Expand

By using Binomial Theorem, the expression $\left(x+\frac{1}{x}\right)^{6}$ can be expanded as

$\left(x+\frac{1}{x}\right)^{6}={ }^{6} C_{0}(x)^{6}+{ }^{6} C_{1}(x)^{5}\left(\frac{1}{x}\right)+{ }^{6} C_{2}(x)^{4}\left(\frac{1}{x}\right)^{2}$

$+{ }^{6} \mathrm{C}_{3}(\mathrm{x})^{3}\left(\frac{1}{\mathrm{x}}\right)^{3}+{ }^{6} \mathrm{C}_{4}(\mathrm{x})^{2}\left(\frac{1}{\mathrm{x}}\right)^{4}+{ }^{6} \mathrm{C}_{5}(\mathrm{x})\left(\frac{1}{\mathrm{x}}\right)^{5}+{ }^{6} \mathrm{C}_{6}\left(\frac{1}{\mathrm{x}}\right)^{6}$

$=x^{6}+6(x)^{5}\left(\frac{1}{x}\right)+15(x)^{4}\left(\frac{1}{x^{2}}\right)+20(x)^{3}\left(\frac{1}{x^{3}}\right)+15(x)^{2}\left(\frac{1}{x^{4}}\right)+6(x)\left(\frac{1}{x^{5}}\right)+\frac{1}{x^{6}}$

$=x^{6}+6 x^{4}+15 x^{2}+20+\frac{15}{x^{2}}+\frac{6}{x^{4}}+\frac{1}{x^{6}}$