The Bring-Jerrard Quintic Equation, its Algebraic Solution by Expansion

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1 Researh & Reviews: Journal of Applied Siene and Innovations The Bring-Jerrard Quinti Equation, its Algebrai Solution by Expansion Samuel Bonaya Buya* Mathematis/Physis teaher at Ngao girls, Seondary Shool, Kenya Researh Artile Reeived date: /6/17 Aepted date: 14/7/17 Published date: /7/17 *For Correspondene Samuel BB, Mathematis/Physis teaher at Ngao girls, Seondary Shool, Kenya Keywords: Solving the Bring-Jerrard quinti equation by expansion, Algebrai solution of the Bring-Jerrard quinti equation, preoupied mathematiians for enturies, Quinti equations are part of polynomial equations ABSTRACT I present a method of solving the Bring-Jerrard quinti equation by expansion. The aim of this researh is to ontribute further to the knowledge of quinti equations. The quest for a formula for the quinti equation has preoupied mathematiians for many enturies. The general quinti equation in its trinomial form is alled the Bring-Jerrard quinti equation. The Bring Jerrard equation has two parameters. The objetive of this presentation is to present the expansion method as a viable tool for solving quinti equations and higher degree polynomials. In this researh I split the unknown quantity in the quinti equation into three unknowns and expand and subsequently ontrat to the lowest possible form. The ontration of the expanded form involves fatoring out the three unknown quantities to give the equation a solvable form. Three simultaneous equations in the three unknowns are then formed from the expanded quinti equations. The solution of the three equations yields an algebrai solution of the Bring-Jerrard quinti equation. Bakground and Literature Review INTRODUCTION The objetive of this paper is to add further to the researh into the solution quinti of equations that has preoupied mathematiians for enturies. Quinti equations are part of polynomial equations. From literature [1, ] polynomial equations were first investigated more than four thousand years ago. The general quinti equation takes the form x + ax + ax + ax + ax+ a = Abel and Ruffini showed that proved that it impossible to solve over a field of rational numbers, Rosen [] Bring [4] and Jerrard [] have shown that the equation an be redued to a simple form with two parameters. That is, x px q + + = The equation above is now alled the Bring-Jerrard quinti equation, a trinomial form of the general quinti equation. Glashan [6], Young [7], Runge [8] showed that some form of the Bring-Jerrard quinti equation is solvable in radials. Spearman and William [9] ame with similar results. Many others ontributed in various ways. Motlotle [1], in his 11 master s thesis managed to present a formula for solving the Bring-Jerrard quinti equation using the Newton s sum formula. In his ontribution Motlotle onviningly argued that Abel s impossibility proof has been misonstrued by many as meaning that no general algebrai solution of the quinti equation is attainable. He showed that suh a formula in unattainable only within a field of Rational numbers. He then moved on to deriving a formula. The problem is that the approah he used is too involving that it dampens any hope for oming up with similar formulae for solving higher degree polynomial equations. RRJASI Volume 1 Issue July, 17 16

2 Researh & Reviews: Journal of Applied Siene and Innovations In this paper the unknown x of the Bring-Jerrard quinti equation is substituted by the unknowns u, v, w, that is x=u+v+w. The resulting expanded quinti equation is then fatorized to a solvable form. Three equations are then identified from the expanded quinti equation and are used to obtain and algebrai solution of the quinti equation. The approah used is simple and an be used to derive formulae for higher degree polynomials. Statement of the Problem Could there be other methods other the onventional ones that an yield an algebrai solution to the quinti equation? Could there be a viable approah by whih the quinti equations an be split into some three solvable simultaneous equations? Is there a viable method by whih the unknown in the algebrai equation is parameterized and subsequently put in a form suh that it an degenerate to solvable forms? If the unknown is replaed by three unknowns, what final form should the resulting expansion take to ahieve an algebrai solution? I seek to present a method of solving the Bring-Jerrard quinti equation by whih three derivative and degenerate equations are solved. Objetives The main objetive of this paper is to derive the algebrai solution of the quinti equation by a method in whih a set of three simultaneous equations is solved. The speifi objetives inlude: Expansion of subsequent quinti equation after replaing the unknown by three unknown parameters Simplifying the ahieved expansion to ahieve a solvable form. Split the simplified solvable form into three solvable equations to ahieve a general solution of the quinti equation. Solution of the Cubi Equation METHODOLOGY Before solving the quinti equation, I will attempt to derive the ubi equation using the proposed method. Consider the ubi equation, x bx + + = If we take x=u+v then, ( u v) b( u v) = 4 The expanded form of is: u u v uv v b u v = an be simplified to take the form: u v u v b uv + + ( + )( + ) + = 6 Take ( u v)( b uv) And + + = 7 u v + + = 8 From., b = uv 9 Substituting 6 into 7 and simplifying: u b + u = One of the roots of 8 is b 4 7 u = RRJASI Volume 1 Issue July, 17 17

