Expand the given expression ( x + 1 x )^6 - Vidyadip

Question

Expand the given expression ${\left( {x + \frac{1}{x}} \right)^6}$

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Answer

Using binomial theorem for the expansion of ${\left( {x + \frac{1}{x}} \right)^6}$ we have
${\left( {x + \frac{1}{x}} \right)^6}$= ${ = ^6}{C_0}{(x)^6}{ + ^6}{C_1}{(x)^5}\left( {\frac{1}{x}} \right)$${ + ^6}{C_2}{(x)^4}{\left( {\frac{1}{x}} \right)^2}{ + ^6}{C_3}{(x)^3}{\left( {\frac{1}{x}} \right)^3}$
${ + ^6}{C_4}{(x)^2}{\left( {\frac{1}{x}} \right)^4}{ + ^6}{C_5}(x){\left( {\frac{1}{x}} \right)^5}$${ + ^6}{C_6}{\left( {\frac{1}{6}} \right)^6}$
$ = {x^6} + 6 \cdot {x^5} \cdot \frac{1}{x} + 15 \cdot 4{x^4} \cdot \frac{1}{{{x^2}}} + $$20 \cdot {x^3} \cdot \frac{1}{{{x^3}}} + 15 \cdot {x^2} \cdot \frac{1}{{{x^4}}} + 6 \cdot x \cdot \frac{1}{{{x^5}}} + \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}}}$

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