Find the domain of each rational expression. See Example 1. (x³ - 1) / (x - 1)

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Identify the rational expression given: \(\frac{x^{3} - 1}{x - 1}\).

Recall that the domain of a rational expression includes all real numbers except those that make the denominator zero.

Set the denominator equal to zero and solve for \(x\): \(x - 1 = 0\) which gives \(x = 1\).

Exclude \(x = 1\) from the domain because it makes the denominator zero and the expression undefined.

Therefore, the domain is all real numbers except \(x = 1\), which can be written as \(\{x \in \mathbb{R} \mid x \neq 1\}\).

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주요 개념

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

Domain of a Rational Expression

The domain of a rational expression includes all real numbers except those that make the denominator zero. Since division by zero is undefined, identifying values that cause the denominator to be zero is essential to determine the domain.

Factoring Polynomials

Factoring polynomials helps simplify expressions and identify common factors. For example, the numerator x³ - 1 can be factored using the difference of cubes formula, which aids in simplifying the expression and analyzing the domain.

Difference of Cubes Formula

The difference of cubes formula states that a³ - b³ = (a - b)(a² + ab + b²). Applying this to x³ - 1 allows factoring the numerator as (x - 1)(x² + x + 1), which is useful for simplifying the rational expression and understanding restrictions on the domain.