Problem 55 - Elementary and Intermediate Algebra

Question

Find the LCD of pair of rational expressions. \(\frac{-2 x}{x^{2}-1}, \frac{5 x}{x+1}\)

Step-by-Step Solution

Answer

The LCD is \(x^2 - 1\).

Step 1: Factor Denominators

First, factor the denominators of each rational expression. The denominator \(x^2 - 1\) can be factored as \((x-1)(x+1)\) using the difference of squares formula. The denominator of the second expression is already in its simplest form, \(x+1\).

Step 2: Identify Unique Factors

Determine the unique factors from the denominators. From \((x-1)(x+1)\), the factors are \(x-1\) and \(x+1\). From the second denominator, the factor is \(x+1\). Thus, the unique factors that will constitute the LCD are \(x-1\) and \(x+1\).

Step 3: Construct the LCD

Multiply all the unique factors to construct the LCD. For these rational expressions, multiply \((x-1)\) and \((x+1)\) to get the least common denominator: \((x-1)(x+1)\).

Step 4: Present the LCD

Since \((x-1)(x+1)\) was already factored in Step 1 as \(x^2 - 1\), the LCD can also be expressed as \(x^2 - 1\). Thus, the least common denominator for the given expressions is \(x^2 - 1\).

Key Concepts

Rational Expressions

A rational expression is similar to a fraction, but instead of just numbers in the numerator and denominator, it can include variables and algebraic expressions. These are helpful when solving equations that involve polynomials. For example, the expression \( \frac{-2x}{x^2-1} \) is a rational expression. Here, \(-2x\) is the numerator, and \(x^2-1\) is the polynomial in the denominator.

• Rational expressions must have non-zero denominators; otherwise, they become undefined.
• When working with multiple rational expressions, it's crucial to find common denominators to add, subtract, or compare them.

Recognizing rational expressions is the first step toward handling complex algebraic fractions effectively.

Factoring Denominators

Factoring the denominators is an essential step in simplifying rational expressions and finding a common denominator. Factoring involves breaking down a polynomial into simpler expressions that, when multiplied together, give the original polynomial. In our example:

• The denominator \(x^2 - 1\) can be factored as \((x-1)(x+1)\) using the difference of squares formula.
• The second expression has a denominator of \(x+1\), which is already factored.

Always look for patterns like difference of squares, perfect square trinomials, or common factors to make factoring easier. This step is crucial for simplifying rational expressions and combining them under a single denominator.

Difference of Squares

The difference of squares is a particular pattern of factoring that is very useful when dealing with polynomial expressions. It applies when you have a subtraction (")minus" -) between two perfect squares. The formula is:
\(a^2 - b^2 = (a-b)(a+b)\)

• In our example, the expression \(x^2 - 1\) fits this pattern because \(x^2\) is a perfect square and \(1\) is \((1)^2\).
• Using the difference of squares formula, we factor \(x^2 - 1\) as \((x-1)(x+1)\).

Recognizing this pattern makes it much easier to factor polynomials and simplify expressions. It is one of the fundamental identities in algebra used frequently in rational expressions.

Unique Factors

Identifying unique factors is vital in finding the Least Common Denominator (LCD) of rational expressions. The unique factors are the distinct building blocks that appear in the factored forms of all denominators involved. To find them, follow these steps:

• List out each factored form of the expressions involved. From \(x^2-1 = (x-1)(x+1)\), we get factors \(x-1\) and \(x+1\).
• The second denominator \(x+1\) is already simplified.
• Identify and combine all distinct (unique) factors from these lists to construct the LCD.

In the given problem, \(x-1\) and \(x+1\) are the unique factors needed for the LCD. Consolidating these yields the overall least common denominator. Recognizing unique factors helps in consolidating denominators and simplifies complex algebraic expressions.