<s>[INST] Determine all zeros of the polynomial $X^4 - 2X^3 - X^2 + 2X + 1 \in \mathbb C[X]$. This is Exercise 14 on page 110 of Analysis I by Amann and Escher.
The hint given is as follows: multiply the polynomial by $1/X^2$ and substitute $Y = X - 1/X$.
If I attempt this, I get the following:
\begin{align*}
X^4 - 2X^3 - X^2 + 2X + 1 &= 0\\
\Rightarrow X^2 - 2X - 1 + \frac{2}{X} + \frac{1}{X^2} &= 0.
\end{align*}
My problem is that I don't understand how to make the suggested substitution. I'm wondering if there is something obvious I'm missing.
I appreciate any help.
 [/INST] $y^2=x^2-2+1/x^2$, so the expression after multiplying by $1/x^2$ can be rewritten
$$y^2-2y+1=(y-1)^2=0$$
So $y=1$, or $x-1/x=1$ or $x^2-x-1=0$. Solving this quadratic gives $x=\frac{1\pm\sqrt5}2$, and it is easy to check that $x=0$ doesn't satisfy the original equation, so we have found all solutions.
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