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. Step 1: Recognize that the expressions in Column I are in the form of a difference or sum of cubes. The general formulas for factoring these are: $a^3-b^3=(a-b)(a^2+ab+b^2)$ and $a^3+b^3=(a+b)(a^2-ab+b^2)$.
2. Step 2: Identify the values of $a$ and $b$ for each expression in Column I. For example, in $8x^3-27$, $a=2x$ and $b=3$.
3. Step 3: Apply the appropriate formula to factor each expression in Column I. For $8x^3-27$, use the difference of cubes formula: $(2x-3)(4x^2+6x+9)$.
4. Step 4: Match the factored form from Step 3 with the options in Column II. For $8x^3-27$, the factored form $(2x-3)(4x^2+6x+9)$ corresponds to option B.
5. Step 5: Repeat Steps 2-4 for the remaining expressions in Column I to find their corresponding factored forms in Column II.

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 and solving equations. Common techniques include identifying common factors, using the difference of squares, and applying special formulas like the sum or difference of cubes.

Sum and Difference of Cubes

The sum and difference of cubes are specific algebraic identities that allow for the factoring of expressions in the form of a³ ± b³. The formulas are a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²). Recognizing these forms is crucial for efficiently factoring polynomials like 8x³ - 27 and 8x³ + 27.

Polynomial Degree and Leading Coefficient

The degree of a polynomial is the highest power of the variable in the expression, which determines its general shape and behavior. The leading coefficient is the coefficient of the term with the highest degree. Understanding these concepts helps in predicting the number of roots and the end behavior of the polynomial, which is useful when matching polynomials with their factored forms.