A) \[\left( {{x}^{4}}+1-{{x}^{2}} \right)\,\,\left( {{x}^{2}}+1+x \right)\,\,\left( {{x}^{2}}+1-x \right)\]
B) \[\left( {{x}^{4}}+1-{{x}^{2}} \right)\,\,\left( {{x}^{2}}-1+x \right)\,\,\left( {{x}^{2}}+1+x \right)\]
C) \[\left( {{x}^{4}}-1+{{x}^{2}} \right)\,\,\left( {{x}^{2}}-1+x \right)\,\,\left( {{x}^{2}}+1+x \right)\]
D) \[\left( {{x}^{4}}-1+{{x}^{2}} \right)\,\,\left( {{x}^{2}}+1-x \right)\,\,\left( {{x}^{2}}+1+x \right)\]
Correct Answer: A
Solution :
\[{{x}^{8}}+1+{{x}^{4}}={{({{x}^{4}})}^{2}}+{{(1)}^{2}}+{{x}^{4}}\] \[={{({{x}^{4}}+1)}^{2}}-2{{x}^{2}}+{{x}^{4}}={{({{x}^{4}}+1)}^{2}}-{{({{x}^{2}})}^{2}}\] \[=({{x}^{4}}+1+{{x}^{2}})({{x}^{4}}+1-{{x}^{2}})\] \[=[{{({{x}^{2}})}^{2}}+{{(1)}^{2}}+{{x}^{2}}]\,\,[{{x}^{4}}-{{x}^{2}}+1]\] \[=[{{({{x}^{2}}+1)}^{2}}-2{{x}^{2}}+{{x}^{2}}]\,\,[{{x}^{4}}-{{x}^{2}}+1]\] \[=[{{({{x}^{2}}+1)}^{2}}-{{(x)}^{2}}]\,\,[{{x}^{4}}-{{x}^{2}}+1]\] \[=({{x}^{2}}+1+x)({{x}^{2}}+1-x)({{x}^{4}}-{{x}^{2}}+1)\] \[=({{x}^{2}}+x+1)({{x}^{2}}-x+1)({{x}^{4}}-{{x}^{2}}+1)\]
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