Finding the roots of an octic

Question

I'm trying to solve a problem, but it involves finding the exact roots of the octic polynomial

$$x^8+4x^7-10x^6-54x^5+9x^4+226x^3+125x^2-301x-269$$

How can I find the roots of an octic? Wolfram Alpha just gives me the rounded values. Not the exact ones.

Answer 1

The root $x=1.87178\dots$ is a root of the quadratic,

$$4x^2+2\Big(2-\sqrt{5}+\sqrt{13-4\sqrt{5}}\Big)x-\Big(7+5\sqrt{5}+\sqrt{118-50\sqrt{5}}\Big)=0$$

which can then be solved using the well-known quadratic formula.

You can factor your octic into two quartics over $\sqrt{5}$, then those quartics into four quadratics over $\sqrt{13\pm4\sqrt{5}}\,$ (like the above), but you'll need Mathematica or Maple for a quick result.