The value of 
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if $x+\frac{1}{x}+2=0$, then the value of x ³³ + x ³² + x ¹³ + x ¹² + x + 1 is

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Please find below the solution to the asked query:

We have :  $x + \frac{1}{x} + 2 = 0$ , So

$$\begin{gathered} \Rightarrow \frac{x^2 + 1 + 2 x}{x}= 0 \\ \Rightarrow x^2 + 2 x+ 1= 0 \\ \Rightarrow x^2 + x + x + 1= 0 \\ \Rightarrow x\left( x +1 \right) +1\left( x +1 \right)= 0 \\ \Rightarrow \left(x + 1\right)\left( x +1 \right)= 0 \end{gathered}$$
So, We get :

x  = - 1 and  - 1 

Then we find value of : x³³ + x³² + x¹³ + x¹² + x +  1 by substituting value of ' x ' ( As we calculated above and get x = - 1 )

$\Rightarrow$ ( - 1 )³³ + ( - 1 )³² + ( - 1 )¹³ + ( - 1 )¹² + ( - 1 ) +  1

$\Rightarrow$ ( - 1 ) + ( 1 )+ ( - 1 ) + ( 1 ) + ( - 1 ) +  1                        ( We know : ( - 1 )^{odd number} = - 1 and ( - 1 )^{even number} = 1 )

$\Rightarrow$ - 1 + 1 - 1 + 1 - 1 + 1

$\Rightarrow$0                                                                                         ( Ans )


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