Problem 71 - College Algebra with Modeling and Visualization

Question

Find the \(L C D\) for the rational expressions. $$ \frac{3}{2}, \frac{x}{2 x+1}, \frac{x}{2 x-4} $$

Step-by-Step Solution

Answer

The LCD is \(2(2x + 1)(2x - 4)\).

Step 1: Identify Denominators

To find the Least Common Denominator (LCD), first identify the denominators of the given rational expressions. The denominators are: 2, \(2x + 1\), and \(2x - 4\).

Step 2: Factor the Denominators

Next, factor each of the denominators if possible. The denominator 2 is already in simplest form. The expressions \(2x + 1\) and \(2x - 4\) cannot be factored further with integers since they are linear polynomials.

Step 3: List Unique Factors

List all unique factors present in each denominator. Here, the factors are 2, \(2x + 1\), and \(2x - 4\).

Step 4: Determine the LCD

The Least Common Denominator is the product of all unique factors from the denominators. Multiply these factors together to determine the LCD: \(LCD = 2 \cdot (2x + 1) \cdot (2x - 4)\).

Key Concepts

Rational Expressions

Rational expressions are similar to fractions but involve variables in their numerators, denominators, or both. They are defined as the quotient of two polynomials, with the denominator not being zero. Understanding rational expressions is key in algebra because they frequently appear in equations and functions.​

• The simplest form of a rational expression is one where the numerator and denominator have no common factors other than 1.
• Simplifying is crucial as it makes equations easier to solve and expressions easier to work with.
• To simplify, factor both numerator and denominator, then cancel out common factors.
• Caution: Always check for restrictions on the variable. Set the denominator not equal to zero to find excluded values for the variable.

When solving problems with rational expressions, finding a common denominator often plays a critical role in adding, subtracting, or comparing them. The least common denominator (LCD) is the smallest polynomial that all denominators divide evenly. To find the LCD, identify all unique factors or terms from the denominators involved, and then calculate their product.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This process is immensely helpful when simplifying rational expressions or finding denominators. Factoring requires identifying patterns and applying various methods based on the polynomial's structure.

• Common Methods: Look for methods like finding common factors, using the difference of squares, or applying the quadratic formula for quadratics.
• Linear Polynomials: When dealing with linear polynomials such as \(2x+1\) or \(2x-4\), factorization isn't applicable in the traditional sense, as these expressions are already simplified and require investigation for common factors.
• Complex Cases: For higher-degree polynomials, methods like synthetic division or grouping may be employed.

Consider every denominator separately and ensure it’s in its simplest factorized form before attempting to find the LCD. This is crucial for combining rational expressions as demonstrated in the solution.

Linear Polynomials

Linear polynomials are expressions of the form \(ax+b\), where \(a\) and \(b\) are constants and\(x\) represents a variable. They are the simplest type of polynomial functions, forming a straight line when graphed.

• Properties: A notable feature is that linear polynomials cannot be factored over the integers unless they share a common factor.
• Examples: For \(2x + 1\) and \(2x - 4\), there are no factorable components within the integers, maintaining their initial form.
• Graphing: Each linear polynomial corresponds to a straight line where \(a\) is the slope, and \(b\) is the y-intercept.

In the context of finding the least common denominator (LCD) for rational expressions, acknowledging these linear polynomials as factors can simplify the process. Since they cannot be broken down further, they are directly used in the product calculation for the LCD, ensuring all terms are accounted for without performing any unnecessary operations.