CONCEPT PREVIEW Work each problem. Match each polynomial in Column I with its factored form in Column II. I II a. 8x³ - 27 A. (3 - 2x) (9 + 6x + 4x²) b. 8x³ + 27 B. (2x - 3) (4x² + 6x + 9) c. 27 - 8x³ C. (2x + 3) (4x² - 6x + 9)

Verified step by step guidance

1. Recognize that each polynomial in Column I is a difference or sum of cubes. Recall the formulas for factoring cubes: for difference of cubes, \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), and for sum of cubes, \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
2. Identify \(a\) and \(b\) for each polynomial by expressing the terms as cubes. For example, \$8x^3\( can be written as \)(2x)^3\( and \)27\( as \)3^3$.
3. Apply the difference of cubes formula to \$8x^3 - 27\( and \)27 - 8x^3$, noting the order of subtraction affects the signs in the factors.
4. Apply the sum of cubes formula to \$8x^3 + 27\(, carefully substituting \)a = 2x\( and \)b = 3$ into the formula.
5. Match each factored form from Column II to the corresponding polynomial in Column I by comparing the factors obtained from the formulas with the given options.

Key Concepts

Sum and Difference of Cubes

The sum and difference of cubes are special polynomial forms that factor into binomial and trinomial products. For example, a³ + b³ factors as (a + b)(a² - ab + b²), and a³ - b³ factors as (a - b)(a² + ab + b²). Recognizing these forms helps in quickly factoring cubic expressions.

Identifying 'a' and 'b' in Cubic Expressions

To factor cubic polynomials using sum or difference of cubes, it is essential to identify the cube roots of each term. For instance, in 8x³, the cube root is 2x, and in 27, it is 3. Correctly determining these values allows proper application of the factoring formulas.

Matching Factored Forms to Original Polynomials

After factoring, it is important to verify the factored form by expanding it back to the original polynomial. This ensures the correct pairing between the polynomial and its factorization, especially when multiple similar options are given, as in matching problems.