How do you solve 5th degree polynomials?

Here is a summary of posts that should address related questions:

1. Reducing the general quintic to Bring-Jerrard form.
2. Solving (1) using elliptic functions.
3. Reducing the general quintic to Brioschi form.
4. Solving (3) using trigonometric and special functions.
5. Solving the Brioschi form using $R(q)$ (or the Rogers-Ramanujan continued fraction).

As mentioned above, no general formula to find all the roots of any 5th degree equation exists, but various special solution techniques do exist. My own favourite: - By inspection, see if the polynomial has any simple real solutions such as x = 0 or x = 1 or -1 or 2 or -2. If so, divide the poly by (x-a), where a is the found root, and then solve the resultant 4th degree equation by Ferrari's rule. - If no obvious real root exists, one will have to be found. This can be done by noting that if f(p) and f(-p) have different signs, then a root must lie between x=p and x= -p. We now try the halfway point between p and -p, say q. We then repeat the above procedure, continually decreasing the interval in which the root can be found. When the interval is small enough, we have found a root. - This is the bisection method; when such a root has been isolated we divide the polynomial by that root, producing a 4th degree equation which can again be solved by Ferrari or any another method.