Problem 73 - Algebra for College Students

Question

Find the indicated value for each given rational expression, if possible. $$W(b)=\frac{4 b^{3}-1}{b^{2}-b-6}, W(-2)$$

Step-by-Step Solution

Answer

The expression is undefined at \( b = -2 \).

Step 1: Understand the Expression

The given rational expression is \( W(b) = \frac{4b^3 - 1}{b^2 - b - 6} \). We need to find \( W(-2) \).

Step 2: Substitute the Value

Substitute \( b = -2 \) into the expression: \[ W(-2) = \frac{4(-2)^3 - 1}{(-2)^2 - (-2) - 6} \]

Step 3: Simplify the Numerator

Calculate the numerator:\[ 4(-2)^3 - 1 = 4(-8) - 1 = -32 - 1 = -33 \]

Step 4: Simplify the Denominator

Calculate the denominator:\[ (-2)^2 - (-2) - 6 = 4 + 2 - 6 = 0 \]

Step 5: Check for Undefined Expression

Since the denominator is 0, the expression is undefined.This means \( W(-2) \) does not exist.

Key Concepts

Rational Functions

A rational function is expressed as the ratio of two polynomials. The general form is: \[ f(x) = \frac{P(x)}{Q(x)} \] Here, \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) is not zero. In our exercise, the rational function given is \( W(b) = \frac{4b^3 - 1}{b^2 - b - 6} \). Rational functions are useful in many applications including engineering and science. They can represent real-world scenarios where one quantity is dependent on another. For example, speed or concentration analysis. The key to understanding rational functions is to manage both the numerator and the denominator effectively. Let’s unpack further to understand more about concepts like undefined expressions and polynomial simplification. By understanding these concepts well, you can master rational functions!

Undefined Expressions

An expression in mathematics becomes undefined when it leads to an impossible or undefined operation. This typically happens when the denominator of a fraction equals zero. As seen in our exercise:
\[ W(b) = \frac{4b^3 - 1}{b^2 - b - 6} \] We substituted \( b = -2 \):
\[ W(-2) = \frac{4(-2)^3 - 1}{(-2)^2 - (-2) - 6} \] By simplifying the denominator:
\[ (-2)^2 - (-2) - 6 = 4 + 2 - 6 = 0 \] Our denominator is zero, making \( W(-2) \) undefined.
An undefined expression occurs whenever dividing by zero, as division by zero has no meaning in conventional arithmetic. Thus, solutions to rational expressions must always confirm that the denominator is not zero at the evaluated point.

Polynomial Simplification

Simplifying polynomials involves combining like terms and factoring where possible. This is crucial when working with rational functions to make expressions easier to evaluate. For example, in \( W(b) \):
\[ W(b) = \frac{4b^3 - 1}{b^2 - b - 6} \] The steps for solving include:

• Understanding the function
• Substituting values into the function
• Simplifying the numerator and the denominator

For our specific case:
Numerator: \[ 4(-2)^3 - 1 = 4(-8) - 1 = -32 - 1 = -33 \] Denominator: \[ (-2)^2 - (-2) - 6 = 4 + 2 - 6 = 0 \] Factors and simplification are often used in polynomial simplification. Factoring can further break down polynomials to reveal roots and zeros, and aid in simplifying rational expressions.
Polynomial simplification is foundational in calculus and algebra, forming the basis for solving more complex mathematical problems.