Problem 5 - Algebra 2 and Trigonometry

Question

List the values of the variables for which the rational expression is undefined. \(\frac{a+2}{a b}\)

Step-by-Step Solution

Answer

The expression is undefined for \(a = 0\) or \(b = 0\).

Step 1: Understand the Expression

The rational expression given is \( \frac{a+2}{a b} \). In a rational expression, division by zero is undefined. Therefore, we need to find the values of the variables that would make the denominator \(a b\) equal to zero.

Step 2: Set the Denominator to Zero

We set the denominator of the rational expression, \(a b\), equal to zero to find which values of \(a\) and \(b\) will make the expression undefined.\[a b = 0\]

Step 3: Identify the Variable Conditions

For the product \(a b\) to be zero, either \(a = 0\), \(b = 0\), or both must be true. This means the expression \( \frac{a+2}{a b} \) is undefined when \(a = 0\) or \(b = 0\).

Step 4: Conclusion

Any value of \(a = 0\) or \(b = 0\) will make the rational expression undefined. Therefore, \(a = 0\) and \(b = 0\) are the values for which the expression is undefined.

Key Concepts

Division by Zero

Division by zero is a concept that makes many mathematical expressions undefined. A divide-by-zero situation occurs when the denominator in a fraction equals zero. Having a zero in the denominator is a problem because there is no number that you can multiply by zero to get a non-zero value. This contradiction makes division by zero an impossible operation in mathematics. That's why, in rational expressions, it's very important to identify any values of variables that cause the denominator to be zero and thus make the expression undefined.

Rational Expressions

A rational expression is similar to a fraction, but instead of having numbers in the numerator and denominator, it has polynomials. This means rational expressions can include variables, as seen in the expression \( \frac{a+2}{a b} \). When working with rational expressions, our main goal is to ensure that the expression remains defined. This involves taking care of the denominator.- The numerator can be any value, but the denominator must not be zero.- Since the denominator in rational expressions can contain variables, determining when the expression is undefined often requires solving an equation, setting the denominator equal to zero to find problematic values for the variables. By doing this, you make sure to exclude these values from the solution or domain of your expression to avoid undefined situations.

Undefined Expressions

Undefined expressions occur when certain operations, such as division by zero, take place within a mathematical expression. These expressions do not have a valid result. In the context of rational expressions, this happens when the denominator of the expression is zero.- To find out when a rational expression is undefined, you set the denominator equal to zero and solve for the variables.- As shown in the original exercise, the expression \( \frac{a+2}{a b} \) becomes undefined when \( a = 0 \) or \( b = 0 \). This is because either value will make the denominator zero, resulting in an undefined expression.Understanding when expressions become undefined helps in determining the domain of an expression and avoiding computational errors in mathematics. Always take extra care to check for undefined values when working with rational expressions.