Factor each polynomial. See Examples 5 and 6. (b+3)³-27

Verified step by step guidance

1. Recognize that the expression \( (b+3)^3 - 27 \) is a difference of cubes, since \(27\) can be written as \$3^3$.
2. Recall the difference of cubes formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
3. Identify \(a = (b+3)\) and \(b = 3\) in the expression \( (b+3)^3 - 3^3 \).
4. Apply the formula: write the factorization as \(((b+3) - 3)((b+3)^2 + (b+3)(3) + 3^2)\).
5. Simplify each factor: simplify \(((b+3) - 3)\) to \(b\), expand and simplify the quadratic expression inside the second factor.

Key Concepts

Difference of Cubes

The difference of cubes formula states that a³ - b³ = (a - b)(a² + ab + b²). It is used to factor expressions where two perfect cubes are subtracted. Recognizing the structure 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 expressions like (b+3)³ and 27, since 27 = 3³. Understanding how to rewrite terms as cubes is essential for correct factoring.

Polynomial Factoring Techniques

Factoring polynomials involves breaking down expressions into simpler factors. Techniques include factoring out common terms, grouping, and special formulas like difference of cubes. Mastery of these methods helps simplify and solve polynomial equations.