In a manufacturing analysis, the difference in the volume of two containers is represented by the expression $(n + 3)^3 - 125n^3$. To factor this expression using the pattern $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, which of the following correctly identifies the values that should be used for $a$ and $b$?

An operations manager is analyzing a volume difference expression, $(n + 3)^3 - 125n^3$, to optimize storage space. To factor this expression using the difference of cubes identity $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, match each formula component with its correct algebraic representation.

A data analyst is simplifying a capacity model represented by the expression $(n + 3)^3 - {}125n^3$. True or False: When applying the difference of cubes identity $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, the middle term of the trinomial factor, $ab$, is correctly represented by the product $(n + 3)({}5n)$.

Logistics Capacity Modeling

A supply chain analyst needs to simplify a capacity difference model, represented by the expression $(n + 3)^3 - 125n^3$. Recall the algebraic procedure for factoring the difference of cubes. Arrange the steps required to factor this expression completely into the correct sequential order.

A packaging engineer is factoring the volume difference expression $(n + 3)^3 - 125n^3$ using the difference of cubes pattern $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. After recalling the substitution values $a = n + 3$ and $b = 5n$, the first factor $(a - b)$ simplifies to the binomial ____.

Analyzing Storage Capacity with Difference of Cubes