Problem 8 - Algebra 2 and Trigonometry

Question

List the values of the variables for which the rational expression is undefined. \(\frac{b+3}{b^{2}+b-6}\)

Step-by-Step Solution

Answer

The values of \(b\) that make the expression undefined are \(b = -3\) and \(b = 2\).

Step 1: Identify the Denominator

The expression can be undefined if the denominator is equal to zero. So the first step is to identify the denominator, which is \(b^2 + b - 6\).

Step 2: Set the Denominator to Zero

To find the values that make the expression undefined, set the denominator equal to zero: \(b^2 + b - 6 = 0\).

Step 3: Factor the Quadratic Expression

Factor the quadratic expression: \(b^2 + b - 6 = (b + 3)(b - 2)\).

Step 4: Solve Each Factor

Set each factor equal to zero and solve for \(b\):\[(b+3) = 0 \Rightarrow b = -3\] \[(b-2) = 0 \Rightarrow b = 2\]

Step 5: List the Undefined Values

The rational expression is undefined for the values of \(b = -3\) and \(b = 2\) because they make the denominator zero.

Key Concepts

Undefined Expressions

In mathematics, especially when working with rational expressions, recognizing when an expression becomes undefined is crucial. A rational expression is the ratio of two polynomials. Sometimes, the denominator of the expression can be zero which makes the expression undefined. For instance, suppose we have a rational expression like \(\frac{b+3}{b^{2}+b-6}\). Here, the expression becomes undefined if the denominator \(b^{2}+b-6\) equals zero. This is because division by zero is impossible in mathematics.
To find the values where the expression is undefined, you set the denominator equal to zero and solve for the variable. These values are not part of the domain of the rational expression. Understanding undefined expressions helps in sketching graphs or solving rational equations effectively.

Factoring Quadratics

Factoring quadratics is a key step in solving problems related to quadratic equations and rational expressions. It involves rewriting the quadratic expression as a product of two binomials. This process is important for identifying potential undefined values in rational expressions.
To factor a quadratic expression like \(b^2 + b - 6\), you look for two numbers that multiply to give the constant term (-6) and add to give the linear coefficient (1). For \(b^2 + b - 6\), these numbers are +3 and -2. Hence, it can be factored as \((b + 3)(b - 2)\). The ability to factor quadratics easily helps in simplifying complex expressions and solving equations quickly.

Solving Equations

Once a quadratic expression is factored, solving the equation becomes a straightforward task. Solving equations often requires setting each factor equal to zero and finding the value of the variable that satisfies the equation.
For the quadratic \((b + 3)(b - 2) = 0\), you solve it by setting each factor to zero separately:

• \( b + 3 = 0 \) implies \( b = -3 \)
• \( b - 2 = 0 \) implies \( b = 2 \)

Thus, solving equations allows you to find the specific points or values for which the expression or function might change behavior, such as becoming undefined through a division by zero.

Denominator Zero

When dealing with rational expressions, particularly crucial is understanding the consequences when the denominator becomes zero. This condition renders the entire expression undefined, and typically indicates locations for breaks or holes if graphed.
To avoid undefined expressions, identify where a denominator like \(b^2 + b - 6\) equals zero by solving the equation \(b^2 + b - 6 = 0\). After factoring and finding \(b = -3\) and \(b = 2\), you know the rational expression is undefined at these points because dividing by zero is mathematically unfeasible.
Knowing how the denominator behaves in rational expressions not only helps prevent undefined outputs but also aids in understanding the function's domain and potential discontinuities.