Question

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

Answer 1

${(x+\frac{1}{x})}^6$
${=}^{6}C+0\left(x{)}^6{+}_1^C\right(x{)}^5\left(\frac{1}{x}\right){+}^6{C}_2\left(x{)}^4{\left(\frac{1}{x}\right)}^2{+}^6{C}_3\right(x{)}^3{\left(\frac{1}{x}\right)}^3+$

${}^6{C}_4\left(x{)}^2{\left(\frac{1}{x}\right)}^4{+}^6{C}_5\right(x){\left(\frac{1}{x}\right)}^5{+}^6{C}_6{\left(\frac{1}{x}\right)}^6$
$=x^6+6(x{)}^5\left(\frac{1}{x}\right)+15(x{)}^4\left(\frac{1}{x^2}\right)+20(x{)}^3\left(\frac{1}{x^3}\right)+15(x{)}^2\left(\frac{1}{x^4}\right)+6\left(x\right)\left(\frac{1}{x^5}\right)+\frac{1}{x^6}$
$=x^6+6x^4+15x^2+20+\frac{15}{x^2}+\frac{6}{x^4}+\frac{1}{x^6}$