Question

Expand the following:
${(x+\frac{1}{x})}^6$

Answer 1

Using binomial theorem for the expansion of ${(x+\frac{1}{x})}^6$, we have

${(x+\frac{1}{x})}^6=6_{C_0}(x{)}^6+6_{C_1}(x{)}^5\left(\frac{1}{x}\right)+6_{C_2}(x{)}^4{\left(\frac{1}{x}\right)}^2+6_{C_3}(x{)}^3\left(\frac{1}{x}\right)+6_{C_4}(x{)}^2{\left(\frac{1}{x}\right)}^4+6_{C_5}(x){\left(\frac{1}{x}\right)}^5+6_{C_6}{\left(\frac{1}{x}\right)}^6$

$=x^6+6.x^5.\frac{1}{x}+15.x^4.\frac{1}{x^2}+20.x^3.\frac{1}{x^3}+15.x^2.\frac{1}{x^4}+6.x.\frac{1}{x^5}+\frac{1}{x^6}$

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