6. Let $x_{1}$ and $x_{2}$ be the largest roots of the polynomials $f(x)=1-x-4 x^{2}+x^{4}$ and $g(x)=16-8 x-16 x^{2}+x^{4}$ respectively. Find $\frac{x_{1}}{x_{2}}$. Justify your solution.

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# Solution.

1st method

Notice that $g(2 x)=16 f(x)$. Then $x_{1}-$ is a root of $f(x)$ if and only if $2 x_{1}-$ is a root of $g(x)$. Therefore, $\frac{x_{1}}{x_{2}}=\frac{1}{2}$.

2nd method

Comparing 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 twice the roots of the polynomial $f(x)$. Hence, it follows that $\frac{x_{1}}{x_{2}}=\frac{1}{2}$.

Answer: 0.5 .