Problem 32 - Elementary and Intermediate Algebra

Question

Find all real numbers for which the rational expression is undefined. See Example 2. $$ \frac{x^{3}-x^{2}}{15} $$

Step-by-Step Solution

Answer

The expression is defined for all real numbers; there are no values of \(x\) that make it undefined.

Step 1: Identify the Denominator

To determine when a rational expression is undefined, we start by examining its denominator. The expression given is \(\frac{x^{3}-x^{2}}{15}\). The denominator here is 15, which is a nonzero constant.

Step 2: Determine when Denominator is Zero

Since the denominator of the expression is a constant 15, it is never equal to zero. A rational expression is undefined when its denominator equals zero, but since 15 is never zero, the expression is defined for all real numbers.

Step 3: Conclusion

After evaluating the denominator, we conclude that there are no restrictions on the value of \(x\) for this rational expression. Therefore, \(\frac{x^{3}-x^{2}}{15}\) is defined for all real numbers.

Key Concepts

Undefined Rational Expressions

Rational expressions are similar to fractions, where you have a numerator and a denominator. These expressions become undefined when their denominator is equal to zero. This is because division by zero is undefined in mathematics. To check if a rational expression is undefined, you begin by looking at the denominator of the expression.

• If the denominator can be zero for some value of the variable, then the expression is undefined for that specific value.
• If the denominator cannot be zero for any value of the variable, the expression is defined for all values of the variable.

In the given exercise, the rational expression is \(\frac{x^{3}-x^{2}}{15}\). Here, the denominator is a constant value, 15, which is never zero. Thus, the expression remains defined for all values of \(x\).

Constant Denominator

In mathematics, a constant denominator is a number that does not change regardless of the value of the variable in the numerator. It remains the same across the entire expression. This is significant because it ensures that the denominator never reaches a problematic value, such as zero.In our example, the rational expression has 15 as its denominator, and:

• Since 15 is non-zero and does not depend on \(x\), the expression cannot be undefined due to a zero denominator.
• Expressions with constant, non-zero denominators are always defined for all values of \(x\).

Thus, with a constant denominator, analyzing the domain of the expression simplifies substantially because there are no restrictions needed to avoid division by zero.

Real Numbers

Real numbers encompass all rational and irrational numbers that can be plotted on a number line. They include whole numbers, integers, fractions, and decimals — essentially any number you can think of that does not involve imaginary or complex components.

• In this exercise, the question pertains to finding real numbers for which the expression is undefined. However, given the constant denominator, there are no restrictions.
• Since the denominator does not affect the real number domain, every real number is part of the expression's domain.

In conclusion, the rational expression \(\frac{x^{3}-x^{2}}{15}\) is defined for all real numbers, as the denominator never becomes zero across the entire number line of real numbers.