The General Quintic Equation, its Solution by Factorization into Cubic and Quadratic Factors

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1 Reserch & Reviews: Journl of Applied Science nd Innovions The Generl Quinic Equion, is Soluion by Fcorizion ino Cubic nd Qudric Fcors Smuel Bony Buy* Mhemics/Physics echer Ngo girls, Secondry School, Keny Reserch Aricle Received de: /5/7 Acceped de: 5//7 Published de: 6//7 *For Correspondence Smuel BB, Mhemics/Physics echer Ngo girls, Secondry School, Keny E-mil: Keywords: Rdicl or lgebric soluion of he generl quinic equion; Algebric soluion of he generl sexic nd sepic equions ABSTRACT I presen mehod of solving he generl quinic equion by fcorizing ino uxiliry qudric nd cubic equions. The im of his reserch is o conribue furher o he knowledge of quinic equions. The ques for formul for he quinic equion hs preoccupied mhemicins for mny cenuries. The monic generl equion hs six prmeers. The objecives of his presenion is o, one, seek fcorized form of he generl quinic equion wih wo exogenous nd four indigenous prmeers, wo, o express he wo exogenous prmeers of fcorizion s funcion of he originl prmeers of he generl quinic equion. In he process of fcorizion wo solvble simulneous polynomil equions conining wo exogenous nd four originl prmeers re formed. Ech of he exogenous prmeers is reled o he coefficiens of he generl quinic equion before proceeding o solve he uxiliry qudric nd cubic fcors. The success in obining generl soluion by he proposed mehod implies h he Tschirnhusen rnsformion is no needed in he serch for rdicl soluion of higher degree generl polynomil equions. As wy forwrd fcorized form for soluion of he generl sexic nd sepic equions will be presened o pve wy for heir lgebric soluion. INTRODUCTION The objecive of his pper is o dd furher o he reserch ino he soluion quinic of equions h hs preoccupied mhemicins for cenuries. The opic under invesigion is very imporn in lgebr nd hs pplicions in compuer science in generl. Invesigion of lgebric soluion of higher degree polynomils is prominen problem in mhemics. An exr mile will be ken o obin he lgebric soluion of he generl quinic equion, s deprure from he originl emp of obining he soluion of he rinomil quinic. Th is, mehod will be sough of solving he generl quinic equion wihou recourse o he Tschirnhusen rnsformion. The reson for doping his line of pursui is o esblish bsis upon which lgebric soluion of higher degree generl polynomils cn be found [, ]. The presumpion of his pper is h he qudric equion soluion mehod nd soluion mehods of Ferrri nd Crdno nd Lodvico Ferrri re sufficien in he pursui of rdicl soluion of generl higher degree lgebric equions. On he bsis of his ssumpion fcorizion mehod of soluion will be sough for in solving he quinic equion. The Bring pproch in he serch of soluion of he quinic ws hrough edious rnsformion h elimines hree inermedie prmeers of he generl quinic equion. The pproch h will be employed in his reserch is fcorizion in which wo unknown prmeers re inroduced nd reled o he originl prmeers of originl unfcorized quinic equion. The oher hree prmeers will be originl prmeers of he quinic equion. The mehod employed should be ble o be exended o higher degree generl polynomil equions. BACKGROUND For hree cenuries before he nineeenh cenuries mhemicins emped o obin he generl lgebric soluion of RRJASI Volume Issue July, 7 9

