Question

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

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{\left(\dfrac{x-1}{x}\right)\left(\dfrac{x+1}{x}\right)\left(\dfrac{x^2+1}{x}\right)}$

$\underline{\textbf{To find:}}$

$\textsf{The product of}$

$\mathsf{\left(\dfrac{x-1}{x}\right)\left(\dfrac{x+1}{x}\right)\left(\dfrac{x^2+1}{x}\right)}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{\left(\dfrac{x-1}{x}\right)\left(\dfrac{x+1}{x}\right)\left(\dfrac{x^2+1}{x}\right)}$

$\mathsf{Using,}$

$\boxed{\mathsf{(a-b)(a+b)=a^2-b^2}}$

$\mathsf{=\left(\dfrac{x^2-1^2}{x^2}\right)\left(\dfrac{x^2+1}{x}\right)}$

$\mathsf{=\left(\dfrac{x^2-1}{x^2}\right)\left(\dfrac{x^2+1}{x}\right)}$

$\textsf{Once again using,}$

$\boxed{\mathsf{(a-b)(a+b)=a^2-b^2}}$

$\mathsf{=\dfrac{(x^2)^2-1^2}{x^3}}$

$\mathsf{=\dfrac{x^4-1}{x^3}}$

$\underline{\textbf{Answer:}}$

$\textsf{The product of}$

$\mathsf{\left(\dfrac{x-1}{x}\right)\left(\dfrac{x+1}{x}\right)\left(\dfrac{x^2+1}{x}\right)\;is\;\;\dfrac{x^4-1}{x^3}}$

Answer 2

Given: Given expression is

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

To find: We have to find the product.

Solution:

To find the product of the above expression we have to follow the below steps-

We know that,

$(x - y)(x + y) = ( {x}^{2} - {y}^{2} )$

Putting the formula in the above expression we get-

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

Multiplying the result with the rest of the expression we get the final product of the expression.

$( {x}^{2} - \frac{1}{ {x}^{2} } )( {x}^{2} + \frac{1}{x} ) \\ = {x}^{4} - \frac{1}{ {x}^{2} } \times {x}^{2} + {x}^{2} \times \frac{1}{x} - \frac{1}{ {x}^{2} } \times \frac{1}{x} \\ = {x}^{4} - 1 + x - \frac{1}{ {x}^{3} }$

Thus, the product of the above expression is ${x}^{4} - 1 + x - \frac{1}{ {x}^{3} }$.