Is this polynomial solvable by radicals? 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
$$
16(x^6-9x^4-4x^3+27x^2-36x-23) =
\left( i\sqrt {3}\sqrt [3]{2}-2\,\sqrt {3}-\sqrt [3]{2}-2\,x
 \right)  \\
\left( i\sqrt {3}\sqrt [3]{2}+2\,x+2\,\sqrt {3}+\sqrt [3]{2}
 \right)  \left( i\sqrt {3}\sqrt [3]{2}-2\,\sqrt {3}+\sqrt [3]{2}+2\,x
 \right)  \\
\left( i\sqrt {3}\sqrt [3]{2}+2\,\sqrt {3}-\sqrt [3]{2}-2\,x
 \right)  \left( x+\sqrt {3}-\sqrt [3]{2} \right)  \left( -x+\sqrt {3}
+\sqrt [3]{2} \right) 
$$
