# Using binomial theorem, expand each of the following:

Question:

Using binomial theorem, expand each of the following:

$\left(x^{2}-\frac{3}{x}\right)^{7}$

Solution:

To find: Expansion of $\left(x^{2}-\frac{3 x}{7}\right)^{7}$

Formula used: (i) ${ }^{n} C_{r}=\frac{n !}{(n-r) !(r) !}$

(ii) $(a+b)^{n}={ }^{n} C_{0} a^{n}+{ }^{n} C_{1} a^{n-1} b+{ }^{n} C_{2} a^{n-2} b^{2}+\ldots \ldots+{ }^{n} C_{n-1} a b^{n-1}+{ }^{n} C_{n} b^{n}$

We have, $\left(x^{2}-\frac{3 x}{7}\right)^{7}$

$\Rightarrow\left[{ }^{7} C_{0}\left(x^{2}\right)^{7-0}\right]+\left[{ }^{7} C_{1}\left(x^{2}\right)^{7-1}\left(-\frac{3 x}{7}\right)^{1}\right]+\left[{ }^{7} C_{2}\left(x^{2}\right)^{7-2}\left(-\frac{3 x}{7}\right)^{2}\right]+$

$\left[{ }^{7} C_{3}\left(x^{2}\right)^{7-3}\left(-\frac{3 x}{7}\right)^{3}\right]+\left[{ }^{7} C_{4}\left(x^{2}\right)^{7-4}\left(-\frac{3 x}{7}\right)^{4}\right]+\left[{ }^{7} C_{5}\left(x^{2}\right)^{7-5}\left(-\frac{3 x}{7}\right)^{5}\right]+$

$\left[{ }^{7} C_{6}\left(x^{2}\right)^{7-6}\left(-\frac{3 x}{7}\right)^{6}\right]+\left[{ }^{7} C_{7}\left(-\frac{3 x}{7}\right)^{7}\right]$

$\Rightarrow\left[\frac{7 !}{0 !(7-0) !}\left(x^{2}\right)^{7}\right]-\left[\frac{7 !}{1 !(7-1) !}\left(x^{2}\right)^{6}\left(\frac{3 x}{7}\right)\right]+\left[\frac{7 !}{2 !(7-2) !}\left(x^{2}\right)^{5}\left(\frac{9 x^{2}}{49}\right)\right]-$

$\left[\frac{7 !}{3 !(7-3) !}\left(x^{2}\right)^{4}\left(\frac{27 x^{3}}{343}\right)\right]+\left[\frac{7 !}{4 !(7-4) !}\left(x^{2}\right)^{3}\left(\frac{81 x^{4}}{2401}\right)\right]-\left[\frac{7 !}{5 !(7-5) !}\right.$

$\left.\left(x^{2}\right)^{2}\left(\frac{243 x^{5}}{16807}\right)\right]+\left[\frac{7 !}{6 !(7-6) !}\left(x^{2}\right)^{1}\left(\frac{729 x^{6}}{117649}\right)\right]-\left[\frac{7 !}{7 !(7-7) !}\left(\frac{2187 x^{7}}{823543}\right)\right]$

$\Rightarrow\left[1\left(x^{14}\right)\right]-\left[7\left(x^{12}\right)\left(\frac{3 x}{7}\right)\right]+\left[21\left(x^{10}\right)\left(\frac{9 x^{2}}{49}\right)\right]-\left[35\left(x^{8}\right)\left(\frac{27 x^{3}}{343}\right)\right]+$

$\left[35\left(x^{6}\right)\left(\frac{81 x^{4}}{2401}\right)\right]-\left[21\left(x^{4}\right)\left(\frac{243 x^{5}}{16807}\right)\right]+\left[7\left(x^{2}\right)\left(\frac{729 x^{6}}{117649}\right)\right]-$

$\left[1\left(\frac{2187 x^{7}}{823543}\right)\right]$

$\Rightarrow x^{14}-3 x^{13}+\left(\frac{27}{7}\right) x^{12}-\left(\frac{135}{49}\right) x^{11}+\left(\frac{405}{343}\right) x^{10}-$

$\left(\frac{729}{2401}\right) x^{9}+\left(\frac{729}{16807}\right) x^{8}-\left(\frac{2187}{823543}\right) x^{7}$

Ans) $x^{14}-3 x^{13}+\left(\frac{27}{7}\right) x^{12}-\left(\frac{135}{49}\right) x^{11}+\left(\frac{405}{343}\right) x^{10}-\left(\frac{729}{2401}\right) x^{9}+\left(\frac{729}{16807}\right) x^{8}-$

$\left(\frac{2187}{823543}\right) x^{7}$