Factorise $x^{4} + x^{2} + 1$.

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

Solution

Add and subtract $x^{2}$: $x^{4}+x^{2}+1 = x^{4}+2x^{2}+1 - x^{2} = (x^{2}+1)^{2} - x^{2} = (x^{2}+1-x)(x^{2}+1+x) = (x^{2}-x+1)(x^{2}+x+1)$.

Asked in: IMO

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