Factor each polynomial. See Example 7. (3x+4)³-1

Verified step by step guidance

1. Recognize that the expression \( (3x+4)^3 - 1 \) is a difference of cubes, since \$1\( can be written as \)1^3$.
2. Recall the difference of cubes formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
3. Identify \(a = 3x + 4\) and \(b = 1\) in the expression.
4. Apply the formula: write the factorization as \(((3x + 4) - 1)((3x + 4)^2 + (3x + 4)(1) + 1^2)\).
5. Simplify each factor: first factor becomes \((3x + 3)\), and expand the second factor by squaring and multiplying terms inside the parentheses.

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 cube is subtracted from another. Recognizing this pattern helps simplify polynomials like (3x + 4)³ - 1 by identifying a = (3x + 4) and b = 1.

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of simpler polynomials. This process simplifies expressions and solves equations. Understanding how to factor special forms, such as cubes or squares, is essential for breaking down complex polynomials efficiently.

Binomial Expansion and Recognition

Binomial expansion involves expressing powers of binomials, like (a + b)³, in expanded form. Recognizing the structure of binomials and their powers helps in identifying patterns for factoring. This skill aids in reversing expansions to factor expressions like (3x + 4)³ - 1.