Problem 23 - Elementary Algebra

Question

For the following problems, find the domain of each of the rational expressions. $$ \frac{6 a}{a^{3}(a-5)(7-a)} $$

Step-by-Step Solution

Answer

Answer: The domain of the given rational expression is \(a \in \mathbb{R} \setminus \{0, 5, 7\}\).

Step 1: Identify the denominator

The denominator of the given rational expression is: $$ a^{3}(a-5)(7-a) $$

Step 2: Find the values of a that make the denominator zero

We need to find when \(a^{3}(a-5)(7-a) = 0\). This equation is true if any of the factors are equal to zero: $$ a^3 = 0 \\ a - 5 = 0 \\ 7 - a = 0 $$ Now, we solve these equations for a.

Step 3: Solve for a

From the equations in Step 2: $$ a^3 = 0 \Rightarrow a = 0 \\ a - 5 = 0 \Rightarrow a = 5 \\ 7 - a = 0 \Rightarrow a = 7 $$

Step 4: Find the domain

The rational expression will be undefined if the denominator is equal to zero. As we found that the denominator equals zero when \(a=0\), \(a=5\), or \(a=7\), we need to exclude these values of a from the domain. Therefore, the domain of the given rational expression is: $$ a \in \mathbb{R} \setminus \{0, 5, 7\} $$

Key Concepts

Understanding Domain of Rational Expressions

When working with rational expressions, one critical concept is understanding the domain. The domain refers to all possible values that the variable in the expression can take without causing the expression to be undefined. In rational expressions, values that make the denominator zero have to be excluded from the domain since division by zero is not defined in mathematics.

To find the domain, perform the following steps:

• Identify the denominator of the rational expression.
• Set each factor in the denominator equal to zero to find critical values.
• Solve these equations to find the values that make the denominator zero.
• Exclude these values from the set of all real numbers.

After these steps, you will have identified the domain as all real numbers except those found to zero the denominator. It's important to always start with finding the denominator since it significantly influences the domain.

Role of Denominators in Rational Expressions

Denominators are crucial in rational expressions because they determine where the expression might be undefined.

Each term in the denominator can affect the range of valid inputs for the expression. When dealing with rational expressions, the denominator is typically a polynomial. It could be as simple as a single variable or as complex as a polynomial equation.

As in our example, the denominator is composed of several factors: - \( a^3 \) - \( a-5 \) - \( 7-a \). Each of these factors contributes potential zero points by which division by zero occurs. Letting any of these terms equal zero causes the whole denominator to be zero. This is why finding zeros of each factor is necessary for determining the domain.

Factoring to Simplify Domains

Factoring is a very useful technique in simplifying rational expressions and in finding the domain. The process of factoring involves breaking down complex polynomials into simpler parts or factors. This simplifies the expression and makes it easier to see which values need to be excluded from the domain.

In our example, the denominator is already factored as: - \( a^3 \) which simplifies to \( a \times a \times a \) - \( (a - 5) \) - \( (7 - a) \). Factoring allows us to identify each distinct factor that can result in the denominator equating to zero. Once all factors are clearly defined, it becomes straightforward to solve for variable values that make the denominator zero.

With these values identified, the expression simplifies in terms of defining the domain by thoroughly removing these specific values.