Question : A complete factorisation of $(x^4+64)$ is:
Option 1: $(x^2+8)^2$
Option 2: $(x^2+8)$$(x^2-8)$
Option 3: $(x^2-4x+8)(x^2-4x-8)$
Option 4: $(x^2+4x+8)$$(x^2-4x+8)$
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Correct Answer: $(x^2+4x+8)$$(x^2-4x+8)$
Solution :
Given: $(x^4+64)$
We know that the algebraic identities are $(x^2+y^2)=(x+y)^2-2xy$ and $(x^2-y^2)=(x+y)(x-y)$.
So, $(x^4+64)=(x^2)^2+8^2$
$=(x^2+8)^2-2x^2\times8$
$=(x^2+8)^2-(4x)^2$
$=(x^2+4x+8)(x^2-4x+8)$
Hence, the correct answer is $(x^2+4x+8)(x^2-4x+8)$.
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