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

Verified step by step guidance

1. Identify the rational expression given: \(\frac{12}{x^2 + 5x + 6}\).
2. 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.
3. Set the denominator equal to zero to find these excluded values: \(x^2 + 5x + 6 = 0\).
4. Factor the quadratic expression in the denominator: \(x^2 + 5x + 6 = (x + 2)(x + 3)\).
5. Solve each factor set to zero: \(x + 2 = 0\) gives \(x = -2\), and \(x + 3 = 0\) gives \(x = -3\). These values are excluded from the domain.

Key Concepts

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 helps identify values that make the denominator zero by setting each factor equal to zero, which is essential for determining the domain restrictions.