2 Reserch & Reviews: Journl of Applied Science nd Innovions polynomil equions of degree five nd bove wihou success. The effor o solving hese higher degree equions culmined o he Abel-Ruffini heorem. Simply sed Abel-Ruffini heorem is n emp o nswer why here is no lgebric formul in erms of coefficiens for he roos of polynomil equions of degree five nd bove. Glois Theory only provided mens for solving some specific forms of higher degree polynomil equions bu filed o come wih generl formul. Bck in he sixeenh cenury Frncois Vièe, French mhemicin, se ino course he ph of solving higher degree polynomil equions by reling is coefficiens o he roos vi elemenry symmeric equions. Louis Lgrnge inroduced Resolvens nd hey were sysemiclly used by Glois s ool for seeking specific soluion of higher degree polynomils. The Cylev Resolven is Resolven for he mximl resolvble Glois group in degree five. In he seveneenh cenury (666 Sir Isc Newon found he Newon ideniies, hough hey were found erlier by Alber Girrd in 69. Thus he Newon ideniies re lso known s Newon-Girrd formule. Newon s ideniies rele he k-h power of ll roos of polynomil o is coefficiens. Newon s ideniies hve mny pplicions in Glois nd group heory. Newon-Girrd nd Vièe formule re bsiclly formule reling he coefficiens of polynomil o is roos. Formule connecing roos nd coefficiens of polynomils hve been used o seek lgebric soluions of polynomil equions. Semen of he Problem Could here be n effecive fcorizion echnique h cn be used o solve higher degree polynomil? If such echnique is possible wh is he form should i ke for esy soluion of higher degree polynomil equions? In doping fcorizion s mehod of solving higher degree polynomil equions, is i possible o reduce he newly inroduced prmeers o wo o complemen he originl prmeers of he polynomil? Is i possible o come up wih fcorizion common lgorihm h cn be used o solve higher degree generl polynomil equions? Is he Glois Theory pproch he mos effecive nd complee pproch of seeking for soluion of higher degree polynomil equions? Generl Objecive OBJECTIVES The generl objecive of his reserch is o presen mehod in which he generl quinic equion cn be solved nd o esblish fcorizion lgorihm h cn be used o solve higher degree generl lgebric equions. Specific Objecives To esblish he mos effecive fcorizion pproch h cn be used o solve quinic nd higher degree polynomil equions To esblish fcorizion echnique h involves solving wo simulneous equions To find he lgebric soluion of he generl quinic equion. To esblish he bsis of lgebric soluion of polynomil equions of degree greer hn five Reserch Quesions Is Glois Theory complee heory? Could here be missing links nd connecions in Glois Theory? During he second decde of his cenury i ws shown h lgebric soluions h les lgebric soluions of fifh degree do exis, [,,6 ]. Furhermore Cmille Jordn showed h ny lgebric equion my be solved using modulr funcions. One such formul ws chieved by Crl Johnness Thome in 87. The formul ws chieved Tschirnhus rnsformion. The rel prcicl pplicion of he formul however is very difficul becuse of he complexiy of he relevn hyperellipic inegrls nd higher genus he funcions. These pproches, however, do no provide lgebric soluions bu hey hve les shown h oher pproches do exis for solving higher degree polynomil equions. Jusificion Algebr is essenil in he sudy of mhemics, he sciences nd engineering. Algebr merges in ll res of mhemics nd hs mny pplicions in geomery, compuer progrmming nd so on. Algebr plys n imporn role in physics ec. Scope The pper is simple enough nd is wihin he mhemicl rech of suden who hs done mhemics high school. RRJASI Volume Issue July, 7

