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

Verified step by step guidance

1. Recognize that the expression is in the form of a difference of cubes: \(a^3 - b^3\), where \(a^3 = 27\) and \(b^3 = (m+2n)^3\).
2. Identify \(a\) and \(b\) by taking the cube root of each term: \(a = 3\) and \(b = m+2n\).
3. Use the difference of cubes formula: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).
4. Substitute \(a = 3\) and \(b = m+2n\) into the formula: \((3 - (m+2n))(3^2 + 3(m+2n) + (m+2n)^2)\).
5. Simplify the expression inside the parentheses: \((3 - (m+2n))(9 + 3(m+2n) + (m+2n)^2)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions, solving equations, and understanding polynomial behavior. Common techniques include identifying common factors, using special products, and applying the distributive property.

Difference of Cubes

The expression 27 - (m + 2n)^3 represents a difference of cubes, which can be factored using the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2). In this case, 27 is 3^3 and (m + 2n)^3 is the cube of (m + 2n). Recognizing this pattern allows for efficient factoring.

Binomial Expansion

The binomial expansion refers to the expansion of expressions raised to a power, such as (a + b)^n. In the context of the given polynomial, understanding how to expand (m + 2n)^3 is crucial for applying the difference of cubes formula correctly. The expansion results in a polynomial that can be further manipulated during the factoring process.