To determine the values for which the rational expression $$\frac{x + 4}{x^2 + 5x + 6}$$ is undefined, follow the procedure by setting the denominator equal to $$0$$ and solving for the variable $$x$$:

$$x^2 + 5x + 6 = 0$$

Factor the quadratic equation:
$$(x + 2)(x + 3) = 0$$

Set each factor to zero:
$$x + 2 = 0$$ or $$x + 3 = 0$$
$$x = -2$$ or $$x = -3$$

Thus, the rational expression is undefined for the values $$x = -2$$ and $$x = -3$$.

Example of Finding Undefined Values for a Rational Expression

To determine the domain of a rational function, follow this three-step procedure:

Step 1. Set the denominator expression equal to zero.
Step 2. Solve the resulting equation to find the values that make the denominator zero.
Step 3. State the domain as the set of all real numbers excluding the specific values found in Step 2.

Procedure for Determining the Domain of a Rational Function

To determine the specific values for which a rational expression is undefined, follow these steps:
1. Set the expression in the denominator equal to $$0$$.
2. Solve the resulting equation for the variable.

The solutions obtained from this equation represent the values that make the original rational expression undefined, as they would result in division by zero.

Claude

A rational expression is undefined when its denominator evaluates to $$0$$. This occurs because division by zero is mathematically undefined. While the numerator of a rational expression can be $$0$$, the denominator must never equal $$0$$. Identifying these undefined values is crucial before performing operations with rational expressions, ensuring that any algebraic solutions found later are actually allowed.

Undefined Rational Expression

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Procedure for Determining Undefined Rational Expressions

When working with rational expressions, it is typically assumed that all numerical values that would make the denominator equal to $$0$$ are excluded. Even if the restrictions are not explicitly written for each rational expression, it must be kept in mind that the denominator can never be zero.

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