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

검증된 단계별 안내

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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주요 개념

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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.

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