966), 536-42.

Mason, T. E., "Note onthe roots of algebraic IS] Carmichael, R. D. and equations", Bulletin ofthe American MathematicalSociety, 21(1914), 14-22. einer Fej6, L., "Ueber die Wurzel vomkleinsten absoluten Betrage [6] 65(1908), 413-23. algebraischen Gleichung",Mathematische Annalen, method of successive [7] Ford, L. R., "One solutionof equations by the approximations", AmericanMathematical Monthly,32(1925)., 272-87.

solution of the quartic",American Mathe- 18] Grant, J. D., "A graphical matical Monthly,40(1933), 31-32. types of equa- 191 Kennedy, E. C., "Concerningthe solution of certain tions", American MathematicalMonthly, 39(1932),535-37. th4ore'me de M. Picard", 1101 Landau, E., "Sur quelquesgeneralisations du Annales de L'Ecole NormaleSuperieure, (3), 24(1907), 179-201. 174(1922), [11] Montel, Paul, "Sur untheor'eme d'algebre ", Comptes Rendus, 850-52. (See also pp. 1220-1222 in samevolume.) solution [12] Dldenburger; R., "Practicalcomputational methods in the of eqdations",American MathematicalMonthly, 55(1948), 335-42. cubic equa- [13] Pierce, T. A., "An approximationto the least root of a units in pure cubic tion with application tothe determination of fields", Bulletin of theAmerican Mathematical Society,32(1926), 263-68. cubic and r141 Running, T. R., "Graphicalsolutions of the quadratic, Monthly, 28(1921) biquadratic equations",American Mathematical 415-23. 1151 Van Vleck, E. B., "A sufficient condition for the maximum number of imaginary roots of an equation of nth degree", Annals of Mathe- matics, (2) 4(1903), 191-92.

[16] Walsh, J. W., "An inequality for the roots of an algebraic equation", Annals of Mathematics, (2) 25(1924), 285-86.

117] Wrd.erstrass, K., "Uber die analytische Dars elibarheit sogenaunter willkurlicher Funktionen einer reellen Veranderlichen", Berliner Berichte, 1885.

[181 Williams, K. P., "Note concerning the roots of an equation") Bulletin, American Mathematical Society, 28(1922), 39I -96.

The best single source for a history of equations and their solu- tions is

[193 Smith, D. E., History of Mathematicy, Vol. II, Ginn and Company, New York, 1925.

58 Appendix C: Bibliography

Achieser, N. I:, "On the asymptotic expressions of the best approximation of certain rational' unctions through polynomials", Proceedings of the Kharkov Mathematical Society, 1932. (In Russian)

Bernstein, S. N., "Sur unepropriete des fonctions entie'res", Comptes Rendus, 176(1923),

Conkwright, N. B., Introduction to Theory of Equations, Ginn and Company, Boston, MT

Dickson, L. E. New First Course in the Theory of Equations, John Wiley& Sons, Inc., New ,York, 1939.

Gummer, C. F., "The relative distribution of the real rootsof a system of polynomials", Transactions of the American MathematicalSociety, 23(1922), 265-82

Kazarinoff, N. D., Analytic Inequalities, Holt, Rinehart and Winston, Inc., New York, 1961.

Kellogg, O. D., "A necessary condition that all the roots of an alge- braic equation be real", Annals of Math., (2),9(1907-8), 97-98.

Scarborough, J. B., Numerical Mathematical Analysis, 6th edition, The John Hopkins Press, Baltimore, 1966.

This book contains a good presentation of various numerical techniques associated with approximation theory.

Van Vleck, E. B., "On the location of roots of polynomials and entire functions", Bulletin of the American Mathematical Society,35(1929), 648-83.

This article is an expository presentation of material, related to ap- proximations and is an excellent summary of previous research.It con- tains a very lengthy bibliography(144 entries)on this related material,' the location of roots, inequalities involving roots, upper and lower bounds on roots, etc. Van Vleck did not discuss the finding of roots of a. polynomial by approximation from roots of a derived polynomial.

Van Vleck, E: B., "On limits to the absolute value of the rootsof a po1ynomial", Bulletin de la Societe Mathematique,53(1925), 105-25.

59