CONCEPT PREVIEW Which of the following is the correct factorization of x⁴ - 1? A. (x² - 1) (x² + 1) B. (x² + 1) (x + 1) (x - 1) C. (x² - 1)² D. (x - 1)² (x + 1)²

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

Apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\), where \(a = x^2\) and \(b = 1\). This gives \(x^4 - 1 = (x^2 - 1)(x^2 + 1)\).

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

Combine the factors to express the full factorization as \((x - 1)(x + 1)(x^2 + 1)\).

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

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주요 개념

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Difference of Squares

The difference of squares is a factoring technique where an expression of the form a² - b² can be factored into (a - b)(a + b). For example, x⁴ - 1 can be seen as (x²)² - 1², allowing it to be factored into (x² - 1)(x² + 1).

Further Factorization of Quadratic Expressions

Some quadratic expressions, like x² - 1, can be further factored if they are also differences of squares. Since x² - 1 = (x - 1)(x + 1), recognizing this allows complete factorization of higher-degree polynomials into linear factors.

Understanding Polynomial Factorization and Exponents

Polynomial factorization involves breaking down expressions into products of simpler polynomials. Recognizing powers and exponents, such as x⁴ being (x²)², helps in applying factoring formulas correctly and identifying equivalent factorizations.