How do you solve 5th degree polynomials? I looked on Wikipedia for a formula for roots of a 5th degree polynomial, but it said that by Abel's theorem it isn't possible. The Abel's theorem states that you can't solve specific polynomials of the 5th degree using basic operations and root extractions.
Can you find the roots of a specific quintic with only real irrational roots (e.g. $f(x)=x^5+x+2$) using other methods (such as logarithms, trigonometry, or convergent sums of infinite series, etc.)?
Basically, how can the exact values of the roots of such functions be expressed other than a radical (since we know that for some functions it is not a radical)?
If no, is numerical solving/graphing the only way to solve such polynomials?
Edit: I found a link here that explains all the ways that the above mentioned functions could be solved.
 A: 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.
A: Here is a summary of posts that should address related questions:


*

*Reducing the general quintic to Bring-Jerrard form.

*Solving (1) using elliptic functions. 

*Reducing the general quintic to Brioschi form.

*Solving (3) using trigonometric and special functions.

*Solving the Brioschi form using $R(q)$ (or the Rogers-Ramanujan continued fraction). 

A: The explicit elementary numbers are generated from the rational numbers by applying finite numbers of $\exp$, $\ln$ and/or radicals.
Clearly, the elementary numbers contain all algebraic numbers and the explicit elementary numbers contain all explicit algebraic numbers (the numbers representable by radicals).
Chow proves in [Chow 1999] his
Corollary 1:
"If Schanuel's conjecture is true, then the algebraic numbers in" the explicit elementary numbers "are precisely the roots of polynomial equations with integer coefficients that are solvable in radicals."
That means, the quintics that are not solvable in radicals cannot be solved by elementary numbers (means by applying elementary functions).
