Question

find the product of (x - 1/x) , (x+1/x) , (x^2+1/x^2), (x^4+1/x^4)​

Answer 1

using the identity (a+b)(a-b)=a^2 b^2 in all cases

$( \frac{x + 1}{x} )( \frac{x - 1}{x} )( {x}^{2} + \frac{1}{ {x}^{2} })( {x}^{4} + \frac{1}{ {x}^{4} } )$

$= ( {x}^{2} - \frac{1}{ {x}^{2} } )( {x}^{2} + \frac{1}{ {x}^{2} } )( {x}^{2} + \frac{1}{ {x}^{2} } )$

$= ({( {x}^{2} })^{2} - ( \frac{1}{ {x}^{2} } {)}^{2} )({x}^{4} + \frac{1}{ {x}^{4} } )$

$= ( {x}^{4} - \frac{1}{ {x}^{4} } )( \frac{x + 1}{ {x}^{4} } )$

$= ( {x}^{4}{)}^{2} - ( \frac{1}{ {x}^{4} } {)}^{2}$

$= > {x}^{2} - \frac{1}{ {x}^{8} }$

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