b) \( (3 x+2 y)^{3} \)

Did you know the expansion of \( (a + b)^{n} \) can be easily achieved using the binomial theorem? For \( (3x + 2y)^{3} \), we take the coefficients from the binomial expansion, giving us terms like \( \binom{3}{k} (3x)^{3-k} (2y)^{k} \) for \( k = 0, 1, 2, 3 \). It’s an algebraic formula that allows us to compute powers of sums swiftly and beautifully! Now, let’s explore a practical application: in fields like physics and engineering, expanding polynomials helps in modeling complex relations, such as load distributions or signal processing. By expanding expressions like \( (3x + 2y)^{3} \), you can derive formulas that simplify calculations or lead to simulations that forecast systems behavior. It’s not just math; it’s a doorway to understanding the world around us!