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

Verified step by step guidance

1. Identify the rational expression: $\frac{12}{x^2+5x+6}$.
2. Recognize that the domain of a rational expression is all real numbers except where the denominator is zero.
3. Set the denominator equal to zero: $x^2+5x+6=0$.
4. Factor the quadratic equation: $(x+2)(x+3)=0$.
5. Solve for $x$ to find the values that make the denominator zero: $x=-2$ and $x=-3$. Exclude these values from the domain.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. The key characteristic of rational expressions is that they can be simplified, added, subtracted, multiplied, or divided, provided that the denominator is not zero. Understanding rational expressions is essential for determining their domain, as the values that make the denominator zero must be excluded.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational expressions, the domain is restricted by the values that cause the denominator to equal zero, as division by zero is undefined. Identifying the domain involves solving the equation set by the denominator to find these restricted values.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. This technique is crucial for finding the roots of the polynomial, which helps in determining the values that make the denominator zero. In the given expression, factoring the quadratic in the denominator will reveal the specific x-values that must be excluded from the domain.