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Chapter 8

Polynomial Equations with Degrees 3 and 4

 With the help of some theorems, that were mentioned in the earlier chapters of the tutorial and with some methods that we previously used, we managed to solve some polynomial equations. It has been proved that the solution of an equation with degree > 4 is not always possible. Hence the only equations that can always be solved are the ones with degrees less than five.

The general form of a polynomial equation of degree 3: takes the form of  (1), when we divide its terms with and set .
Doing the following transformation:

the equation (1) gets the form:


Now doing the transformation:

the equation (2) takes the form:


If the roots of the second degree equation (3) with respect to , then solving one(*) of binomial equations:

we can find three values  for z .

Assigning these values to the transformation (M2) we find three values for y, from which with the help of (M1) we find the roots  of (1)

(*) : it does not matter which one of the two we are going to solve. At the end we will find the same roots
  of the equation (1).


Solve the equation : .


We must bring the equation to the form (1).

It is : 

Equation (3) becomes: from which we get the following:

and with the help of (M2) we find:

Finally (M1) gives :