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

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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.

Recall the difference of cubes factoring formula: $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$.

Identify $x = 5$ and $y = (4a - b)$ in the expression \$125 - (4a - b)^3$.

Apply the formula to factor the expression as: $(5 - (4a - b)) \left(5^2 + 5(4a - b) + (4a - b)^2\right)$.

Simplify the factors inside the parentheses by performing the arithmetic and expanding $(4a - b)^2$ to complete the factorization.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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