To find the coefficient of \(x\) in the expansion of \(\left( 3x - \frac{2}{x} \right)^5\), we use the binomial theorem:
\(\left( a + b \right)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
Here, \(a = 3x\) and \(b = -\frac{2}{x}\), and \(n = 5\).
We need the term where the power of \(x\) is 1. In the general term:
\(T_k = \binom{5}{k} (3x)^{5-k} \left(-\frac{2}{x}\right)^k\)
The power of \(x\) in \(T_k\) is given by:
\((5-k) - k = 5 - 2k\)
We set \(5 - 2k = 1\) to find \(k\):
\(5 - 2k = 1\)
\(2k = 4\)
\(k = 2\)
Substitute \(k = 2\) into the general term:
\(T_2 = \binom{5}{2} (3x)^{3} \left(-\frac{2}{x}\right)^2\)
\(= \binom{5}{2} \cdot 3^3 \cdot x^3 \cdot \frac{4}{x^2}\)
\(= 10 \cdot 27 \cdot 4 \cdot x\)
\(= 1080x\)
Thus, the coefficient of \(x\) is 1080.
