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You cannot prove it because it is not true. Some quintics ARE solvable.
For example, if all the coefficients (in a quintic in x) sum to 0 then (x - 1) is a factor. So one solution is x = 1 and you are left qith a quartic. If the sum of odd coeffs equals the sum of even coeffs then (x + 1) is a factor. So, in some cases, at least, quintics are solvable.
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How do you prove the general form of the quintic is not solvable?
A general quintic can be solved using numeric methods. It may be an approximate solution but then even the solution to x2 = 2, in decimal terms, is approximate.
How do you prove ax5 plus bx4 plus cx3 plus dx2 plus ex plus f equals 0 is not solvable?
This is called the Abel-Ruffini theorem.
What is a sentence using the word solvable?
I can give you several example sentences.That problem is just not solvable.I think that's a solvable situation if we work together.Is the math problem solvable?
Is every solvable group abelian?
No.
Are all equations solvable?
maybe maybe not
