1. \( 3 p^{3}-24 \)

To factor the expression \( 3p^{3} - 24 \), you can first pull out the common factor of 3: \( 3(p^{3} - 8) \). Now, the remaining expression \( p^{3} - 8 \) is a difference of cubes, which can be factored further using the formula \( a^{3} - b^{3} = (a-b)(a^{2} + ab + b^{2}) \). Here, \( p^{3} - 2^{3} \) can be modeled with \( a = p \) and \( b = 2 \). So, we have: \( 3(p - 2)(p^{2} + 2p + 4) \). The final factored expression is \( 3(p - 2)(p^{2} + 2p + 4) \). To see the real-world application of factoring expressions like this, consider physics problems where you need to solve for unknown variables. Factoring can simplify complex equations into manageable forms, allowing you to quickly find critical values—like how long it takes for an object to hit the ground or how to adjust the parameters of a project in engineering design. A common mistake when factoring polynomials is neglecting the importance of a common factor. Always look for the greatest common factor first before diving into more complex factoring methods like the difference of cubes. This step can simplify your work and prevent errors in simplifying your expressions later on!