ORMAT Topics Related to the Solution of Some Algebraic Equations

ORMAT Topics related to the solution of some Algebraic Equations

Deb Kumar Basu*

Abstract

In this paper the properties and solubility of a particular type of algebraic equations has been studied and also comparison has been made between their different solution methods.

1. Introduction

We consider algebraic equations of the type

,

and outline some methods of solution for the special cases for n= 3 and n = 4, which correspond to equations of degree 5 and 7 respectively. In this paper different solution methods has been developed as well as comparison between methods has been studied also. Author has obtained a system of equations each of which involves single unknown in the case of . Finally some comment has been passed for the method on the solution of this type of equations developed by S. Ramanujan in the year 1912.

2. Polynomial of fifth degree

Proposition 2. Any quintic (5th degree) polynomial expression can be expressed as a sum of three fifth powered linear expression.

Proof: The problem is to find A, B, C, p,q,r, for some given a0 , a1, a2, a3, a4, a5 such that

is satisfied.

We first develop the system of equations by expanding the R. H. S of the above equation, the initial system (2.1) is as follows,

(2.1)

Next we outline the methods of solution.

Solution 1.

A new system (2.2) can be developed by the following way ,

(2.2)

Some other system (2.3) can also be developed by the way

(2.3)

System (2.4) can be developed from (2.2) by the similar way

or

--Eq(2.4.1)

similarly we can get

or

Using Eq(2.4.1) , it can be said that

,

or,

provided and are both non zero.

Now system (2.5) may be developed from (2.3) by the way,

or

or

or

-- Eq(2.5.1)

where

which we get from the third eqn. of (2.1) , provided is nonzero

hence

provided , is nonzero.

Again from the equation

,

we get also

or

-- Eq(2.5.2)

or

.

However alternatively one can obtained from system (2.2) by the following way,

Simplifying we get

or

-- Eq(2.5.3)

Hence,

for the simplification of the above equation (like system (2.3)) we use the identity,

In this way we can get and from the equation

or

hence can be found using the known values of from any three equations of system (2 .1)

Alternative solution of system (2.1)

New system (2.6) can be developed by the following way,

Now System (2.7) can be developed by the way,

From system (2.7) we get ,

obviously the above equation is a biquadratic (generally) involves only p and following the method adopted above , we can similarly get other two biquadratic equation involving only q and only r . Obviously all of these are same in form due to the symmetric structure of the equation system (2.1). This means that we can get some biquadratic eqn., whose roots are . If so, situation may arise that it would be difficult to determine which root should be accepted or which one should be rejected. However by any mean one can get the values of from the first three equation of system (2.1) using the known values of (may be by Crammer Method / Vendermonde Determinant Rule).

3. Polynomial of degree seven

Proposition 3. Any Seventh degree Polynomial can be expressed as a sum of four seventh powered linear expression.

In other words the problem is:

For some given one can find , such that

satisfies.

Proof. Expanding , we get the initial system. The Initial system (3 .1) is as follows:

The new system (3 .2) can be developed by the way,

Now the system (3.3) can be developed by the way,

The system (3.4) can be developed by the following way,

Simplifying we get,

or

where

Similarly we get,

or

or

Taking analogy from the above equations in system .4 we can easily get,

,

i.e., we get

System .4 implies,

or

It means that the above equation involving onlyis of ten degree in order (possibly) and hence obviously not easy to solve even in the simpler cases. However From the symmetrical structure of the equation system .1, expectedly one can find similar equation of and from the known values of , the values of can be found using the first four of the equations of the initial system .1 by Crammer rule.

4. Polynomial of degree three

Proposition 4. However it may be noted that any Cubic Polynomial can be expressed as a sum of two cubes (i.e third power of linear expression) and hence by this way any third degree equation can be solved completely. In other words we can get

for some given such that

satisfies. Solution methods are similar to the procedures mentioned in proposition. 2. and proposition. 3. It is worth to mention that the method is a standard one and very often referred in the Classical Algebra text books. Here the corresponding equations are

5. Ramanujan’s Method

We conclude this section by presenting an method of Srinivas Ramanujan for the solution of such types of equations (system.4) (or even more general types) which was published in the journal of Indian Mathematical Society in the year 1912 . Actually he defined an infinite power series expression with some specifically mentioned terms containing , which is related to a fractional expression involving in the way that

-- Eq(5 .1)

satisfies. Let us define the series , for some complex , by the way that are given and ………….etc. are determined by the equation

It means that , s (for ) can be found in terms of i.e

, from solution of (system.4) by determining (at least implicitly).

Now starting from the series , and assuming convergence condition , i.e < 1 ,< 1 , we get

Morover this form can be expressed as the compact form such that

…. Eq(5.2)

holds . Also one can choose from given satisfying the above relation. Comparing between Eq(5.1) and Eq(5.2) , we get

--Eq (5.3)

Again comparing between like powers of of the equality ,

one can find ,

-- Eq(5.4)

Setting , one can get easily, and hence by the separation of partial fraction method one can find from (Eq .4).

The method can be generalized also for higher order Ap equations involving more unknowns.

Proposition 4a. For given we can get A , B , C, p, q, r such that

satisfies.

Since one can choose so that

and that will be found by factorizing into linear factors of z and then using Separation of Partial Fraction method from the following equation:

Similarly,

Proposition 4b. For given , we can find so that

satisfies.

Obviously in this case a biquadratic expression is to be resolved into linear factors of z. It follows that the method is not applicable in higher order cases since in general a quintic (5th degree) polynomial cannot be reduced into product of linear factors (in the sense of general solvability of polynomials).

* Academic Research Worker

email id :

Acknowledgement. Author like to express his gratitude and thanks to Dr. Kalyan Chakraborty, Associate Professor, Department of Mathematics, Scottish Church College, Kolkata-700006 for his valuable suggestions and help, without these the paper would not be completed . Encouragement and suggestions from Dr. Smarajit Bose , Professor , Statistics Unit , I.S.I , Kolkata are also recognized .

References--- (For prerequisites)

1.  Higher Algebra --Bernard and Child .

2.  Theory of Equation ---Burnside and Panton .

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