Example 1. Let $x+y=1, x^{2}+y^{2}=2$, find the value of $x^{7}+y^{7}$. (Japanese University Entrance Exam Question, 1979)
From formula (1), we can get,
$$
x^{4}+y^{4}=\left(x^{2}+y^{2}\right)^{2}-2 x^{2} y^{2} \text {. }
$$

From formula (2), we can get,
$$
x y=-\frac{1}{2}\left(1^{2}-2\right)=-\frac{1}{2} \text { . }
$$

Thus, \( x^{7}+y^{7}=\left(x^{3}+y^{3}\right)\left(x^{4}+y^{4}\right)-x^{4} y^{3} \)
$$
\begin{array}{l}
-y^{4} x^{3} \\
= {[(x+y)-3 x(x+y)] } \\
{\left[\left(x^{2}+y^{2}\right)^{2}-2 x^{2} y^{2}\right] } \\
-(x y)^{3}(x+y)=\frac{71}{8} .
\end{array}
$$