Find the domain of each rational expression. See Example 1. 12 / (x² + 5x + 6)

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Identify the rational expression given: \(\frac{12}{x^{2} + 5x + 6}\).

Recall that the domain of a rational expression excludes values of \(x\) that make the denominator equal to zero, because division by zero is undefined.

Set the denominator equal to zero to find these excluded values: \(x^{2} + 5x + 6 = 0\).

Factor the quadratic expression in the denominator: \(x^{2} + 5x + 6 = (x + 2)(x + 3)\).

Solve each factor equal to zero to find the values to exclude from the domain: \(x + 2 = 0 \Rightarrow x = -2\) and \(x + 3 = 0 \Rightarrow x = -3\). Therefore, the domain is all real numbers except \(x = -2\) and \(x = -3\).

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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 values of the variable for which the expression is defined. Since division by zero is undefined, values that make the denominator zero must be excluded from the domain.

Factoring Quadratic Expressions

Factoring a quadratic expression involves rewriting it as a product of two binomials. This helps identify the roots of the quadratic, which are the values that make the expression equal to zero, crucial for finding restrictions in the domain.

Setting the Denominator Not Equal to Zero

To find the domain of a rational expression, set the denominator not equal to zero and solve for the variable. The solutions to this inequality indicate values that must be excluded from the domain to avoid division by zero.