Factor each polynomial. See Examples 5 and 6. 8-a³

Verified step by step guidance

1. Recognize that the expression \$8 - a^3\( is a difference of cubes because \)8\( can be written as \)2^3\( and \)a^3$ is already a cube.
2. Recall the formula for factoring a difference of cubes: \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\).
3. Identify \(x = 2\) and \(y = a\) in the expression \$8 - a^3$.
4. Apply the difference of cubes formula: write \((2 - a)(2^2 + 2 \cdot a + a^2)\).
5. Simplify the terms inside the second parenthesis to get \((2 - a)(4 + 2a + a^2)\).

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 one term is a cube subtracted from another. Recognizing this pattern helps in breaking down polynomials like 8 - a³, since 8 is 2³.

Identifying Perfect Cubes

A perfect cube is a number or expression raised to the third power, such as 8 (2³) or a³. Identifying perfect cubes in a polynomial is essential before applying the difference of cubes formula. This step ensures the correct factorization approach.

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 squares or cubes. Mastery of these methods is crucial for simplifying expressions and solving equations.