Suppose that \( \sqrt{5} \) is a zero of \( p(x)=x^{3}-\sqrt{5} x^{2}-25 x+25 \sqrt{5} \) and \( q(x)=x^{3}-5 x^{2}-5 x+25 \)

Did you know that the roots of polynomials can often lead to fascinating patterns and relationships? For example, in the case of the polynomial \( p(x) \), the presence of \( \sqrt{5} \) as a root indicates that it might have some hidden symmetry or connection to quadratic functions or can lead to simplifications in related expressions. This is a delightful layer to explore when searching for other roots or factoring the polynomial! Let's put our thinking caps on about real-world applications! Polynomials like \( p(x) \) and \( q(x) \) appear not just in algebra textbooks, but also in fields like physics and engineering. They can represent trajectories, optimize designs, or model natural phenomena by describing curves. For example, if you were to model the height of a projectile over time, the equations can often be cubic, just like these, making them pivotal in predicting outcomes in the real world!