Factor each polynomial. See Example 7. (5x-2)³-8

Verified step by step guidance

1. Recognize that the expression \( (5x - 2)^3 - 8 \) is a difference of cubes, since \$8\( can be written as \)2^3$.
2. Recall the difference of cubes formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
3. Identify \(a = (5x - 2)\) and \(b = 2\) in the expression.
4. Apply the formula: write the factorization as \(((5x - 2) - 2)((5x - 2)^2 + (5x - 2)(2) + 2^2)\).
5. Simplify each factor: first simplify \(((5x - 2) - 2)\), then 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 helps simplify polynomials like (5x - 2)³ - 8 by identifying a = (5x - 2) and b = 2.

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of simpler polynomials. This process simplifies expressions and solves equations. Factoring techniques include recognizing special products like difference of squares, sum/difference of cubes, and factoring by grouping.

Exponents and Powers

Understanding exponents is crucial for identifying perfect cubes and manipulating expressions like (5x - 2)³. Exponents indicate repeated multiplication, and recognizing powers helps in applying formulas such as the difference of cubes for factoring.