Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Find the least common denominator of the expressions.\(\frac{10}{x+5}, \frac{x+4}{x-7}, \frac{x+5}{x^{2}-2 x-35}\)

Step-by-Step Solution

Answer

The least common denominator of the given fractions is \(x^{2}-2x-35\).

Step 1: Factorizing the denominators

Factorizing the denominators gives the following: \((x+5)\), \((x-7)\), and \((x-7)(x+5)\). It's important to factorize as it simplifies the process of finding the least common multiple (LCM).

Step 2: Determining the LCM

The least common multiple (LCM) of a group of numbers is the smallest number that is a multiple of each of the numbers. From the factorized denominators, it can be seen that \((x+5)\), and \((x-7)\) are common to all fractions. Therefore, the LCD is \((x-7)(x+5)\).

Step 3: Expressing the LCD

Finally, the LCD should be expressed in a simplified form, which is \(x^{2}-2x-35\)

Key Concepts

Factoring Denominators

Factoring denominators is a crucial first step in working with rational expressions. When you look at a denominator, especially one that involves polynomials, you aim to break it down into simpler multiplicative factors.

• This simplifies the process of manipulation and calculation involving the expressions.
• Take the expression from the exercise: \(x^2-2x-35\). This polynomial can be factored into \((x-7)(x+5)\).

Breaking down denominators in this way - helps in identifying common factors, - makes discovering the Least Common Multiple (LCM) more straightforward, and - ultimately ensures accuracy in further operations with the expressions. Having a strong understanding of factoring can save time and effort in algebraic processes involving rational expressions.

Least Common Multiple

The Least Common Multiple, or LCM, is vital when working with fractions, including rational expressions. It is defined as the smallest expression that is a multiple of each member in a given set of expressions.

• Finding the LCM involves identifying all unique factors that appear in any of the denominators and using the highest power of each.
• In our rational expression example, you first identify the factors from the denominators: \((x+5)\), \((x-7)\), and \((x-7)(x+5)\).

Notice that \((x+5)\) and \((x-7)\) individually appear in the denominators, as well as both combined as another denominator. This means the LCM will simply be \((x-7)(x+5)\) because it contains both factors fully, subsuming any duplicate appearance. Understanding and finding the LCM helps in performing operations such as addition and subtraction with rational expressions.

Rational Expressions

Rational expressions are fractions that have polynomials in both the numerator and the denominator. These expressions behave similarly to regular fractions, and many operations on rational expressions parallel those you'd perform on numerical fractions.

• To manage them, we often need a common denominator, as in the process of finding the Least Common Denominator (LCD).
• The LCD ensures a uniform base for addition, subtraction, or comparison of rational expressions.

For example, when dealing with \( \frac{10}{x+5} \), \( \frac{x+4}{x-7} \), and \( \frac{x+5}{x^2-2x-35} \), you first need a shared denominator for simplification or operations. Rational expressions can appear daunting at first, but when approached with factoring and carefully finding common elements, they become more manageable. Remember, simplifying the factors and finding a common denominator are key steps in working efficiently with rational expressions.