Question
Using binomial theorem, expand: $\left(x^2-\frac{2}{x}\right)^7$.
✓
Answer
To find: Expansion of $\left(x^2-\frac{3 x}{7}\right)^7$
Formula used: ${ }^n C_r=\frac{n!}{(n-r)!(r)!}$
We know that, $(a+b)^n={ }^n C_0 a^n+{ }^n C_1 a^{n-1} b+{ }^n C_2 a^{n-2} b^2+\ldots \ldots .+{ }^n C_{n-1} a^{n-1}+{ }^n C_n b^n$
Here We have, $\left(x^2-\frac{3 x}{7}\right)^7$
$\begin{array}{l}\Rightarrow\left[{ }^7 C _0\left( x ^2\right)^{7-0}\right]+\left[7 C _1\left( x ^2\right)^{7-1}\left(-\frac{3 x }{7}\right)^1\right]+\left[7 c_2\left(x^2\right)^{7-2}\left(-\frac{3 x}{7}\right)^2\right]+\left[7 C _3\left( x ^2\right)^{7-3}\left(-\frac{3 x }{7}\right)^3\right]+\left[7 C _4\left( x ^2\right)^{7-4}\left(-\frac{3 x }{7}\right)^4\right] \\ +\left[7 C _5\left( x ^2\right)^{7-5}\left(-\frac{3 x }{7}\right)^5\right]+\left[7 C _6\left( x ^2\right)^{7-6}\left(-\frac{3 x }{7}\right)^6\right]+\left[7 C _7\left(-\frac{3 x }{7}\right)^7\right] \\ \Rightarrow\left[\frac{7!}{0!(7-0)!}\left(x^2\right)^7\right]-\left[\frac{7!}{1!(7-1)!}\left(x^2\right)^6\left(\frac{3 x}{7}\right)\right]+\left[\frac{7!}{2!(7-2)!}\left(x^2\right)^5\left(\frac{9 x^2}{49}\right)\right]-\left[\frac{7!}{3!(7-3)!}\left(x^2\right)^4\left(\frac{27 x^3}{343}\right)\right] \\ +\left[\frac{7!}{4!(7-4)!}\left(x^2\right)^3\left(\frac{81 x^4}{2401}\right)\right]-\left[\frac{7!}{5!(7-5)!}\left(x^2\right)^2\left(\frac{243 x^5}{16807}\right)\right]+\left[\frac{7!}{6!(7-6)!}\left(x^2\right)^1\left(\frac{729 x^6}{117649}\right)\right]-\left[\frac{7!}{7!(7-7)!}\left(\frac{2187 x^7}{823543}\right)\right] \\ -\left[\frac{7!}{7!(7-7)!}\left(\frac{2187 x^7}{823543}\right)\right]+\left[21\left(x^{10}\right)\left(\frac{9 x^2}{49}\right)\right]-\left[35\left(x^8\right)\left(\frac{27 x^3}{343}\right)\right] \\ +\left[35\left(x^6\right)\left(\frac{81 x^4}{2401}\right)\right]-\left[21\left(x^4\right)\left(\frac{243 x^5}{16807}\right)\right]+\left[7\left(x^2\right)\left(\frac{729 x^6}{117649}\right)\right]-\left[1\left(\frac{2187 x^7}{823543}\right)\right] \\ \Rightarrow x^{24}-3 x^{13}+\left(\frac{27}{7}\right) x^{12}-\left(\frac{135}{49}\right) x^{11}+\left(\frac{405}{343}\right) x^{10}-\left(\frac{729}{2401}\right) x^9+\left(\frac{729}{16807}\right) x^8-\left(\frac{2187}{823543}\right) x^7 \\ x^{14}-3 x^{13}+\left(\frac{27}{7}\right) x^{12}-\left(\frac{135}{49}\right) x^{11}+\left(\frac{405}{343}\right) x^{10}-\left(\frac{729}{2901}\right) x^9+\left(\frac{729}{12807}\right) x^8-\left(\frac{2187}{825353}\right) x^7\end{array}$
