How To Find The LCD Of Rational Expressions

In mathematics, particularly when dealing with rational expressions, finding the least common denominator (LCD) is a crucial step in performing operations such as addition and subtraction. The LCD is the smallest multiple that the denominators of a given set of fractions have in common. This article will guide you through the process of finding the LCD for rational expressions, using the example provided: $\frac{3-x}{3 x^2-14 x-5}$ and $\frac{3 x+2}{2 x^2-13 x+15}$.

Understanding the Importance of LCD

The least common denominator plays a vital role when you need to add or subtract fractions. Just as you need a common denominator to combine numerical fractions, you also need a common denominator to combine rational expressions. The LCD ensures that the fractions have the same base, allowing you to combine the numerators correctly. It simplifies the process and helps avoid errors. In essence, finding the LCD is a foundational skill for manipulating and simplifying rational expressions, making it essential for more advanced algebraic operations and problem-solving.

Step-by-Step Guide to Finding the LCD

1. Factor the Denominators Completely

The first and most critical step in finding the LCD is to factor each denominator completely. Factoring breaks down the polynomials into their simplest multiplicative components, which helps identify common factors and build the LCD. For the given expressions, we have the denominators $3x^2 - 14x - 5$ and $2x^2 - 13x + 15$. Factoring these quadratic expressions involves finding two binomials that multiply to give the original quadratic.

Let's factor the first denominator: $3x^2 - 14x - 5$.

To factor this quadratic, we look for two numbers that multiply to the product of the leading coefficient (3) and the constant term (-5), which is -15, and add up to the middle coefficient (-14). These numbers are -15 and 1. We can rewrite the middle term using these numbers:

$3x^2 - 14x - 5 = 3x^2 - 15x + x - 5$

Now, we factor by grouping:

$3x^2 - 15x + x - 5 = 3x(x - 5) + 1(x - 5)$

Factoring out the common binomial $(x - 5)$, we get:

$3x(x - 5) + 1(x - 5) = (3x + 1)(x - 5)$

So, the factored form of the first denominator is $(3x + 1)(x - 5)$.

Now, let's factor the second denominator: $2x^2 - 13x + 15$.

Similarly, we need two numbers that multiply to the product of the leading coefficient (2) and the constant term (15), which is 30, and add up to the middle coefficient (-13). These numbers are -10 and -3. Rewriting the middle term using these numbers gives:

$2x^2 - 13x + 15 = 2x^2 - 10x - 3x + 15$

Factoring by grouping:

$2x^2 - 10x - 3x + 15 = 2x(x - 5) - 3(x - 5)$

Factoring out the common binomial $(x - 5)$, we get:

$2x(x - 5) - 3(x - 5) = (2x - 3)(x - 5)$

Thus, the factored form of the second denominator is $(2x - 3)(x - 5)$.

2. Identify Unique Factors

Once the denominators are completely factored, the next step is to identify all the unique factors present in both expressions. This involves looking at each factored denominator and noting down each distinct factor. If a factor appears multiple times in any single denominator, you only need to include it once when compiling the list of unique factors. For the factored denominators $(3x + 1)(x - 5)$ and $(2x - 3)(x - 5)$, we identify the following unique factors:

• $(3x + 1)$
• $(x - 5)$
• $(2x - 3)$

Each of these factors is a distinct component that must be considered when constructing the LCD. Recognizing these unique factors is crucial because the LCD must include each factor to ensure that it is divisible by both original denominators. This step sets the stage for building the LCD by ensuring that all necessary components are accounted for.

3. Determine the Highest Power of Each Unique Factor

After identifying the unique factors, it’s essential to determine the highest power of each factor that appears in any of the denominators. This is crucial because the LCD must be divisible by each denominator, and including the highest power ensures this condition is met. For the given expressions, the factored denominators are $(3x + 1)(x - 5)$ and $(2x - 3)(x - 5)$.

• The factor $(3x + 1)$ appears once in the first denominator and not at all in the second. Thus, the highest power of $(3x + 1)$ is 1.
• The factor $(x - 5)$ appears once in both denominators. Therefore, the highest power of $(x - 5)$ is 1.
• The factor $(2x - 3)$ appears once in the second denominator and not at all in the first. Consequently, the highest power of $(2x - 3)$ is 1.

In this case, each unique factor appears with a power of 1. However, in other scenarios, some factors might appear multiple times within a single denominator (e.g., $(x + 2)^2$). In such cases, it’s necessary to take the highest exponent of the factor to ensure the LCD can accommodate each denominator’s structure. Determining the highest power for each unique factor is a critical step in constructing an accurate LCD, setting the foundation for simplifying and performing operations on rational expressions.

4. Construct the LCD

The final step in finding the least common denominator (LCD) is to construct it by multiplying together each unique factor raised to its highest power. This ensures that the LCD is the smallest expression that is divisible by all the original denominators. For the expressions with denominators factored as $(3x + 1)(x - 5)$ and $(2x - 3)(x - 5)$, we identified the unique factors as $(3x + 1)$, $(x - 5)$, and $(2x - 3)$, each with a highest power of 1.

To construct the LCD, we multiply these factors together:

LCD = $(3x + 1)^1 * (x - 5)^1 * (2x - 3)^1$

This simplifies to:

LCD = $(3x + 1)(x - 5)(2x - 3)$

Therefore, the LCD for the given rational expressions is $(3x + 1)(x - 5)(2x - 3)$. This expression is the smallest common multiple of the denominators and can be used to combine the rational expressions when adding or subtracting them. By systematically following these steps—factoring, identifying unique factors, determining highest powers, and constructing the LCD—you can efficiently find the LCD for any set of rational expressions, facilitating further algebraic manipulation and simplification.

Answer

Therefore, the LCD for the given rational expressions $\frac{3-x}{3 x^2-14 x-5}$ and $\frac{3 x+2}{2 x^2-13 x+15}$ is:

$(3x + 1)(x - 5)(2x - 3)$