x8+ax3+b=0

Simplest examples of irreducible and solvable by radicals octic trinomials, x⁸+ax³+b, where a and b are integers. Checked range: |a|, |b| ≤ 23,700,000. Excluded trivial cases where a or b are equal zero.

Basic notes:

- if x⁸+ax³+b=0 is solvable then x⁸-ax³+b=0 is too. More generally x⁸+ax³+b=0 is solvable if and only if x⁸+ac⁵x³+bc⁸=0 is solvable (c is non-zero rational number).

- if x⁸+ax³+b=0 is solvable then x⁸+(a/b)x⁵+(1/b)=0 is too. And in more general: x⁸+(ac³/b)x⁵+(c⁸/b)=0

See incredible explicit formulas for the roots of the solvable polynomials.