Work each problem. Which of the following is the correct complete factorization of x4−$$1x^4-1$$? A. (x2−$$1)(x2+1)(x^2-1)(x^2$+1) B. (x2+1)(x+1)(x−1)(x^2$+1)(x+1)(x-1) C. (x2−1)2(x^2$-1)^2 D. (x$$−$$1)2(x+1)2(x-1)^2$(x+1)^2$$

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 9

1장, 문제 9

Work each problem. Which of the following is the correct complete factorization of $x^{4} - 1$ ?
A. $(x^{2} - 1) (x^{2} + 1)$
B. $(x^{2} + 1) (x + 1) (x - 1)$
C. $open paren x squared minus 1 close paren squared$
D. $(x - 1)^{2} (x + 1)^{2}$

검증된 단계별 안내

Recognize that the expression $x^4 - 1$ is a difference of squares because it can be written as $(x^2)^2 - 1^2$.

Apply the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$, so $x^4 - 1 = (x^2 - 1)(x^2 + 1)$.

Notice that $x^2 - 1$ is itself a difference of squares and can be factored further as $(x - 1)(x + 1)$.

Combine the factors to write the complete factorization as $(x - 1)(x + 1)(x^2 + 1)$.

Compare this factorization with the given options to identify the correct one.

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주요 개념

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

Difference of Squares

The difference of squares is a factoring technique where an expression of the form a^2 - b^2 can be factored into (a - b)(a + b). This is essential for factoring x^4 - 1 because it can be seen as (x^2)^2 - 1^2.

Factoring Quadratic Expressions

After applying the difference of squares once, the resulting quadratic expressions may be factored further if possible. For example, x^2 - 1 can be factored again using the difference of squares into (x - 1)(x + 1).

Complete Factorization

Complete factorization means breaking down a polynomial into irreducible factors over the real numbers. For x^4 - 1, this involves factoring repeatedly until no further factorization is possible, ensuring the expression is fully simplified.

Work each problem. Which of the following is the correct complete factorization of x4−1x^4-1? A. (x2−1)(x2+1)(x^2-1)(x^2+1) B. (x2+1)(x+1)(x−1)(x^2+1)(x+1)(x-1) C. (x2−1)2(x^2-1)^2 D. (x−1)2(x+1)2(x-1)^2(x+1)^2