\( * \) Term involuing \( x^{2} \) in the expanson of \[ \left(2 x-\frac{1}{2 x}\right)^{12} \]

To find the term involving \( x^{2} \) in the expansion of \( \left(2x - \frac{1}{2x}\right)^{12} \), we can use the binomial theorem. The general term in the expansion of \( (a + b)^n \) is given by: \[ T_{r+1} = \binom{n}{r} \cdot a^{n-r} \cdot b^{r} \] In this case, \( a = 2x \), \( b = -\frac{1}{2x} \), and \( n = 12 \). We want to find the term involving \( x^{2} \), which means we need to find the term where the power of \( x \) is 2. Let's find the term involving \( x^{2} \) in the expansion of \( \left(2x - \frac{1}{2x}\right)^{12} \) using the binomial theorem. Simplify the expression by following steps: - step0: Evaluate the power: \(\left(2x-\frac{1}{\left(2x\right)}\right)^{12}\) - step1: Multiply the terms: \(\left(2x-\frac{1}{2x}\right)^{12}\) - step2: Subtract the terms: \(\left(\frac{4x^{2}-1}{2x}\right)^{12}\) - step3: Evaluate the power: \(\frac{\left(4x^{2}-1\right)^{12}}{4096x^{12}}\) The expanded form of \( \left(2x - \frac{1}{2x}\right)^{12} \) is \( \frac{(4x^{2}-1)^{12}}{4096x^{12}} \). To find the term involving \( x^{2} \), we need to look for the term where the power of \( x \) is 2. In this case, the term involving \( x^{2} \) is \( \frac{1}{4096} \cdot (4x^{2}-1)^{12} \). Therefore, the term involving \( x^{2} \) in the expansion of \( \left(2x - \frac{1}{2x}\right)^{12} \) is \( \frac{1}{4096} \cdot (4x^{2}-1)^{12} \).