3 Reserch & Reviews: Journl of Applied Science nd Innovions The pper is lso of gre vlue furhering he undersnding of mehods of solving lgebric equions of higher degree. The pper is lso of gre vlue in undergrdue mhemics, e.g. in mrices where knowledge of mehods of solving polynomil equions re highly needed. Inroducion LITERATURE REVIEW The objecive of his pper is o dd o he reserch on lgebric soluion of higher degree polynomil equion. Mny reserchers of mhemics nd he physicl sciences hve been occupied on he subjec for cenuries. Indeed lierure indices h polynomil equions hve been invesiged for more hn four housnd yers. The elegn nd prcicl noion h we use ody in deermining he roos of polynomil equions developed in he beginning of he fifeenh cenury. Cubic nd quric polynomil equions were comprehensively solved in he sixeenh cenury. Three cenuries ler he quinic equion could no be solved. In 8 Niel s Abel published his impossibiliy heorem s n nswer o he persisen lgebric insolvbiliy of generl quinic equions nd higher degree polynomils equions. Erlier, in 798, Crl Friedrich Guss in secion 59 of his book eniled Disquisiiones Arihmeic conjecured (published in 8 bou he impossibiliy of rdicl soluion of he quinic equion. Abel s impossibiliy heorem ws lmos proved by Polo Ruffini in 799. Abel s heorem did no sufficienly did no provide necessry nd sufficien condiions on which quinic specific quinic (or higher equion were solvble in rdicls. I ws Glois Theory which compleed h which lcked in Abel s heorem. Glois Theory (Published in 86 suggesed h he impossibiliy heorem ws sricly sronger hn he resul of Abel-Ruffini heorem. As In Sewr wroe for ll h Abel's mehods could prove, every priculr quinic equion migh be soluble, wih specil formul for ech equion. Glois Theory dmpened ny hope of obining rdicl soluion of lgebric equions of degrees greer hn four. Bring rdicls were inroduced by Erlnd Bring nd roos of he Bring-Jerrrd quinic equion my be expressed in erms of Bring rdicls. Mny oher chrcerizions of he Bring rdicl were developed by oher mhemicins. Chrles Hermie published he firs known soluion of he generl quinic equion in erms of ellipic modulr funcions in 858. Some ohers like Frncesco Brioschi nd Leopold Kronecker derived similr resuls round he sme ime. Cmille Jordn showed h ny polynomil equion my be solved by use of modulr funcions. Crl Johnnes Thome chieved such soluion in 87. In Edwrd Thbo Molole conribued o he firs lgebric soluion of he Bring-Jerrrd quinic equion. Buy S.B. mde furher conribuions o lgebric soluion of higher degree polynomil equions beween nd 7 Theoreicl Review/Concepul Frmework From he heoreicl nd concepul viewpoin, polynomil equions re buil ou of sums nd producs. Ech of coefficiens of he polynomil codes wihin i he sum nd produc of roos in chrcerisic wy depending on he degree of he erm i is ssocied wih. The coefficien ched o he degree zero erm is solely buil from he roos of he polynomil equion. Mehods of solving polynomil equions lgebriclly involve exrcing he roos from he coefficiens hrough finie process of muliplicion ddiion nd subrcion. I hs been shown h coefficien of polynomil cn be reled o generl roo L nd oher coefficiens. This mens prmeerizion cn be doped in which one of he prmeers forming coefficien ched o given erm of given degree is lso roo. This mens lso h he polynomil cn be linerly fcorized bsed on his relionship. This concep is no mde use of in conemporry lgebr nd in Glois Theory. The formule of qudric, cubic nd quric equions hve bsiclly similr srucure. Mehods of solving cubic equion bsiclly involve reducing i o depressed cubic nd mking subsiuions for he unknown h conver i degree six equion of qudric form. Mehods of he solving quric equion bsiclly involve fcorizing o wo qudric fcors nd correling he coefficiens of he fcors o he coefficiens of he originl quric equion. The nex sep is o solve he uxiliry qudric equions wih he correled coefficiens. In boh cses here is sep involving polynomil equion wih qudric form. The resolven pproch iniied by Lgrnge nd exended by Glois ws successful in he qudric, cubic nd quric equions. I involves reling he roos of he polynomil equion o he coefficiens vi he elemenry symmeric polynomils resuling from expnsion of he liner fcorized form of he polynomil. In soluion of polynomil equions of degree greer hn five he mehod his sng nd reches ded end (even wih he id of Newon s ideniy. The inbiliy of Glois resolven (nd modern field nd group pproch o obin generl soluion of higher degree polynomil does no men heir insolvbiliy. I only reflecs he limiions of he pproch nd ool used. Oher pproches hve been shown o work effecively [-8]. CRITIQUE OF THE EXISTING LITERATURE RELEVANT TO THE STUDY Srenghs of Glois Theory The resolven pproch nd Glois Theory hve gre success in obining lgebric soluion of polynomil equions of degree less hn for. I provides remedil nd lgebric soluions of some specific forms of higher degree polynomil equions. RRJASI Volume Issue July, 7

