Work each problem. Which of the following is the correct factorization of x3+8? A. (x+2)3 B. (x+2)(x2+2x+4) C. (x+2)(x2-2x+4) D. (x+2)(x2-4x+4)

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 10

1장, 문제 10

Work each problem. Which of the following is the correct factorization of x³+8?
A. (x+2)³
B. (x+2)(x²+2x+4)
C. (x+2)(x²-2x+4)
D. (x+2)(x²-4x+4)

검증된 단계별 안내

Recognize that the expression $x^3 + 8$ is a sum of cubes, since $8$ can be written as \$2^3$. So, we have $x^3 + 2^3$.

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

Identify $a = x$ and $b = 2$ in the formula, then substitute these values into the factorization formula.

Write the factorization as $(x + 2)(x^2 - (x)(2) + 2^2)$, which simplifies to $(x + 2)(x^2 - 2x + 4)$.

Compare this factorization to the given options to determine which one matches the expression $(x + 2)(x^2 - 2x + 4)$.

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주요 개념

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

Sum of Cubes Formula

The sum of cubes formula states that a³ + b³ = (a + b)(a² - ab + b²). This identity helps factor expressions where two cubes are added together, such as x³ + 8, by identifying a = x and b = 2.

Polynomial Factorization

Polynomial factorization involves rewriting a polynomial as a product of simpler polynomials. Recognizing special forms like sum or difference of cubes allows efficient factorization, which simplifies solving or analyzing polynomial expressions.

Identifying Correct Factorization

After applying the sum of cubes formula, it is important to match the resulting factors with the given options. This requires careful substitution and verification of each term to ensure the factorization is accurate and corresponds to the original polynomial.