Problem 23 - Algebra A Combined Function

Question

Find the \(L C D\) for each list of rational expressions. $$ \frac{8 x^{2}}{(x-6)^{2}}, \frac{13 x}{5 x-30} $$

Step-by-Step Solution

Answer

The LCD is \(5(x-6)^2\).

Step 1: Identify the Denominators

First, note the denominators of each rational expression. The denominators are: \((x-6)^{2}\) and \(5x-30\).

Step 2: Factor the Denominators

Factor each denominator as much as possible. For \((x-6)^{2}\), it is already factored as a square. For \(5x-30\), factor out the common factor of 5 to get \(5(x-6)\).

Step 3: Determine the Least Common Multiple (LCM) of the Denominators

The LCM must include each distinct factor present in any of the denominators, raised to the highest power they occur in any one of the denominators. We have the factors \((x-6)\) and 5 in this case. Thus, the LCM is \(5(x-6)^2\).

Step 4: Form the Least Common Denominator (LCD)

The Least Common Denominator (LCD) of the rational expressions is the LCM of their denominators. Therefore, the LCD is \(5(x-6)^2\).

Key Concepts

Factoring Polynomials

One of the first steps to solving rational expressions is factoring polynomials that you come across in the denominators. Factoring can simplify expressions and make them easier to work with when finding the least common denominator (LCD). A polynomial is an expression involving variables and coefficients. To factor it means to break it down into simpler parts or factors that can be multiplied together to give the original polynomial.

In our example, we had two denominators: \((x-6)^2\) and \(5x-30\). The expression \((x-6)^2\) is already factored because it's just the square of \((x-6)\). For the expression \(5x-30\), you can factor out the greatest common factor, which is 5, resulting in \(5(x-6)\).

Here's how you can tackle factoring:

• Look for a common factor in all terms.
• Use factorization techniques like grouping or the quadratic formula, if necessary.
• Verify your factorization by expanding it to ensure it matches the original expression.

Understanding factoring is crucial for simplifying and manipulating polynomials in rational expressions.

Least Common Multiple

Finding the least common multiple (LCM) of the denominators is essential to determine the least common denominator for rational expressions. The LCM of two or more numbers is the smallest number that is evenly divisible by each of them. When it comes to expressions, we look at each factor within the polynomials.

In the given exercise, we start with denominators \((x-6)^2\) and \(5(x-6)\). To determine the LCM, we consider all distinct factors across these denominators:

• \((x-6)\)
• 5

Then, we take each factor at its highest power present in any denominator. Thus, the LCM becomes \(5(x-6)^2\), including the full multiplicity of \((x-6)^2\) from the first denominator.

To summarize:

• Identify all distinct factor bases in the denominators.
• For each base, use the highest exponent found in any denominator's factorization.
• Combine these to form the LCM.

This process ensures that the LCM can accommodate all denominators involved.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Understanding how to manipulate these expressions is crucial in algebra. This often involves simplifying the expressions, finding common denominators, and performing arithmetic operations on them.

In our exercise, we were tasked with finding the least common denominator (LCD) of two rational expressions: \(\frac{8x^2}{(x-6)^2}\) and \(\frac{13x}{5x-30}\). Start by acknowledging each part of the expression:

• The numerator: a polynomial that can often be simplified or factored.
• The denominator: a polynomial where factoring plays a key role in operations like finding the LCD.

By factoring these denominators, as done in our solution, it becomes possible to identify and derive the least common denominator easily. The LCD makes it possible to add, subtract, or compare rational expressions effectively.

Remember, the techniques applied to manipulate rational expressions are foundational for advanced algebraic operations, ensuring you can handle expressions with different denominators without losing accuracy or consistency.