Question

Factorize
$x ^{2} + \frac{1}{x {}^{2} } + 2 - 2x - \frac{2}{x}$
​

Answer 1

Answer

• ( x + ¹/ₓ ) ( √x - ¹/√ₓ )²

Given

$\bullet \;\;\; \rm x^2+\dfrac{1}{x^2}+2-2x-\dfrac{2}{x}$

To Find

• Value of given factorization

Formula Used

$\bullet \;\;\; \rm \bigg(x+\dfrac{1}{x}\bigg)^2=x^2+\dfrac{1}{x^2}+2\\\\\bullet \;\;\; \rm \bigg(x+\dfrac{1}{x}-2\bigg)=\bigg(\sqrt{x}-\dfrac{1}{\sqrt{x}}\bigg)^2$

Solution

$\rm x^2+\dfrac{1}{x^2}+2-2x-\dfrac{2}{x}\\\\\implies \rm \bigg(x^2+\dfrac{1}{x^2}+2\bigg)-2\bigg(x+\dfrac{1}{x}\bigg)\\\\\implies \rm \bigg(x^2+\dfrac{1}{x^2}+2.x.\dfrac{1}{x}\bigg)-2\bigg(x+\dfrac{1}{x}\bigg)\\\\\implies \rm \bigg(x+\dfrac{1}{x}\bigg)^2-2\bigg(x+\dfrac{1}{x}\bigg)\\\\\implies \rm \bigg(x+\dfrac{1}{x}\bigg)\bigg(x+\dfrac{1}{x}-2\bigg)\\\\\implies \rm \bigg(x+\dfrac{1}{x}\bigg)\bigg((\sqrt{x})^2+\bigg(\dfrac{1}{\sqrt{x}}\bigg)^2-2.\sqrt{x}.\dfrac{1}{\sqrt{x}}\bigg)$

$\implies \rm \bigg(x+\dfrac{1}{x}\bigg)\bigg(\sqrt{x}-\dfrac{1}{\sqrt{x}}\bigg)^2$

Answer 2

ANSWER :-

We have to factorize $x ^{2} + \frac{1}{x {}^{2} } + 2 - 2x - \frac{2}{x}$

For solving these problems we must know the formula -

{(a +b )}^{2} = {a}^{2} + {b}^{2} + 2ab

${x}^{2} + \frac{1}{ {x}^{2} } + 2 - 2x - \frac{2}{x} \\ \\ We \: know \: the \: formula - \\ \\ {(a +b )}^{2} = {a}^{2} + {b}^{2} + 2ab \\ \\ Now, \: compare \: this \: formula \: to \: {x}^{2} + \frac{1}{ {x}^{2} } + 2 - 2x - \frac{2}{x} \\ \\ After \: comparing, \: we \: can \: write - \\ \\ ({x + \frac{1}{x}) }^{2} - 2(x + \frac{1}{x} ) \\ \\ Taking \: common (x + \frac{1}{x} ) \\ \\ = >(x + \frac{1}{x} )[(x + \frac{1}{x} ) - 2] - - - - - - - (1) \\ \\ Now, \\ \: again \: compare \: [(x + \frac{1}{x} ) - 2] \: to \: the \: {(a +b )}^{2} = ({a}^{2} + {b}^{2} + 2ab) \\ \\ so, \: we \: can \: write \: it \: \: as \\ ( \sqrt{x} + \frac{1}{ \sqrt{x} })^{2}\: = [(x + \frac{1}{x} ) - 2] \\ \\ Now, \: equation \: (1) \: can \: be \: written \: as - - \\ \\ = > (x + \frac{1}{x} )( \sqrt{x} + \frac{1}{ \sqrt{x} } )^{2} \\ \\ And \: \: the \: \: factorized \: form \: of \: {x}^{2} + \frac{1}{ {x}^{2} } + 2 - 2x - \frac{2}{x} \: is \\ (x + \frac{1}{x} )( \sqrt{x} + \frac{1}{ \sqrt{x} } )^{2}$