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Term Independent of x

Find the coef...

Question

Find the coefficient of $x^{2}$ in the expansion of $(x - 1)^{3} {(\frac{1}{x} + 2 x)}^{6}$

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Solution

$\begin{matrix} {(x - 1)}^{3} {(\frac{1}{x} + 2 x)}^{6} \Rightarrow (^{3} c_{0} x^{3} -^{3} c_{1} x^{2} +^{3} c_{2} x^{1} +^{3} c_{3}) + (^{6} c_{0} \frac{1}{x^{6}} +^{6} c_{1} (2 x) . \frac{1}{x^{6}} +^{6} c_{2} {(2 x)}^{2} (\frac{1}{x^{4}}) +^{6} c_{3} {(2 x)}^{3} . \frac{1}{x^{3}} +^{6} c_{4} {(2 x)}^{4}) \Rightarrow (x^{3} - 3 x^{2} + 3 x - 1) (\frac{1}{x^{6}} + \frac{6 \times 2}{x^{4}} + \frac{60}{x^{2}} + 160 + 20 x^{2}) \Rightarrow - 480 x^{2} - 20 x^{2} = - 500 x^{2} \end{matrix}$
Hence, coefficient of $x^{2}$ is $(- 500)$

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