Formula used: ${ }^n C_r=\frac{n!}{(n-r)!(r)!}$
We know that, $(a+b)^n={ }^n C_0 a^n+{ }^n C_1 a^{n-1} b+{ }^n C_2 a^{n-2} b^2+\ldots \ldots .+{ }^n C_{n-1} a^{n-1}+{ }^n C_n b^n$
Here We have, $\left(x^2-\frac{3 x}{7}\right)^7$
$\begin{array}{l}\Rightarrow\left[{ }^7 C _0\left( x ^2\right)^{7-0}\right]+\left[7 C _1\left( x ^2\right)^{7-1}\left(-\frac{3 x }{7}\right)^1\right]+\left[7 c_2\left(x^2\right)^{7-2}\left(-\frac{3 x}{7}\right)^2\right]+\left[7 C _3\left( x ^2\right)^{7-3}\left(-\frac{3 x }{7}\right)^3\right]+\left[7 C _4\left( x ^2\right)^{7-4}\left(-\frac{3 x }{7}\right)^4\right] \\ +\left[7 C _5\left( x ^2\right)^{7-5}\left(-\frac{3 x }{7}\right)^5\right]+\left[7 C _6\left( x ^2\right)^{7-6}\left(-\frac{3 x }{7}\right)^6\right]+\left[7 C _7\left(-\frac{3 x }{7}\right)^7\right] \\ \Rightarrow\left[\frac{7!}{0!(7-0)!}\left(x^2\right)^7\right]-\left[\frac{7!}{1!(7-1)!}\left(x^2\right)^6\left(\frac{3 x}{7}\right)\right]+\left[\frac{7!}{2!(7-2)!}\left(x^2\right)^5\left(\frac{9 x^2}{49}\right)\right]-\left[\frac{7!}{3!(7-3)!}\left(x^2\right)^4\left(\frac{27 x^3}{343}\right)\right] \\ +\left[\frac{7!}{4!(7-4)!}\left(x^2\right)^3\left(\frac{81 x^4}{2401}\right)\right]-\left[\frac{7!}{5!(7-5)!}\left(x^2\right)^2\left(\frac{243 x^5}{16807}\right)\right]+\left[\frac{7!}{6!(7-6)!}\left(x^2\right)^1\left(\frac{729 x^6}{117649}\right)\right]-\left[\frac{7!}{7!(7-7)!}\left(\frac{2187 x^7}{823543}\right)\right] \\ -\left[\frac{7!}{7!(7-7)!}\left(\frac{2187 x^7}{823543}\right)\right]+\left[21\left(x^{10}\right)\left(\frac{9 x^2}{49}\right)\right]-\left[35\left(x^8\right)\left(\frac{27 x^3}{343}\right)\right] \\ +\left[35\left(x^6\right)\left(\frac{81 x^4}{2401}\right)\right]-\left[21\left(x^4\right)\left(\frac{243 x^5}{16807}\right)\right]+\left[7\left(x^2\right)\left(\frac{729 x^6}{117649}\right)\right]-\left[1\left(\frac{2187 x^7}{823543}\right)\right] \\ \Rightarrow x^{24}-3 x^{13}+\left(\frac{27}{7}\right) x^{12}-\left(\frac{135}{49}\right) x^{11}+\left(\frac{405}{343}\right) x^{10}-\left(\frac{729}{2401}\right) x^9+\left(\frac{729}{16807}\right) x^8-\left(\frac{2187}{823543}\right) x^7 \\ x^{14}-3 x^{13}+\left(\frac{27}{7}\right) x^{12}-\left(\frac{135}{49}\right) x^{11}+\left(\frac{405}{343}\right) x^{10}-\left(\frac{729}{2901}\right) x^9+\left(\frac{729}{12807}\right) x^8-\left(\frac{2187}{825353}\right) x^7\end{array}$
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