# 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.

• I don't think Abel's theorem states that you can't solve specific polynomials (consider the specific polynomial $(x-1)(x-2)(x-3)(x-4)(x-5)$ for example). Abel's theorem states that there is no general formula (i.e. no analogue of the quadratic formula) that will work for all quintic equations. – Sam Weatherhog Dec 2 '15 at 1:00
• You can solve a quintic equation in terms of roots only when it's Galois group is solvable. – Sam Weatherhog Dec 2 '15 at 1:01
• @SamWeatherhog there are specific polynomials that cannot be solved in the described way. Of course not every polynomial is such a polynomial, only specific ones. – quid Dec 2 '15 at 1:02
• Did you see the section "Beyond radicals" on the WIkipedia page? – quid Dec 2 '15 at 1:04
• @quid I think I'm missing your point. Is there an error in what I said? – Sam Weatherhog Dec 2 '15 at 2:47