| In this paper, we mainly discuss zeros of real quaternionic polynomials and its factoring theory. The main results are stated as follows.1) New Methods and Comparison: Two traditional methods and two new methods to solve real quaternionic polynomial equations are introduced and compared during the analysis process. Those new methods are based on Jacobson's theory and its characteristic structure which make the problem equivalent to a 2 x 2 matrix polynomial equation with diagonalizable solutions. Furthermore, we express a real quaternionic polynomial with four general polynomials which have real coefficients, thus build direct relation between real quaternionic polynomials and general polynomials. In addition, we improve Niven's method and our new method to solve the equation more quickly.2) Complete Solution Structure: We obtain that the whole solutions of real quaternionic polynomial equation are just composed of some quaternion equivalence classes and some isolated points. Some results before become natrual corollaries.3) Final Form of Factoring: A real quaternionic polynomial can be finally factored into an irreducible real quaternionic polynomial and some linear or quadratic polynomials with real efficients. We also can prove that those polynomials with real efficients are GCF(greatest common factor) of the four general polynomials mentioned above. This can simplify our methods obviously.4) Relations between Factoring and Solution Structure: We build the corresponding relation between factoring and the complete solution structure. Zeros of irreducible real quaternionic polynomial are the isolated point solutions of the original polynomial equation. Zeros of linear polynomials with real efficients are the real solutions of the original polynomial equation. Zeros of quadratic polynomials with real efficients(A < 0) are the equivalence class solutions of the original polynomial equation.5) Explicit Expressions of the Solutions of Quadratic Equation: Using the new improved method, we obtain the explicit expressions of the solutions of quadratic equation and some sufficient and necessary conditions of the solutions.6) Diagonalizable Solution of Matrix Polynomial Equation: We can obtain all of the diagonalizable solutions of standard matrix polynomial equation by using the same method as to solve real quaternionic polynomial equation.Our important new work is: expressing the solutions of the standard quaternionicpolynomial equation with general polynomial; obtaining the complete structure of the zeros' set of the standard quaternionic polynomial equation for the first time; giving the final form of the factoring of the standard quaternionic polynomial for the first time and building the 1-1 relations between the factoring and the solution structure; completing the analysis of the solutions of the quadratic equation by using the new method and also obtaining the diagonalizable solutions of standard matrix polynomial equation.To show that new methods are not only right but also easy to apply, we give some examples which are diifferent from each other to solve in Chapter Four, using some math software such as MATLAB, Mathematica. |
