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Solution to function with power greater than 4

roni

New member
Apr 21, 2018
20
Why we can't use radical to solve an equations with power greater than 4?
 

Cbarker1

Active member
Jan 8, 2013
255
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)