4 Reserch & Reviews: Journl of Applied Science nd Innovions Weknesses I does no provide for mehods of obining generl rdicl soluion generl polynomil equions of degrees greer hn four. The Glois resolven pproch becomes compliced o levels of indeermincy in endevors for seeking generl soluion of higher degree generl polynomil equions. Glois Theory is oo bsrc nd even misleding by purporing lgebric insolubiliy of higher degree generl polynomil equion. The Glois nd Lgrnge Resolvens complice he effor o obin formule for generl polynomil equions of degree five nd bove. Glois Theory does no provide for oher possible connecions beween he coefficiens of polynomil nd coefficiens of uxiliry polynomil equions in cses of non-liner fcorizion. I hevily relies on solving he elemenry symmeric equions vi Glois resolven nd Newon s sum formul. Glois Theory permis ouded conceps of irreducible quinics. As long coefficiens of uxiliry fcors cn be conneced o hose of he generl polynomil h cnno be he cse. Exisence of irreducible quinics would cully mens insolvbiliy in he generl sense. In reliy generl quinic equion re lgebriclly solvble [5,9]. Glois Theory does no consider possible fcorized forms h cn yield generl lgebric soluions. I hs does no ccommode oher possible lgebric correlions beween roos nd coefficiens h cn yield he required resul. In ny polynomil here is for exmple connecion beween generl roo L nd is coefficiens which forms he bsis of is liner fcorizion [,]. In ny polynomil here exiss correlion beween given coefficien, generl roo L nd oher coefficiens. When such connecions re esblished we ge n effecive ool h cn be used o obin n lgebric soluion []. Opporuniies Glois Theory needs o incorpore solvble fcorized form for higher degree polynomils o enble correlion of he coefficiens of he uxiliry polynomil equions o hose of he polynomil whose soluion is being sough. Correling he coefficiens of he uxiliry equion o hose of he min polynomil is n indirec wy of solving he elemenry symmeric equions of he min polynomil [,5]. Thre Mehod If Glois Theory fils o incorpore hese opporuniies hen here is hre h i will be obsolee [6-8]. SUMMARY A concepul frmework exiss by which higher degree polynomil equions cn exis. The generl quinic equion is given by: The generl quinic equion kes he form x x x x x = 5 The rinomil or Bring-Jerrrd form of he bove equion is given by 5 x px q A fcorized form of he generl quinic equion cn ke he form below: x x x ( x ( x = = Expnding equion : 5 x x x x ( ( ( ( x = Equing coefficiens: = ( ( 5 = ( = ( ( ( ( ( ( ( 6 = ( ( 7 RRJASI Volume Issue July, 7

5 Reserch & Reviews: Journl of Applied Science nd Innovions Subsiuing 7 ino 6: ( ( ( ( ( ( ( ( ( ( ( = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( = ( ( ( ( =( ( ( ( ( ( ( ( ( ( ( ( =( ( ( ( ( ( ( ( =( ( ( (?? ( ( ( = ( ( ( = = 8 Tke b = 9 c = d = If b = w Then bc w w b c b d = 7 bc w w b c b d = 7 The Crdno soluion of one of he roos of 8 is given by: bc bc bc bc b = b d b d b c b d b d b c We hve succeeded in reling he exogenous prmeers of he uxiliry cubic nd qudric equion. We cn now go on o ge he Cubic nd qudric soluions of he generl qudric equion. The uxiliry cubic equion of he generl quinic equion is given by: ( = x x x 5 RRJASI Volume Issue July, 7

6 Reserch & Reviews: Journl of Applied Science nd Innovions If we ke x= y 6 Then: ( y y ( = 7 7 The Crdno Soluion of equion 5 is given by: ( ( = x ( ( ( ( ( ( We cn ge he remining roos of he uxiliry cubic equion 5 by considering he fcorizion: ( x x ( x Ax C = The expnsion of 9 gives: ( ( x x A x x C Ax xc Equing coefficiens: = xc C = x C Ax = ( ( x A = x Solving he remining roos of he uxiliry cubic equion involves solving he cubic equion below: 9 ( x x x = x x ( ( x x ± x x x x, = The uxiliry qudric equion is given by: ( x x = 6 The remining roos of he generl quric equion re given by: 5 RRJASI Volume Issue July, 7

