Problem 15 - Algebra A Combined Function

Question

Find the \(L C D\) for each list of rational expressions. $$ \frac{19}{2 x}, \frac{5}{4 x^{3}} $$

Step-by-Step Solution

Answer

The LCD is \(4x^3\).

Step 1: Understand the Problem

To find the Least Common Denominator (LCD) of the given rational expressions \( \frac{19}{2x} \) and \( \frac{5}{4x^3} \), we need to identify the denominators and find the least common multiple of these denominators.

Step 2: Identify Denominators

The denominators for the rational expressions are \(2x\) and \(4x^3\).

Step 3: Factor Denominators

Factor the denominators: \(2x = 2 \cdot x\) and \(4x^3 = 2^2 \cdot x^3\).

Step 4: Find Least Common Multiple (LCM)

The LCM of the denominators is found by taking the highest power of each prime factor from the factored denominators. For 2, the highest power is \(2^2\), and for \(x\), the highest power is \(x^3\).

Step 5: Compute the LCD

Multiply these together to find the LCM, giving the LCD: \(2^2 \cdot x^3 = 4x^3\).

Step 6: Verify the LCD

Check that both original denominators \(2x\) and \(4x^3\) divide evenly into the LCD \(4x^3\). \(2x\) divides evenly into \(4x^3\) by \(2x^2\) times, confirming \(4x^3\) is indeed the LCD.

Key Concepts

Rational expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. These expressions are quite common in algebra, and understanding them is crucial for solving various types of math problems. In our example, the expressions are \( \frac{19}{2x} \) and \( \frac{5}{4x^3} \). Here, 19 and 5 are the numerators, while \(2x\) and \(4x^3\) are the denominators, which makes these rational expressions because the denominators are linear and cubic polynomials, respectively.

The key challenge when working with rational expressions, particularly when finding their Least Common Denominator (LCD), is dealing with their polynomial nature. This involves understanding their algebraic structure, which later helps in performing operations like addition, subtraction, and simplification of these expressions.

Rational expressions are essential because they allow us to represent a wide range of mathematical relationships in an approachable way. Mastering these concepts provides a solid foundation for further algebraic manipulation and problem-solving.

Factoring

Factoring is a crucial step in simplifying algebraic expressions and finding the Least Common Denominator (LCD) for rational expressions. In simple terms, factoring involves expressing a number or an expression as a product of its prime factors. By breaking down expressions into their basic elements, factoring makes it easier to handle operations.

In the context of our example with denominators \(2x\) and \(4x^3\), the factoring process looks like this:

• For \(2x\), the factored form is \(2 \cdot x\).
• For \(4x^3\), it factors into \(2^2 \cdot x^3\).

Factoring helps us clearly identify each component of the denominators, setting the stage for finding their Least Common Multiple (LCM). This step simplifies further calculations and provides a clear path toward solving complex algebraic expressions. The ability to factor expressions smoothly is an invaluable skill, making problem-solving in algebra a more straightforward task.

Least Common Multiple (LCM)

The Least Common Multiple (LCM) is the smallest number or polynomial that is a multiple of two or more numbers or expressions. It's an essential concept when working with rational expressions, particularly when you want to add or subtract them, which requires converting them to have the same denominator. To find the LCM of the denominators in our example, we first factor each expression:

• \(2x\) becomes \(2 \cdot x\).
• \(4x^3\) becomes \(2^2 \cdot x^3\).

The LCM is determined by taking the highest power of each factor found in the factored forms. For the number 2, \(2^2\) is the highest power present in the denominators. For the variable \(x\), \(x^3\) is the highest power. Therefore, the LCM is \(2^2 \cdot x^3\), which simplifies to \(4x^3\).

This concept of LCM is foundational, not just for finding the LCD, but also in various applications across different areas of mathematics. It ensures uniformity in expressions, facilitating easier operations and manipulations.