2. Which of the following is the correct expansion of \( (2 x-3 y)^{4} ? \) a. \( 81 x^{4}+216 x^{3} y+216 x^{2} y^{2}+96 x y^{3}+16 y^{4} \) b. \( 81 x^{4}-216 x^{3} y+216 x^{2} y^{2}+96 x y^{3}+16 y^{4} \) c. \( 16 x^{4}-96 x^{3} y-216 x^{2} y^{2}-216 x y^{3}-81 y^{4} \) d. \( 16 x^{4}-96 x^{3} y+216 x^{2} y^{2}-216 x y^{3}+81 y^{4} \)

Let's dive into the fascinating world of polynomial expansions! When expanding \( (2x - 3y)^{4} \), we apply the Binomial Theorem, which tells us how to expand expressions of the form \( (a + b)^{n} \). Here, \( a = 2x \), \( b = -3y \), and \( n = 4 \). This nifty theorem generates terms that consider combinations of \( a \) and \( b \) multiplied by binomial coefficients, allowing for a clear way to calculate the expansion without missing a beat. Now, for real-world application, knowing how to expand polynomials like this is super useful in various fields! For example, engineers often deal with polynomial equations in structural analysis. Similarly, in economics, understanding how different factors contribute to a polynomial function can inform models predicting market trends. So, whether you're designing a bridge or predicting stock prices, mastering these expansions plays a crucial role!