Elementary methods for solving equations of the third degree and fourth degree by ProQuest

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									  Scientia Magna
  Vol. 5 (2009), No. 4, 5-13



Elementary methods for solving equations of the
        third degree and fourth degree
                                      Fatmir M. Shatri


                Department of Mathematics, University of Prishtina, Kosova
                           E-mail: fushashkencore@gmail.com

    Abstract In this paper we will establish some new methods for solving third and fourth
    degree equations.
    Keywords Cardano method, Ferrari method, equation.



§1. Introduction
     In this paper, using new transformations, we will introduce new elementary methods for
solving third and fourth degree equations, which essentially differ from Cardano’s method and
Ferrari’s method. We will present the importance of these transformations through some ex-
amples.


§2. Equations of the third degree
    Let
                                   x3 + px2 + qx + r = 0                                 (2.1)
be an equation of the third degree. Usually, it is solved by the means of Cardano’s method.
Let us start from the following definition:
    Definition 1. The equation of form

                                         X3 + a = 0                                      (2.2)

is called binomial equation of the third degree.
     The following theorem gives the condition to be satisfied by the coefficients of equation
(2.1) in order that (2.1) becomes in the form (2.2).
     Theorem 2. Necessary and sufficient condition in order that equation (2.1) be an equation
of form (2.2) is that its coefficients satisfy the condition

                                        p2 − 3q = 0.                                     (2.3)

    Proof of Theorem 2. (Necessary Condition) Substituting x = m + y, in equation (2.1),
where y is a new variable, we have

              y 3 + (3m + p)y 2 + (3m2 + 2pm + q)y + m3 + pm2 + qm + r = 0.              (2.4)
6                                         Fatmir M. Shatri                                    No. 4


In order that equation (2.4) be a binomial equation of y, it is necessary that its coefficients
satisfy conditions
                              3m + p = 0, 3m2 + 2pm + q = 0.                            (2.5)
From the first equation of (2.5) we have
                                                p
                                             m=− .                                            (2.6)
                                                3
Substituting the value of m from (2.6) in the second equation of (2.5) we obtain condition (2.3).
     (Sufficient Condition). Let the coefficients of equation (2.1) satisfy condition (2.3). Then
for equation (2.1) we have
                                                p2
                                   x3 + px2 + x + r = 0.                                    (2.7)
                                                 3
                   
								
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