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Factoring techniques help break down complex polynomial expressions into simpler parts. Recognizing the type of polynomial you have is crucial for choosing the appropriate technique.In the case of the sum of cubes, you first identify the formula to use, which is: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]Then, express each term as a cube. For example, \(24n^3 + 81p^3\) can be rewritten as \((2n)^3 + (3p)^3\).After expressing each term, substitute \(a\) and \(b\) in the formula. For our example, \(a = 2n\) and \(b = 3p\) gives: \[ (2n)^3 + (3p)^3 = (2n + 3p)((2n)^2 - (2n)(3p) + (3p)^2) \]Lastly, simplify the terms inside the parentheses to get the final factored form: \[(2n + 3p)(4n^2 - 6np + 9p^2)\]

Understanding polynomial expressions is fundamental for factoring. A polynomial is an expression consisting of variables and coefficients, structured in terms of powers. In the given exercise, we have a polynomial with cubic terms: \(24n^3 + 81p^3\).Polynomials can often be factored to simplify them or solve equations. Recognizing special forms like the sum of cubes helps in using specific formulas. Here's a recap of steps:

• Identify the type of polynomial (sum of cubes, in this case).
• Write each term as a cube, \((2n)^3 + (3p)^3\).
• Apply the sum of cubes formula: \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
• Simplify using the values of \(a\) and \(b\).

This structured approach to learning polynomial expressions helps in recognizing patterns and applying the right techniques for factoring and solving problems.