Problem 3 - Algebra 2 and Trigonometry

Question

List the values of the variables for which the rational expression is undefined. \(\frac{5 a^{2}}{3 a}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Denominator

Rational expressions are undefined when their denominators are equal to zero. In the given expression \( \frac{5a^2}{3a} \), the denominator is \( 3a \).

Step 2: Set the Denominator to Zero

To find for which values the expression is undefined, we need to set the denominator equal to zero and solve for \( a \): \[ 3a = 0 \]

Step 3: Solve the Equation

Solve the equation \( 3a = 0 \) by dividing both sides by 3:\[ a = 0 \]

Step 4: Conclusion

The expression \( \frac{5a^2}{3a} \) is undefined when \( a = 0 \). This is because division by zero is not allowed.

Key Concepts

Denominator Identification

In every rational expression, identifying the denominator is crucial. The denominator is the part of the fraction below the line or the expression under the division sign. In our example, the rational expression given is \( \frac{5a^2}{3a} \). Here, the denominator is \( 3a \). Recognizing the denominator is essential because it determines when the entire expression is undefined. Whenever the denominator equals zero, the expression will not yield a valid mathematical outcome. Learn to spot the denominator swiftly in any fraction-based expression you encounter.

• Look below the fraction bar to locate the denominator.
• In mathematical expressions, identify terms being divided as the denominator.

Understanding this concept helps prevent errors, especially when solving for values that make an expression undefined.

Undefined Expressions

Rational expressions become undefined when their denominators equal zero. This topic ties closely with the denominator identification process. In the expression \( \frac{5a^2}{3a} \), we already identified \( 3a \) as the denominator. To seek the points where the expression is undefined, we need to find out when \( 3a = 0 \). Solving for \( a \) provides the value that causes the denominator to become zero, hence making the rational expression undefined.

• Set the denominator equal to zero.
• Solve the resulting equation to find the point of undefined expression.

This process is key to ensuring you avoid values that would disrupt the arithmetic or lead to erroneous interpretations in further problem-solving.

Division by Zero

The concept of division by zero is a fundamental rule in algebra and mathematics overall. It states that any operation dividing by zero is undefined. In the exercise, this is seen with the equation \( 3a = 0 \). Solving this gives \( a = 0 \). Therefore, when \( a = 0 \), substituting it back into the expression \( \frac{5a^2}{3a} \) results in an undefined value because you would be dividing by zero (i.e., 0 in the denominator).

• Division by zero is against mathematical rules.
• Attempting it results in an expression without a numerical meaning.

Always be cautious of denominators and avoid numerical inputs that lead to division by zero to maintain the integrity and validity of your calculations.