<s>[INST] Existence of real roots of a quartic polynomial Question
What is the minimum possible value of $a^{2}+b^{2}$ so that the polynomial $x^{4}+ax^{3}+bx^{2}+ax+1=0$ has at least 1 root?
Attempt
I divided by $x^{2}$ and got $$x^{2}+\frac{1}{x^{2}}+2+a\left(x+\frac{1}{x}\right)+b-2=0$$ by letting $$x+\frac{1}{x}=X$$
the equation becomes:
$$X^{2}+aX+(b-2)=0$$
$$\therefore X=\frac{-a\pm \sqrt{a^{2}-4b+8}}{2}$$ but I am not sure how to continue. If the polynomial has 1 root doesn't that is should be a double root? Are we counting multiplicity or not?
 [/INST] It seems to me that requesting the smallest possible value for $a^2+b^2$ is almost a red herring.

Claim.
For $|a| \leq 4$ and $b > 2 |a| -  2$, the polynomial
  $$
p(x) = x^4 + ax^3 + bx^2 + ax + 1
$$
  has no real zeros.

Proof.
First set $b = 2|a| - 2 + \epsilon$, where $\epsilon > 0$.  If we denote $\pm = \operatorname{sign} a$, so that $a = \pm |a|$, we have
$$
\begin{align}
p(x) &= x^4 + ax^3 + (2|a|-2+\epsilon)x^2 + ax + 1 \\
&= x^4 - 2x^2 + 1 + x(ax^2 + 2|a|x + a) + \epsilon x^2 \\
&= (x^2-1)^2 + ax(x \pm 1)^2 + \epsilon x^2 \\
&= (x \pm 1)^2\left[ (x \mp 1)^2 + ax \right] + \epsilon x^2 \\
&= (x \pm 1)^2\left[ x^2 \pm (|a|-2)x + 1 \right] + \epsilon x^2 \\
&\geq (x \pm 1)^2 \min\left\{x^2 - 2x + 1,x^2+2x+1\right\} + \epsilon x^2 \\
&= (x \pm 1)^2\min\left\{(x-1)^2,(x+1)^2\right\} + \epsilon x^2 \\
&> 0.
\end{align}
$$
Q.E.D.
The points on the boundary of the region
$$
A = \{(a,b) \,\colon |a| \leq 4 \,\,\,\text{and}\,\,\,b > 2 |a| -  2\}
$$
which have the smallest norm are
$$
(a,b) = \left(\pm \frac{4}{5},-\frac{2}{5}\right),
$$
as shown in the following image.

It only remains to demonstrate that one of these points yields a polynomial with a real zero.  In fact they both do:
$$
x^4 + \frac{4}{5} x^3 - \frac{2}{5} x^2 + \frac{4}{5} x + 1
$$
has a zero at $x=-1$ and
$$
x^4 - \frac{4}{5} x^3 - \frac{2}{5} x^2 - \frac{4}{5} x + 1
$$
has a zero at $x=1$.
Thus

The smallest value of $a^2 + b^2$ for which we can find some $a,b\in \mathbb R$ such that $p(x)$ has a real zero is
  $$
\left(\frac{4}{5}\right)^2 + \left(\frac{2}{5}\right)^2 = \frac{4}{5}.
$$

Below is a plot of the $(a,b)$-plane which shows the region (in blue) where the polynomial $p(x)$ has at least one real root.  Note that for $|a| \leq 4$ the boundary of this region is precisely $b = 2|a| - 2$ (shown in black), but for $|a| > 4$ it curves inward.

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