The polynomial
$p(x) = x^6-9x^4-4x^3+27x^2-36x-23$.
has at least one (real, irrational) root that is expressible by radicals (can you find it?).
Are all the roots of $p$ expressible by radicals and if so, how can one find the expressions?
The polynomial
$p(x) = x^6-9x^4-4x^3+27x^2-36x-23$.
has at least one (real, irrational) root that is expressible by radicals (can you find it?).
Are all the roots of $p$ expressible by radicals and if so, how can one find the expressions?
Maple says $$ 16(x^6-9x^4-4x^3+27x^2-36x-23) = \left( i\sqrt {3}\sqrt [3]{2}-2\,\sqrt {3}-\sqrt [3]{2}-2\,x \right) \\ \left( i\sqrt {3}\sqrt [3]{2}+2\,x+2\,\sqrt {3}+\sqrt [3]{2} \right) \left( i\sqrt {3}\sqrt [3]{2}-2\,\sqrt {3}+\sqrt [3]{2}+2\,x \right) \\ \left( i\sqrt {3}\sqrt [3]{2}+2\,\sqrt {3}-\sqrt [3]{2}-2\,x \right) \left( x+\sqrt {3}-\sqrt [3]{2} \right) \left( -x+\sqrt {3} +\sqrt [3]{2} \right) $$