Factor each polynomial. See Examples 5 and 6. 125-(4a-b)³

Verified step by step guidance

1. Recognize that the expression \$125 - (4a - b)^3\( is a difference of cubes, since \)125 = 5^3\( and \)(4a - b)^3$ is already a cube.
2. Recall the difference of cubes factoring formula: \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\).
3. Identify \(x = 5\) and \(y = (4a - b)\) in the expression \$125 - (4a - b)^3$.
4. Apply the formula to factor the expression as: \((5 - (4a - b)) \left(5^2 + 5(4a - b) + (4a - b)^2\right)\).
5. Simplify the factors inside the parentheses by performing the arithmetic and expanding \((4a - b)^2\) to complete the factorization.

Key Concepts

Difference of Cubes Formula

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 the structure allows you to rewrite the polynomial as a product of a binomial and a trinomial.

Identifying Perfect Cubes

To apply the difference of cubes formula, you must identify terms that are perfect cubes. For example, 125 is 5³, and (4a - b)³ is the cube of the binomial (4a - b). Recognizing these helps in rewriting the expression in the form a³ - b³.

Factoring Binomial Cubes

When factoring expressions like (4a - b)³, treat the entire binomial as a single term raised to the third power. This understanding is crucial for correctly applying the difference of cubes formula and expanding or factoring complex polynomial expressions.