A) \[\left( a+\frac{1}{a} \right)\left( {{a}^{2}}+\frac{1}{{{a}^{2}}} \right)-2\left( a-\frac{1}{a} \right)\]
B) \[\left( a-\frac{1}{a} \right)\left( {{a}^{2}}+\frac{1}{{{a}^{2}}}+1 \right)+2\left( a-\frac{1}{a} \right)\]
C) \[\left( a+\frac{1}{a} \right)\left( a-\frac{1}{a} \right)\]
D) \[\left( a-\frac{1}{a} \right)\left( {{a}^{2}}+\frac{1}{{{a}^{2}}}-1 \right)\]
Correct Answer: D
Solution :
| \[{{a}^{3}}-\frac{1}{{{a}^{3}}}-2a+\frac{2}{a}={{a}^{3}}-{{\left( \frac{1}{a} \right)}^{3}}-2\left( a-\frac{1}{a} \right)\] |
| \[=\left( a-\frac{1}{a} \right)\left( {{a}^{2}}+\frac{1}{{{a}^{2}}}+1 \right)-2\left( a-\frac{1}{a} \right)\] |
| \[=\left( a-\frac{1}{a} \right)\left[ {{a}^{2}}+\frac{1}{{{a}^{2}}}+1-2 \right]\] |
| \[=\left( a-\frac{1}{a} \right)\left( {{a}^{2}}+\frac{1}{{{a}^{2}}}-1 \right)\] |
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