I want to find a nice tidy algebraic form for the equation of this polynomial:
It has degree 6, double roots at $x=\pm1$, and it oscillates between $-1$ and $+1$.
Clearly it has the form $f(x) = c(x-1)^2(x+1)^2(x^2-k^2)$ where $k$ is the zero that's roughly $0.32$, and $c$ is an unknown constant. So, all I really need is a nice tidy expression for $k^2$. All my attempts have failed -- I get a mess of surds that my rusty algebra skills cannot handle.
Since $f(0) = -1$, we have $-1 = -ck^2$, so $c = 1/k^2$, which means that $k$ is the only unknown.
If it helps, the numerical value of $k$ is approximately 0.325411344339772081233.
Motivation (if you need some): these kinds of polynomials show up as error functions in certain types of approximation problems.
Edit:
The best I can do is
$$
k^2=\frac{-3+2 \sqrt[3]{\sqrt{2}-1}+3 \left(\sqrt{2}-1\right)^{2/3}}{2
\sqrt[3]{\sqrt{2}-1}}
$$
Any hope of simplifying that??