Factorise $1 - 81x^{4}$.

Factorise $1 - 81x^{4}$.
  1. $(1 - 3x)(1 + 3x)(1 + 9x^{2})$
  2. $(1 - 9x^{2})^{2}$
  3. $(1 - 3x)^{2}(1 + 9x^{2})$
  4. $(1 - 9x^{2})(1 - 9x^{2})$

Solution

$1 - 81x^{4} = 1^{2} - (9x^{2})^{2} = (1-9x^{2})(1+9x^{2})$. Further, $1 - 9x^{2} = (1-3x)(1+3x)$. Final: $(1-3x)(1+3x)(1+9x^{2})$.

Asked in: IMO

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