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

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 90

1장, 문제 90

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

검증된 단계별 안내

Recognize that the expression $ (5x - 2)^3 - 8 $ is a difference of cubes, since $8$ can be written as \$2^3$.

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

Identify $a = (5x - 2)$ and $b = 2$ in the expression.

Apply the formula: write the factorization as $((5x - 2) - 2)((5x - 2)^2 + (5x - 2)(2) + 2^2)$.

Simplify each factor: first simplify $((5x - 2) - 2)$, then 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 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.