Analytical methods for solving polynomial Question is very simple: What analytic  methods (not numerical or approximation methods) do you know to solve this kind of polynomial?: $$x^4-2x^3+x^2-2x+1=0$$
 A: There is, of course, the general solution to the quartic, as noted in a comment.
Another way is to notice that the coefficients are symmetric. We therefore divide the equation by $x^2$ and rearrange the terms to get
$$\left(x^2+\frac{1}{x^2}\right) -2\left(x+\frac 1x\right) + 1=0$$
We let $u=x+\frac 1x$ and note that
$$u^2-2=x^2+\frac 1{x^2}$$
So our equation becomes
$$u^2-2u-1=0$$
We can use this to get two solutions for $u$. We then use our definition of $u$ to get two $x$'s for each $u$, giving us four solutions to the original equation. And all this was done with simple algebra and solving three quadratic equations.
Note that, for real valued $x$, the expression $x+\frac 1x$ has a minimum absolute value of $2$. Thus, for any $u$ where $|u|<2$ the values of $x$ will be non-real. There are two real values of $x$ if $|u|>2$ and only one double-root if $|u|=2$. This answers the question in the comment on showing how many of the solutions are real numbers.
Since you just asked for analytical methods I assume you do not need me to carry this to completion and show the values of $x$.
