Problem 76 - Elementary Algebra

Question

For the following problems, convert the given rational expressions to rational expressions having the same denominators. $$ \frac{2}{x+5}, \frac{4}{x-5} $$

Step-by-Step Solution

Answer

Question: Convert the given rational expressions to have the same denominator and write the final expressions: $\frac{2}{x+5}$ and $\frac{4}{x-5}$. Answer: The converted expressions with the same denominator are: $$\frac{2(x-5)}{(x+5)(x-5)} \quad \text{and} \quad \frac{4(x+5)}{(x+5)(x-5)}.$$

Step 1: Find the LCM of the denominators

To find the LCM of \((x+5)\) and \((x-5)\), we can simply multiply them together since they are distinct linear factors and don't share any common factor except 1. Therefore, the LCM is \((x+5)(x-5)\).

Step 2: Convert the first fraction

To convert the first fraction \(\frac{2}{x+5}\), we will have to multiply both the numerator and the denominator by the missing factor of the LCM which is \((x-5)\). So we get: $$ \frac{2}{x+5} \times \frac{x-5}{x-5} = \frac{2(x-5)}{(x+5)(x-5)} $$

Step 3: Convert the second fraction

To convert the second fraction \(\frac{4}{x-5}\), we will have to multiply both the numerator and the denominator by the missing factor of the LCM which is \((x+5)\). So we get: $$ \frac{4}{x-5} \times \frac{x+5}{x+5} = \frac{4(x+5)}{(x+5)(x-5)} $$

Step 4: Write the final expressions

Now we have both fractions written with the same denominator \((x+5)(x-5)\). The converted expressions are: $$ \frac{2(x-5)}{(x+5)(x-5)} \quad \text{and} \quad \frac{4(x+5)}{(x+5)(x-5)} $$

Key Concepts

LCM (Least Common Multiple)

Understanding the concept of Least Common Multiple (LCM) is crucial when working with rational expressions. In mathematics, the LCM of two or more numbers or expressions is the smallest multiple that is evenly divisible by each of the numbers or expressions. This concept helps us when we aim to add or subtract fractions that have different denominators.

In the context of rational expressions, the denominators can be polynomial expressions rather than simple numbers. This can initially seem intimidating, but the process is quite similar to finding the LCM of numerical values.

• Identify each denominator: This could be just a number or a more complex polynomial expression.
• Determine if there are any common factors. In rational expressions, these are typically factors of polynomials.
• If there are no common factors, you simply multiply the denominators together to get the LCM.

In our example, since (x+5) and (x-5) are distinct and don't share factors, their LCM is simply the product (x+5)(x-5) . This allows us to rewrite the fractions with this common denominator, facilitating their addition or subtraction later.

Fractions

Understanding fractions is key when dealing with rational expressions. A fraction consists of a numerator, which is the top part, and a denominator, which is the bottom part. Fractions essentially represent division of the numerator by the denominator.

In algebra, fractions can also feature variables, and these are known as rational expressions. For example, \( \frac{2}{x+5} \) is a fraction where the denominator is a variable expression. To work effectively with such fractions, especially when adding or subtracting them, we need to convert them to equivalent fractions with a common denominator.

• Find the LCM of the denominators if they are different.
• Adjust the original fractions by multiplying the numerator and denominator by the missing factor from the LCM so each fraction has the same denominator.

By doing this, we handle both arithmetic fractions and algebraic fractions in a similar manner, facilitating operations upon them. In the provided exercise, converting each rational expression to have the same denominator involved multiplying each original fraction by another fraction that equals 1 which ensures that their values remain equal.

Algebra

Algebra is foundational to understanding rational expressions and manipulating them effectively. In algebra, we solve problems by finding unknown values, often represented by variables such as x . Rational expressions are a type of algebraic expression that involve ratios of polynomials.

The algebraic operations, such as addition, subtraction, multiplication, and division apply to rational expressions, much like they do with ordinary numbers. The key in mixing algebra with fractions is to maintain balance — what you do to the numerator, you must also do to the denominator.

• Converting rational expressions to a common denominator often involves multiplying by a form of 1 that's expressed in terms of necessary factors.
• Simplification ensures that we handle the simplest form of an expression for ease of further operations.

In algebra, recognizing distinct linear factors, like (x+5) and (x-5) , helps us navigate the expressions more smoothly. The exercise shows how we apply these principles to ensure both fractions share a common denominator, streamlining them for any further algebraic manipulation.