2. Let $x_{1}$ and $x_{2}$ be the largest roots of the polynomials

$$
\begin{gathered}
f(x)=1-x-4 x^{2}+x^{4} \\
\text { and } \\
g(x)=16-8 x-16 x^{2}+x^{4}
\end{gathered}
$$

respectively. Find $\frac{x_{2}}{x_{1}}$.
Solution:

Notice that $f(-2)>0, f(-1)<0, f(0)>0, f(1)<0$. Therefore, the polynomial $f(x)$ has 4 real roots. Similarly, from the inequalities $g(-4)>0, g(-2)<0, g(0)>0, g(2)<0$ it follows that the polynomial $g(x)$ has 4 real roots.

Comparison of the coefficients of the polynomials

$$
f(x)=1-x-4 x^{2}+x^{4} \text { and } g(x)=16-8 x-16 x^{2}+x^{4}
$$

shows that according to Vieta's formulas, the roots of the polynomial $g(x)$ are double the roots of the polynomial $f(x)$. Hence, it follows that $\frac{x_{2}}{x_{1}}=2$.

Answer: 2.