Factor each polynomial. See Examples 5 and 6. 27-(m+2n)³

Verified step by step guidance

1. Recognize that the expression \$27 - (m + 2n)^3\( is a difference of cubes, since \)27 = 3^3\( and \)(m + 2n)^3$ is already a cube.
2. Recall the difference of cubes factoring formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
3. Identify \(a = 3\) and \(b = (m + 2n)\) in the expression \$27 - (m + 2n)^3$.
4. Apply the formula by substituting \(a\) and \(b\): write the factorization as \((3 - (m + 2n)) \left(3^2 + 3(m + 2n) + (m + 2n)^2\right)\).
5. Simplify inside the parentheses: calculate \$3^2 = 9\(, expand \)3(m + 2n)\( to \)3m + 6n\(, and expand \)(m + 2n)^2$ using the binomial square formula.

Key Concepts

Difference of Cubes Formula

The difference of cubes formula states that a³ - b³ = (a - b)(a² + ab + b²). It is used to factor expressions where one term is a perfect cube subtracted by another perfect cube. Recognizing this pattern allows for straightforward factoring of cubic expressions.

Identifying Perfect Cubes

To apply the difference of cubes formula, each term must be a perfect cube. This involves recognizing numbers or expressions raised to the third power, such as 27 (which is 3³) and (m + 2n)³. Correct identification is essential for proper factoring.

Polynomial Factoring Techniques

Factoring polynomials involves rewriting them as products of simpler polynomials. Techniques include factoring out common terms, grouping, and special formulas like difference of cubes. Mastery of these methods simplifies solving and analyzing polynomial expressions.