Problem 20 - Elementary Algebra

Question

For the following problems, find the domain of each of the rational expressions. $$ \frac{3 b+1}{b(b-4)(b+5)} $$

Step-by-Step Solution

Answer

Answer: The domain of the rational expression is (-∞, -5) ∪ (-5, 0) ∪ (0, 4) ∪ (4, ∞).

Step 1: Identify the factors in the denominator

In this rational expression, the denominator has three factors: b, (b-4), and (b+5).

Step 2: Find the values of b that make each factor equal to zero

To find the values of b that make each factor equal to zero, we need to set each factor equal to zero and solve for b: 1) b = 0 2) b - 4 = 0 => b = 4 3) b + 5 = 0 => b = -5

Step 3: Exclude the values of b that make the denominator equal to zero

Since the expression is undefined when the denominator is equal to zero, the domain of the rational expression will exclude the values of b that we found in Step 2. Therefore, the domain will exclude b = 0, b = 4, and b = -5.

Step 4: Write the domain in interval notation

To represent the domain in interval notation, we can write three separate intervals that exclude the values we found in Step 2: 1) b < -5: (-∞, -5) 2) -5 < b < 0: (-5, 0) 3) 0 < b < 4: (0, 4) 4) b > 4: (4, ∞) Then, combine these intervals using the union symbol (∪) to represent the entire domain: Domain: (-∞, -5) ∪ (-5, 0) ∪ (0, 4) ∪ (4, ∞)

Key Concepts

Interval Notation

Interval notation is a helpful way to express ranges of numbers, especially when discussing domains of functions or expressions. It allows us to succinctly indicate which values are included or excluded.
The notation uses brackets and parentheses:

• Square brackets [ ] indicate that a number is included in the interval.
• Parentheses ( ) indicate that a number is not included.

For example, \[ (-∞, -5) \] means all real numbers less than -5, not including -5 itself.
In our rational expression, the domain is expressed as:\[ (-∞, -5) ∪ (-5, 0) ∪ (0, 4) ∪ (4, ∞) \].
This notation clearly expresses that the values -5, 0, and 4 are excluded from the domain.

Undefined Expressions

Undefined expressions occur in mathematics when an operation cannot be completed. A common reason is division by zero. In rational expressions,
the denominator cannot be zero because division by zero is undefined, leading to an expression that doesn't have a meaningful value.
In our problem, to avoid undefined expressions, we need to ensure that none of the factors in the denominator is zero. This prevents the entire denominator from zeroing out. Excluding these values from the domain ensures mathematical coherence and prevents undefined results.

Denominator Factors

Denominator factors are the expressions or numbers found in the denominator of a fraction. Each factor contributes to the potential for the denominator to become zero, making it crucial to analyze them when finding a domain.
Let's look at our example's denominator factors:

• \( b \)
• \( b - 4 \)
• \( b + 5 \)

Each factor gives a condition for exclusion:

• \( b = 0 \)
• \( b = 4 \)
• \( b = -5 \)

By understanding these factors, we see why these specific values are excluded from the expression's domain.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operators. These expressions can be solely numerical or contain both numbers and variables.
Rational algebraic expressions are fractions where the numerator and/or the denominator are algebraic expressions.
The expression in our example, \(\frac{3b+1}{b(b-4)(b+5)}\), combines multiple algebraic components:

• A linear polynomial \( 3b + 1 \) in the numerator
• A polynomial expression \( b(b-4)(b+5) \) in the denominator

These components must be managed carefully to ensure the expression remains defined and operational, particularly keeping a close eye on the denominator's effects on the domain.