Problem 9 - Algebra for College Students

Question

Find the domain of each rational expression. $$\frac{2 z-5}{7 z}$$

Step-by-Step Solution

Answer

Domain: \(\forall z eq 0\).

Step 1: Identify Rational Expression

Recognize that the given expression is a rational function \(\frac{2z-5}{7z}\). A rational expression is a fraction with polynomials in both the numerator and the denominator. For this exercise, the numerator is \(2z-5\) and the denominator is \(7z\).

Step 2: Determine Denominator Restrictions

The domain of a rational expression is all real numbers except where the denominator is zero. Set the denominator equal to zero and solve: \(7z = 0\).

Step 3: Solve for Zero in the Denominator

Solving \(7z = 0\), we get \(z = 0\). This means \(z = 0\) is not in the domain of the expression.

Step 4: Conclusion: Domain of the Expression

Since the only restriction is that \(z eq 0\), the domain of the rational expression \(\frac{2z-5}{7z}\) is all real numbers except zero.

Key Concepts

Rational Functions

Rational functions are an important concept in algebra. These are fractions where both the numerator and the denominator are polynomials. In the given exercise, we have the rational function \(\frac{2z-5}{7z}\). Here, \2z - 5\ is a polynomial in the numerator, and \7z\ is a polynomial in the denominator. Rational functions can have various forms, but understanding the parts of the fraction is key.

To recognize rational functions:

• A polynomial can be a simple number, variable, or a combination like \(2z - 5\).
• The denominator should never be zero, which impacts the domain.

With this groundwork, you can tackle more complex problems involving these fractions.

Polynomials

Polynomials might sound intimidating, but they are just expressions made up of variables and coefficients. They involve operations like addition, subtraction, and multiplication.

For example, in \(\frac{2z-5}{7z}\), the numerator is \2z - 5\ and the denominator is a single-term polynomial \7z\. Each part plays a vital role in forming the rational function.

In more detail:

• The polynomial \2z - 5\ combines the variable **z** with constants and coefficients.
• The denominator, \7z\, is also a polynomial, albeit simpler.

By breaking down the components of polynomials, the larger structure of rational functions becomes much easier to handle and understand.

Denominator Restrictions

One crucial aspect of rational functions is understanding the restrictions imposed by the denominator. The denominator can never be zero because division by zero is undefined. This means we must find when the denominator equals zero and exclude these values from the domain.

To apply this to our function \(\frac{2z-5}{7z}\):

• Set the denominator equal to zero: \(7z = 0\).
• Solve for \(z\) (in this case, \(z = 0\)).
• Exclude \(z = 0\) from the set of all real numbers.

By identifying these restrictions, you ensure the rational function behaves correctly across its intended domain.

Domain

The domain of a rational function includes all real numbers except those that make the denominator zero. For \(\frac{2z-5}{7z}\), we determined that the denominator is zero when \(z = 0\). Therefore, the domain is all real numbers except zero.
\[Domain: \{ z | z \eq 0 \} \]
Here’s how we calculated it:

• First, identify the polynomial in the denominator.
• Set the denominator equal to zero and solve for the variable.
• Exclude these values from the domain.

By understanding the domain, you ensure that the function is valid and can be correctly used in further calculations and analyses.

Zero Denominator

A zero denominator is problematic because it makes the entire fraction undefined. In the context of rational functions, this means you must always check for these problematic values.

For \(\frac{2z-5}{7z}\), we set the denominator equal to zero and found \(z = 0\). When \(z\) is zero, the fraction \(\frac{2z-5}{7z}\) does not exist.

Steps to avoid zero denominators:

• Identify the denominator
• Set the denominator equal to zero.
• Solve for the variable.
• Exclude these values from your domain.

This simple check ensures you avoid undefined expressions and correctly interpret rational functions in mathematical problems.