Example 4 Find the remainder when $x^{19}+x^{17}+x^{13}+x^{11}+x^{7}+x^{5}+x^{3}$ is divided by $x^{2}-1$.
$\begin{array}{l}\text { Analysis: } x^{2} \equiv 1\left(\bmod \left(x^{2}-1\right)\right), \\ \text { so } x^{19}+x^{17}+x^{13}+x^{11}+x^{7}+x^{5}+x^{3} \equiv \\ 7 x\left(\bmod \left(x^{2}-1\right)\right) .\end{array}$