Factor each polynomial. See Examples 5 and 6. (b+3)3-27

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 80

1장, 문제 80

Factor each polynomial. See Examples 5 and 6. (b+3)³-27

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Recognize that the expression $ (b+3)^3 - 27 $ is a difference of cubes, since $27$ can be written as \$3^3$.

Recall the difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Identify $a = (b+3)$ and $b = 3$ in the expression $ (b+3)^3 - 3^3 $.

Apply the formula: write the factorization as $((b+3) - 3)((b+3)^2 + (b+3)(3) + 3^2)$.

Simplify each factor: simplify $((b+3) - 3)$ to $b$, expand and simplify the quadratic expression inside the second factor.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 allows for straightforward factoring of cubic expressions.

Identifying Perfect Cubes

To apply the difference of cubes formula, each term must be a perfect cube. This involves recognizing expressions like (b+3)³ and 27, since 27 = 3³. Understanding how to rewrite terms as cubes is essential for correct factoring.

Polynomial Factoring Techniques

Factoring polynomials involves breaking down expressions into simpler factors. Techniques include factoring out common terms, grouping, and special formulas like difference of cubes. Mastery of these methods helps simplify and solve polynomial equations.