Find the domain of each rational expression. 3 / (x2 - 5x - 6)

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 16a

1장, 문제 16a

Find the domain of each rational expression. 3 / (x² - 5x - 6)

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Identify the rational expression given: \(\frac{3}{x^2 - 5x - 6}\).

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

Set the denominator equal to zero to find the values to exclude: \(x^2 - 5x - 6 = 0\).

Factor the quadratic expression: \(x^2 - 5x - 6 = (x - 6)(x + 1)\).

Solve each factor equal to zero: \(x - 6 = 0\) gives \(x = 6\), and \(x + 1 = 0\) gives \(x = -1\). These values are excluded from the domain.

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주요 개념

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

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding rational expressions involves knowing how to simplify, evaluate, and find restrictions on the variable to avoid division by zero.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational expressions, the domain excludes values that make the denominator zero, as division by zero is undefined.

Factoring Quadratic Expressions

Factoring quadratics involves rewriting a quadratic polynomial as a product of two binomials. This skill is essential to identify values that make the denominator zero by setting each factor equal to zero and solving for x.