3 Researh & Reviews: Journal of Applied Siene and Innovations Similarly it an be shown that one of the roots of v is given by: b 4 7 v = + + One root of the above ubi equation is given by, 1 b b x = The above method an be extended to provide a solution to the Bring-Jerrard quinti equation. Solution of the Bring-Jerrard Quinti Equation Consider the Bring-Jerrard quinti equation below x px q + + = If we take x= u+ v+ w then, u+ v+ w + b u+ v+ w + = 1 The expanded form of is: 4 v+ w u+ 1u v+ w + 1 u v+ w + b u+ v+ w) + = 1u v+ w + 1u v+ w = 1u v+ w u+ v+ w u v+ w + v+ w u = u v+ w u + v+ w = uv u + v+ w u v+ w u+ v+ w ) Substituting and 6 into 4 ( ) u + v+ w + u+ v+ w) uv u + v+ w u v+ w + 1u v+ w + b = 19.7 an further be simplified to: ( ) u v w u v w u v w u v w u v w b ) = 8 an further be simplified to: ( ) u v w u v w u v w u v w u v w b ) = The equation 9 an be hanged to three equations with three unknown quantities u, v and w. Take: ( ) u+ v+ w) u v+ w u+ v+ w u v+ w + b = That is: u( v w)(( u v w)) u( v w)) b and = u v w = The form the equation 11 is in annot generate an algebrai solution for the quinti equation. Two useful forms are: u v+ w + b= u ( v+ w) + b= If u+ v+ w= s u( v+ w) = t RRJASI Volume 1 Issue July, 17 18

4 Researh & Reviews: Journal of Applied Siene and Innovations We an now introdue the third equation to make the quinti equation solvable. That is if we take: s t = u 7 Then substation of 1 into 11 gives the result: b= u ( v+ w), or b v+ w= u This makes the quinti equation solvable Substituting 16 into 1 and simplifying; 1 1 b u + u = 1 Then equation 17 takes the form: b y + y = 1 If additionally y = z The equation 18 redues to: b z z+ = 7 1 One of the roots of is given by: b 1 b z = b + b b 1 b 1 b 1 b u = From 1, 14 and 1: u+ v+ w u v+ w = u The expansion of. is ( v+ w) + u( v+ w) + u u = u 4u u v+ w= ± Substituting into b 1 b 1 b ± v + w = b b b b b b b b b + + / stwo of the roots of the quinti equation are: / 4 RRJASI Volume 1 Issue July, 17 19

5 Researh & Reviews: Journal of Applied Siene and Innovations x, x 1 1 b 1 b = 1 b 1 b b b b b ± 4 b b b b b 1 b 1 b + We note from 1 1 b + / s = x 7 The other roots are obtained by solving the subsequent auxiliary equations. Using equations 1, 14 and 1 we an alulate the unknown w: s + u us u w = u b b b b b 1 b 1 b 1 b 1 b 1 b w = ( ± 4 1 b 1 b + 1 b 1 b 1 b 1 b / b 1 b 1 b 1 b + + / SUMMARY RECOMMENDATIONS AND CONCLUSION If the unknown quantity in the Bring-Jerrard quinti equation is replaed by three parameters and the resulting expansion simplified to a solvable form it is possible to arrive at an algebrai solution. The algebrai solution is arrived at by solving three simultaneous equations split from the simplified expanded equation. From this treatise it an be onluded that a radial or algebrai solution of the quinti equation is ahievable. Radial solution is loosely taken to mean root form. The researh ontests the generally alaimed notion proposed in Abel-Ruffini theorem and expounded in Galois Theory that quinti equations in general are algebraially indeterminate. The Abel-Ruffini theorem is inomplete sine it didn t put into onsideration the broad spetrum of mathematial possibilities in the for algebrai solution of higher degree polynomials. I reommend a rewrite of Galois Theory to take into aount the urrent reality. The urrent reality is that Quinti equations are in general solvable. REFERENCES 1. Cajori F. A history of mathematis. Amer Math Journal D. J. Struik. A onise history of mathematis. Courier Dover Publiations RRJASI Volume 1 Issue July, 17

6 Researh & Reviews: Journal of Applied Siene and Innovations. Rosen MI. Neils Hendrik Abel and equations of fifth degree. Amer Math. 199; 1: Bring ES. Meletemata some mathematematia the transformation of algebrai fair-tion, Quart J Math. 6, Jerrard GB. An essay on the resolution of equations. Taylor and Franis, Red line ourt, Fleed Street. 189; Glashan JC. Notes on the quinti. Amer J Math. 188;8: Young GP. Solution of solvable irreduible quinti Equations, without the aid of a resolvent sexti. Amer J Math. 188;7: Runge C. Ueber die aufloesbaren gleihungen von der form. Ata Math. 188;7: Spearman B and Williams KS. Charaterization of solvable quintis x+ax +b=. Amer Math. 1994;11: Motlotle ET. The Bring-Jerrard quinti equation, its solution and a formula for the universal gravitational onstant 11. RRJASI Volume 1 Issue July, 17 1