Here are some of the general solutions to some higher order equations (polynomials of degree 5 or higher). A general solution does exist for the quintic (degree 5), however it is extremely complex and takes about 100 times the amount of hard drive space I have. The formulas below I generated using my ToSymbolic package which is mainly based off of the files I found at MathSource. I've left the equations in StandardForm so you can see the use of the higher level functions better. NOTE: I do not guarentee the accuracy of these answers!

1.

To solve this we divide both sides by a. This new equation can then be transformed by the Canonical Transform into the form of x^5-x-p==0 which is the Hermite quintic equation.

Solution

2.

This is the Principle form of the quintic equation and is solved using Klein's method.

Solution | Notebook

3. Quintic in radicals

Here are some of the examples of quintic equations that are solvable in radicals.

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4. Quintic in triginometry. Solving

Here is a quintic equation that I solved using some properties of the sine function. I cannot say that I am the first to do this (because I have been told it can also be done in radicals in the method above), but I have not seen it done anywhere else. I also cannot say that this will work for all values of a, but it does work for some. You can check out how I came upon the solution here.

Solution

5. Differential Resolvant

These were solved using the Differential Resolvant method of finding solutions. They are all in the form of HypergeometricSeries of a single variable. Here's the sextic (degree 6):

| Solution | Notebook

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