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:

where

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).

__Example__

Solve the equation : .

__Solution__

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 :