A Note on Neutrosophic Polynomials and Some of Its Properties

The purpose of this article is to study neutrosophic polynomials i.e. polynomials which are Neutrosophic in nature and study its properties with the help of neutrosophic numbers. Apart from this we discuss different types of neutrosophic polynomials with concrete examples and establish some theorems and results which will be useful for the further study. We also give a solution method to find the approximate roots of a neutrosophic polynomial equation.

In this work we mainly concerned with Neutrosophic polynomials and its types. We also contribute some theorems and results based on Neutrosophic polynomials and try to develop the earlier concept of classical polynomials so that we have a bright scope to use it further and to establish new theories and results related to polynomials which are indeterministic in nature. For practical purpose we include [18][19][20][21].
The present article is arranged in the following manner : Section-2 comprised with Neutrosophic numbers and basic operations on them. In section-3, firstly defined neutosophic polynomials and its types and then deduce some theorems and give examples of each type to justify the results.In the last section(in section-4), the main purpose of the paper is discussed briefly.

Classical Neutrosophic Numbers and basic operations
Here, we give the basic definition of Neutrosophic Numbers and its properties. n . If x , y are real then x y   is called Neutrosophic real number otherwise it is known as Neutrosophic complex number. 3 . , for 0 and 0 ( )

Neutrosophic Polynomials
Firstly we give the notion of neutrosophic polynomials with a concrete example then some results based on it are established.
Definition 3. Neutrosophic polynomial is a polynomial whose coefficients (atleast one of them contain  ) are neutrosophic numbers . If its coefficients are neutrosophic real numbers then it is called neutrosophic real polynomial otherwise it is called neutrosophic complex polynomial. In this work we mainly concerned with neutrosophic real polynomials.
In general any expression of the form   From logical view we give a real example of Neutrosophic polynomial in the following way: We brought a cake from a bakery which has a volume 120 cu.units. Length of the cake is 2 units more than its breadth and its hight is slightly more than its breadth. We consider a,b and c for length, breadth and height respectively. Using the given condition we write problem. So such a situation can be shown as Which is the required neutrosophic polynomial for the above problem.
Now the question arises how we solve this type of equation. To do so we consider some examples.
Example 1. Find the roots of the neutrosophic real polynomial 2 6 (10 Solution. We have, Putting the values of a and b in the above equation we get the values of x which are 5 4 10 , , and 2 Now we check the other properties of the polynomial. 5 4 10 Clearly, 0 2 3 3 6 Clearly, the above neutrosophic polynomial has more than one factorization i.e it is not unique and the number of roots are more than that of the degree of the neutrosophic polynomial.
Let us take one more example to get more concrete result Example 2. Find the roots of the neutrosophic polynomial where a and b are real and 0   .
, 0 6 Therefore, the solutions are 6   In this case it obeys the property of classical polynomial as the degree of the polynomial is equal to the no of roots. So there is no such proper relation between the degree and the number of roots of a neutrosophic polynomial . But one thing is clear from these two examples that number of roots will be either the degree of the polynomial or double of its degree. It is for the reader to verify the roots of the higher degree polynomial to get more concrete results.
Remark 1. Number of roots of a neutrosophic polynomial of degree 1 n  will be n or 2n or 3n……..and so on.
Definition4. A neutrosophic polynomial without any zero coefficient is called a complete Neutrosophic polynomial otherwise it is incomplete.
Definition5. A neutrosophic polynomial whose coefficients are zero and it is represented by 0=0. is called vanishing Neutrosophic polynomial.
Definition6. Two neutrosophic polynomials are said to be equal iff their corresponding coefficients are alike and their corresponding indeterminacies converges to a fixed value.

Division algorithm
Where N Q and N R are respectively the quotient and the remainder and

Synthetic division method
It is a process in which we divide a polynomial of degree n by a binomial under neutrosophic environment. Let, Equating the coefficients of like powers, we have Thus, we establish an easy approach by which we calculate the coefficients of the quotient and the remainder as follows: x   

 
N P x of n-th degree (n 1)  can be vanish for more than n values of x.

Method to express a given neutrosopic real polynomial
is the quotient and ( ) b n   is the remainder.Therefore, the remainder is the last If we continue like this , we obtain, after each successive division ( Thus, the synthetic division method is applied for successive division as at each stage we get the quotient and the remainder.
Solution. By using synthetic division method successively, we have Theorem1. In the neutrosophic polynomial Hence the theorem. Proof. It is straight forward.

Newtons method of approximation
Let   0 N P x  be a given Neutrosophic polynomial equation in x and it has a root nearly equal to 1 Neglecting the square and higher powers of h , it becomes Hence the first approximation of the root will be Then its closer approximation will be Proceeding in this manner we may obtain an approximation which is very close to the root upto any desired degree of accuracy. In this way approximation is usually rapid.
Example7. Find the real root of Solution. It is left for the reader.

Symmetric Neutrosophic Functions of Roots
By symmetric neutrosophic functions we mean those functions which remained unchanged in value when any two of its roots are interchanged. We can find the values of symmetric neutrosophic functions of roots interms of the coefficients.

Conclusions
In this article we have studied neutrosophic real polynomial whose coefficients are neutrosophic numbers . Then we study some of its properties and introduce synthetic method of division to find the quotient and remainder easily. We also give a solution method to find the approximate roots of a neutrosophic polynomial equation. Some concrete examples are given. There is some possibility to use this concept to find the real roots by using different approximation method. Under neutrosophic environment we will use this concept to solve real life problems based on mensuration, trigonometry, geometry etc. We also include some papers in reference section for practical purpose.
Funding: "This research received no external funding" Conflicts of Interest: "The authors declare no conflict of interest."