Factorise $x^{4} + 4$ (Sophie Germain).

Factorise $x^{4} + 4$ (Sophie Germain).
  1. $(x^{2} + 2x + 2)(x^{2} - 2x + 2)$
  2. $(x^{2} + 2)(x^{2} - 2)$
  3. $(x^{2} + 2)^{2}$
  4. $(x + 2)(x^{3} - 2x^{2} + 4x - 8)$

Solution

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

Asked in: IMO

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