Factorise $x^{2} - \dfrac{1}{x^{2}}$ (for $x \ne 0$).

Factorise $x^{2} - \dfrac{1}{x^{2}}$ (for $x \ne 0$).
  1. $\left(x - \dfrac{1}{x}\right)\left(x + \dfrac{1}{x}\right)$
  2. $\left(x - \dfrac{1}{x}\right)^{2}$
  3. $\dfrac{x^{2}-1}{x^{2}}$
  4. $\left(x + \dfrac{1}{x}\right)^{2}$

Solution

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

Asked in: IMO

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