Finding the roots of a polynomial of the form: $w^6-9kw^4+18k^2w^2-6 k^3$

Question

I need to find the roots of this sixth-degree polynomial in $w$:

$$w^6-9\left(\frac{g}{l}\right)w^4+18\left(\frac{g}{l}\right)^2w^2-6 \left(\frac{g}{l}\right)^3$$

I've found this polynomial while solving a problem related to a triple pendulum (for context, $w$ is the frequency of a triple pendulum with equal lengths $l$; and $g$ is gravity) and couldn't manage to proceed.

Answer 1

Let $w=\sqrt{x\frac{g}{l}}$ then the polynomial becomes $$x^3-9x^2+18x-6=0,$$ with roots $$ 0.41577,\quad 2.2943\quad\text{and}\quad 6.2899.$$

Answer 2

With the substitution $w = \sqrt{\frac{g}{\ell}(v + 3)}$, we get the "depressed" cubic equation $$v^3 - 9 v - 6 = 0$$ The discriminant is positive, so the polynomial has three real roots that we can readily express in trigonometric form $$v = 6\cos\left(\frac13\operatorname{arccos}\frac{1}{\sqrt{3}}-\frac{2\pi k}{3} \right) $$ for $k = 0$, $1$, $2$.