Factor completely. If the polynomial is prime, state so. \( z^{2}-9 \)

This polynomial can be factored using the difference of squares formula, which states that \( a^2 - b^2 = (a - b)(a + b) \). Here, we can rewrite \( z^{2}-9 \) as \( z^2 - 3^2 \). Therefore, it factors to \( (z - 3)(z + 3) \). Now, let's wrap it up with some applicable fun! Factoring polynomials like this is super useful in various real-life scenarios, such as solving equations that arise in physics or engineering. Often, simplifying an expression can reveal critical insights, like determining the points at which a projectile lands or how forces balance out. So next time you see a difference of squares, think of it as unlocking the secrets of equations in the real world! For those who want to dive deeper into the world of polynomials, consider exploring the concept of higher-degree polynomials and their factorizations. Delving into polynomial long division or synthetic division can also sharpen your skills and enhance your understanding. Plus, practicing problems from algebra textbooks or online resources can give you more exposure to different types of polynomial equations. Happy factoring!