A) \[(x-y)({{x}^{2}}+xy+{{y}^{2}})(x+y)({{x}^{2}}-xy+{{y}^{2}})\]\[({{x}^{2}}+{{y}^{2}})({{x}^{4}}-{{x}^{2}}{{y}^{2}}+{{y}^{4}})\]
B) \[(x+y)({{x}^{2}}-xy+{{y}^{2}})(x+y)({{x}^{2}}-xy+{{y}^{2}})\]\[({{x}^{2}}+{{y}^{2}})({{x}^{4}}-{{x}^{2}}{{y}^{2}}+{{y}^{4}})\]
C) \[(x+y)({{x}^{2}}+xy-{{y}^{2}})(x+y)({{x}^{2}}-xy+{{y}^{2}})\]\[({{x}^{2}}+{{y}^{2}})({{x}^{4}}-{{x}^{2}}{{y}^{2}}+{{y}^{4}})\]
D) \[(x-y)({{x}^{2}}-xy+{{y}^{2}})(x+y)({{x}^{2}}-xy+{{y}^{2}})\]\[({{x}^{2}}+{{y}^{2}})({{x}^{4}}-{{x}^{2}}{{y}^{2}}+{{y}^{4}})\]
Correct Answer: A
Solution :
We have, \[{{x}^{12}}-{{y}^{12}}\] \[={{({{x}^{6}})}^{2}}-{{({{y}^{6}})}^{2}}=({{x}^{6}}-{{y}^{6}})({{x}^{6}}+{{y}^{6}})\] \[=[{{({{x}^{3}})}^{2}}-{{({{y}^{3}})}^{2}}][{{({{x}^{2}})}^{3}}+{{({{y}^{2}})}^{3}}]\] \[=({{x}^{3}}+{{y}^{3}})({{x}^{3}}-{{y}^{3}})[({{x}^{2}}+{{y}^{2}})({{x}^{4}}+{{y}^{4}}-{{x}^{2}}{{y}^{2}})]\]\[=(x+y)({{x}^{2}}+{{y}^{2}}-xy)(x-y)({{x}^{2}}+{{y}^{2}}+xy)\] \[({{x}^{2}}+{{y}^{2}})({{x}^{4}}+{{y}^{4}}-{{x}^{2}}{{y}^{2}})\]
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