MHB Solution to function with power greater than 4

[roni1]

Why we can't use radical to solve an equations with power greater than 4?

 

[cbarker1]

You can solve certain equations with higher than 4:

1. $x^n-a=0$
2. $x^{2n}+x^n+a=0$
3. $x^{3n}+x^{2n}+x^n+a=0$
4. $x^{4n}+x^{3n}+x^{2n}+x^n+a=0$
5. etc.

where $a \in \Bbb{R}$ and $n\in \Bbb{N}$.

However, the general fifth degree or higher polynomial does not have a radical solution due to a theorem, Abel-Ruffin Theorem. there is no solutions in radical to general polynomial equations of degree five or higher with arbitrary coefficients.

Note: it does not assert some higher-degree polynomials have no solutions... (see at the beginning of the post)