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, ${\left(x+\frac{1}{x}\right)}^6$
$={}^6{C}_0{\left(x\right)}^6+{}^6{C}_1{\left(x\right)}^5\left(\frac{1}{x}\right)+{}^6{C}_2{\left(x\right)}^4{\left(\frac{1}{x}\right)}^2+{}^6{C}_3{\left(x\right)}^3{\left(\frac{1}{x}\right)}^3$
$+{}^6{C}_4{\left(x\right)}^2{\left(\frac{1}{x}\right)}^4+{}^6{C}_5\left(x\right){\left(\frac{1}{x}\right)}^5+{}^6{C}_6{\left(\frac{1}{x}\right)}^6$
$=x^6+6x^4+15x^2+20+15\left(\frac{1}{x^2}\right)+6\left(\frac{1}{x^4}\right)+\frac{1}{x^6}$
$=x^6+6x^4+15x^2+20+\frac{15}{x^2}+\frac{6}{x^4}+\frac{1}{x^6}$.