Question

$\left(x+\frac{1}{x}\right)^{6} =$

• A. $x^{6}+6x^{4}+15x^{2}-20-\frac{15}{x^{2}}+\frac{6}{x^{4}}-\frac{1}{x^{6}}$
• B. $x^{6}+6x^{4}+15x^{2}+20+\frac{15}{x^{2}}+\frac{6}{x^{4}}+\frac{1}{x^{6}}$
• C. $x^{6}+2x^{4}+15x^{2}+20-\frac{15}{x^{2}}-\frac{6}{x^{4}}+\frac{1}{x^{6}}$
• D. $2x^{6}+6x^{4}+15x^{2}-20+\frac{15}{x^{2}}-\frac{6}{x^{4}}+\frac{1}{x^{6}}$

Answer 1

Answer: (B)

We have, $\left(x+\frac{1}{x}\right)^{6} $
$=\,{}^{6}C_{0}\left(x\right)^{6}+\,{}^{6}C_{1}\left(x\right)^{5}\left(\frac{1}{x}\right)+\,{}^{6}C_{2}\left(x\right)^{4}\left(\frac{1}{x}\right)^{2}+\,{}^{6}C_{3}\left(x\right)^{3}\left(\frac{1}{x}\right)^{3}$
$+\,{}^{6}C_{4}\left(x\right)^{2}\left(\frac{1}{x}\right)^{4}+\,{}^{6}C_{5}\left(x\right)\left(\frac{1}{x}\right)^{5}+\,{}^{6}C_{6}\left(\frac{1}{x}\right)^{6}$
$= x^{6}+6x^{4}+15x^{2}+20+15\left(\frac{1}{x^{2}}\right)+6\left(\frac{1}{x^{4}}\right)+\frac{1}{x^{6}}$
$= x^{6}+6x^{4}+15x^{2}+20+\frac{15}{x^{2}}+\frac{6}{x^{4}}+\frac{1}{x^{6}}$.