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प्रश्न
Using binomial theorem, write down the expansions .
(iii) \[\left( x - \frac{1}{x} \right)^6\]
उत्तर
(iii) \[\left( x - \frac{1}{x} \right)^6 \]
\[ = ^{6}{}{C}_0 x^6 \left( \frac{1}{x} \right)^0 - ^{6}{}{C}_1 x^5 \left( \frac{1}{x} \right)^1 +^{6}{}{C}_2 x^4 \left( \frac{1}{x} \right)^2 - ^{6}{}{C}_3 x^3 \left( \frac{1}{x} \right)^3 + ^{6}{}{C}_4 x^2 \left( \frac{1}{x} \right)^4 -^6 C_5 x^1 \left( \frac{1}{x} \right)^5 + ^{6}{}{C}_6 x^0 \left( \frac{1}{x} \right)^6 \]
\[ = x^6 - 6 x^5 \times \frac{1}{x} + 15 x^4 \times \frac{1}{x^2} - 20 x^3 \times \frac{1}{x^3} + 15 x^2 \times \frac{1}{x^4} - 6 x \times \frac{1}{x^5} + \frac{1}{x^6}\]
\[ = x^6 - 6 x^4 + 15 x^2 - 20 + \frac{15}{x^2} - \frac{6}{x^4} + \frac{1}{x^6}\]
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