\( \begin{array}{ll}3.1 .1 & (r+s-1)(r+s) \\ \text { 3.1.2 } & \left(p^{3}-1\right)^{2}-3 p^{3}\left(p^{3}+2\right)-6 p^{6} \\ 3.1 .3 & \left(a^{3}\right)^{2} \times a^{4}\end{array} \)

Did you know that factoring expressions like \( (r+s-1)(r+s) \) can often simplify complex algebraic problems? This is particularly useful when evaluating polynomial equations or integrating functions in calculus. Equations that look intimidating at first can often reveal their secrets through factoring, leading to simpler forms that are easier to work with. Another fun fact is the relevance of polynomial identities in real-world applications. For example, the expression \( (p^3 - 1)^2 - 3p^3(p^3 + 2) - 6p^6 \) can reflect concepts in physics when calculating trajectory paths or modeling population growth in biology. Understanding polynomials helps bridge mathematical concepts with practical situations, making the world seem just a little bit more connected!