Find the domain of each rational expression. (3x + 7) / (4x + 2)(x - 12)

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Identify the rational expression given: \(\frac{3x + 7}{(4x + 2)(x - 12)}\).

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

Set each factor in the denominator equal to zero to find the values to exclude: solve \(4x + 2 = 0\) and \(x - 12 = 0\).

Solve \(4x + 2 = 0\) by isolating \(x\): \(4x = -2\) then \(x = -\frac{1}{2}\).

Solve \(x - 12 = 0\) by isolating \(x\): \(x = 12\). The domain is all real numbers except \(x = -\frac{1}{2}\) and \(x = 12\).

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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 and Setting Denominator Equal to Zero

To find values excluded from the domain, set each factor in the denominator equal to zero and solve for the variable. These solutions indicate points where the expression is undefined and must be excluded from the domain.

Simplifying Rational Expressions

Simplifying rational expressions by factoring helps identify restrictions on the domain clearly. It also aids in understanding the behavior of the expression near excluded values and ensures the domain is accurately described.