3. It is known that the polynomial $f(x)=8+32 x-12 x^{2}-4 x^{3}+x^{4}$ has 4 distinct real roots $\left\{x_{1}, x_{2}, x_{3}, x_{4}\right\}$. The polynomial of the form $g(x)=b_{0}+b_{1} x+b_{2} x^{2}+b_{3} x^{3}+x^{4}$ has roots $\left\{x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}\right\}$. Find the coefficient $b_{1}$ of the polynomial $g(x)$.
Solution: Let the coefficients of the given polynomial (except the leading one) be denoted by $a_{0}, a_{1}, a_{2}, a_{3}$

$$
f(x)=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+x^{4}
$$

Then, according to the problem, we have

$$
f(x)=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+x^{4}=\left(x-x_{1}\right)\left(x-x_{2}\right)\left(x-x_{3}\right)\left(x-x_{4}\right)
$$

Together with the polynomial $f(x)$, consider the polynomial $h(x)$, which has roots $\left\{-x_{1},-x_{2},-x_{3},-x_{4}\right\}$

$$
h(x)=\left(x+x_{1}\right)\left(x+x_{2}\right)\left(x+x_{3}\right)\left(x+x_{4}\right)=a_{0}-a_{1} x+a_{2} x^{2}-a_{3} x^{3}+x^{4}
$$

Consider the polynomial $G(x)=f(x) h(x)$:

$$
\begin{aligned}
& G(x)=\left(x-x_{1}\right)\left(x-x_{2}\right)\left(x-x_{3}\right)\left(x-x_{4}\right)\left(x+x_{1}\right)\left(x+x_{2}\right)\left(x+x_{3}\right)\left(x+x_{4}\right)= \\
&=\left(x^{2}-x_{1}^{2}\right)\left(x^{2}-x_{2}^{2}\right)\left(x^{2}-x_{3}{ }^{2}\right)\left(x^{2}-x_{4}^{2}\right)
\end{aligned}
$$

By substituting the variable $y=x^{2}$, we obtain the required polynomial $g(y)$, since

$$
g(y)=\left(y-x_{1}{ }^{2}\right)\left(y-x_{2}{ }^{2}\right)\left(y-x_{3}{ }^{2}\right)\left(y-x_{4}{ }^{2}\right)
$$

In our case

$$
\begin{gathered}
f(x)=8+32 x-12 x^{2}-4 x^{3}+x^{4} \\
h(x)=8-32 x-12 x^{2}+4 x^{3}+x^{4} \\
g(x)=f(x) h(x)=64-1216 x^{2}+416 x^{4}-40 x^{6}+x^{8}
\end{gathered}
$$

Interregional Olympiad for Schoolchildren based on Departmental Educational Organizations

$$
g(y)=64-1216 y+416 y^{2}-40 y^{3}+y^{4}
$$

Answer: -1216.