7 Reserch & Reviews: Journl of Applied Science nd Innovions ( ( ± x,5 = Fcorized forms for solving Sexic nd quinic equions The generl sexic equion kes he form x x x x x x = A fcorized form for solving sexic equions bsed on he bove resoning is given by: x x x ( x ( x ( ( x = 5 5 The expnsion of equion 9 is given by: 6 5 x 5x x x ( 5 ( ( 5 ( x ( ( 5 ( ( 5 x = Equing coefficiens ( = ( 5 ( = ( 5 ( ( ( ( = 5 ( ( ( ( 5 ( ( 5 ( = Subsiuing ino we obin he equion. ( ( ( ( 5 ( ( ( ( 5 ( 5 ( ( ( ( ( ( 5 ( ( ( ( 5 ( ( ( 5 ( ( ( = ( ( ( 5 To obin he generl soluion of he sexic equion solve equion 6 followed by solving he uxiliry cubic equions using he Crdno formul. In emping o solve equion bove i should be h generl quinic equion in is lgebriclly solvble using he procedures oulined in his pper. In his pper I will no emp o seek soluion for equion. The generl sepic equion kes he form x x x x x x x = A fcorized form of he generl sexic equion is given by: x x x ( 6 5 ( 6 5 ( 5 x ( 5 ( 6 5 x x ( 5 x ( 5 ( 6 5x = 5 Agin he lgebric soluion of he generl sepic involves firs obining he lgebric soluion of prmeers nd. CONCLUSION REACHED AND RECOMMENDATIONS A soluion of he generl quinic equion hs been rrived nd wihou Tschirnhus rnsformion. An lgorihm hs been chieved by which higher degree generl polynomil equions cn be solved lgebriclly.glois Theory requires rewri- 7 9 RRJASI Volume Issue July, 7 5

8 Reserch & Reviews: Journl of Applied Science nd Innovions ing. Abel-Ruffini impossibiliy heorem requires some re-exminion. I recommend h furher reserch ino lgebric soluion of generl lgebric equions using he lgorihm provided in his pper. REFERENCES. Admchik VS, Dvid JJ. Polynomil rnsformions of Tschirnhus, Bring nd Jerrrd. ACM SIGSAM Bullein. ;7:9-9.. Buy SB. A Formul for Solving Generl Quinics: A Foundion for Solving Generl Polynomils of Higher Degrees. Open Science Reposiory Mhemics open-ccess ;e595.. Buy SB. On solvbiliy of higher degree polynomil equions. J Appli Scie Inno, 7;:7-5.. Buy SB. The Bring-Jerrrd Quinic equion, is lgebric soluion by expnsion. J Appli Scie Inno. 7;: Buy SB. The Bring-Jerrrd, Is lgebric soluion by conversion o solvble fcorized form. J Appli Scie Inno, 7;: Buy SB. Algebric equion of he generl sexic equion. Inern J Curr Rese Sruik DJ. A concise hisory of mhemics. Courier Dover Publicions, Dickson L. Algebric heories. Courier Corporion,. 9. Molole ET. The Bring-Jerrrd quinic equion, is soluion nd formul for he universl grviionl consn. Uir unis c.z.. Cjori F. A hisory of mhemics. Amer Mh J, 99. Friedrich GC. Disquisiiones rihmeice. Yle Universiy Press, 966;57.. George PY. Soluion of Solvble irreducible quinic Equions, wihou he Aid of Resolven Sexic. Amer J Mh. 885;7:7-77. Jordn C. Tres subsiuions nd lgebric equions by m. Cmille Jordn. Guhier-Villrs,87.. Rosen MI. Niels Hendrik Abel nd equions of he fifh degree. The Ame mhem mon 995;: Sewr IN. Glois heory. CRC Press, Thome J. Conribuion o he deerminion of... (,,... by he clss modules of lgebric funcions. J pure ppl mh. 869;7:-. 7. Sruik DJ. A concise hisory of mhemics. Courier Corporion,. 8. Vn der W, Brel L. A hisory of lgebr: from l-khwārizmī o Emmy Noeher. Springer Science & Business Medi,. RRJASI Volume Issue July, 7 6

e t dt e t dt = lim e t dt T (1 e T ) = 1

e t dt e t dt = lim e t dt T (1 e T ) = 1 Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie