Find the domain of each rational expression. See Example 1. (x² - 1) / (x + 1)

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Identify the rational expression given: \(\frac{x^{2} - 1}{x + 1}\).

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

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

Solve the equation to find the value(s) to exclude from the domain: \(x = -1\).

Conclude that the domain is all real numbers except \(x = -1\), which can be written in interval notation as \((-\infty, -1) \cup (-1, \infty)\).

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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, as division by zero is undefined. To find the domain, identify values that cause the denominator to be zero and exclude them.

Factoring Polynomials

Factoring polynomials involves rewriting expressions as products of simpler polynomials. For example, x² - 1 factors into (x - 1)(x + 1). Factoring helps simplify expressions and identify zeros of numerator and denominator.

Simplifying Rational Expressions

Simplifying rational expressions means reducing them by canceling common factors in numerator and denominator. However, restrictions on the domain remain based on the original denominator, even if factors cancel out.