In a previous post, the hunt for the roots, I showed how to develop the formula for the second degree polynomial. In this post I want to develop the formulas for the third and forth degree. Unfortunately this means that this whole post will be algebra and nothing else. This formulas were originally developed in the 16 century - you can read more about this in the post linked above.
A little warning - in this post I don't solve numerical examples, so I use letters to denote numbers that would be known in a numerical example, but I freely move them around. This means that if in one line I wrote bx, in the next line I will also write bx even if I should write (b+4)x instead, and the same with the sign of b. To use the formulas you will need to follow the simplification process, and then to aplly the final formula to the result.
Lets start. The idea is to get the general formula for the equation of the form:
Lets suppose that x=u-v (this step is called uglification):
Now to the forth degree. We need to solve an equation of the form:
The first step is to use the same method I used in solving the third degree polynomial, to reduce the problem to:
Now, if only the left side was a square.... Well, it is again time for uglification. Lets look on:
Now, when this will be a perfect square? The answer is simple. We need the discriminant to be equal zero. This means that:
We can now take the root and get the simple equation: