Problem 16 - Algebra for College Students

Question

Which real numbers cannot be used in place of the variable in each rational expression? $$\frac{3 b+1}{b^{2}-3 b-4}$$

Step-by-Step Solution

Answer

Real numbers 4 and -1 cannot be used.

Step 1: Identify the rational expression

The given rational expression is \(\frac{3 b+1}{b^{2}-3 b-4}\).

Step 2: Find the domain

To determine which real numbers cannot be used in place of the variable, find where the denominator is equal to zero. The denominator is \(b^{2}-3b-4\).

Step 3: Set the denominator equal to zero

We need to solve for \(b\) where \(b^{2} - 3b - 4 = 0\).

Step 4: Factor the quadratic equation

\(b^{2} - 3b - 4 = (b-4)(b+1) = 0\). This factors the quadratic equation.

Step 5: Solve for the variable

Set each factor equal to zero: \(b-4 = 0\) and \(b+1 = 0\). This gives \(b = 4\) and \(b = -1\).

Step 6: Interpret the solution

The solution indicates that the rational expression is undefined when \(b = 4\) and \(b = -1\).

Key Concepts

Domain of Rational Expressions

The domain of a rational expression consists of all real numbers except those that make the denominator zero. To find these numbers, look for the values that make the denominator undefined. In our problem, the rational expression is \(\frac{3b + 1}{b^2 - 3b - 4}\). We need to set the denominator equal to zero and solve for the variable to determine these values. These values are excluded from the domain, ensuring you do not divide by zero.

Factoring Quadratics

Factoring quadratics is a crucial step in solving rational expressions. When you factor a quadratic, you rewrite it as a product of two binomials. For our expression, \(b^2 - 3b - 4\) needs to be factored. It can be rewritten as \((b - 4)(b + 1)\). This step simplifies the process of finding the values that make the denominator zero. Successfully factoring quadratics allows you to solve the equation efficiently by breaking it down into simpler parts.

Solving Quadratic Equations

After factoring the quadratic, each factor is set to zero to find the variable's values. For the equation \((b - 4)(b + 1) = 0\), set each factor equal to zero: \(b - 4 = 0\) and \(b + 1 = 0\). Solving these gives \(b = 4\) and \(b = -1\). These values make the original denominator zero and are therefore excluded from the domain. Solving quadratic equations by setting each factor to zero is a fundamental step in determining when rational expressions are undefined.

Undefined Expressions

Rational expressions are undefined when their denominator equals zero. To avoid undefined expressions, exclude these values from the domain. In our example, the values \(b = 4\) and \(b = -1\) make the expression \(\frac{3b + 1}{b^2 - 3b - 4}\) undefined. Understanding when an expression is undefined helps in correctly solving rational expressions and ensures the result is mathematically valid.