How To Find The Lowest Common Denominator Of Rational Expressions
In the realm of mathematics, particularly when dealing with rational expressions, the concept of the lowest common denominator (LCD) is paramount. The LCD serves as the cornerstone for performing various operations, such as adding and subtracting fractions with different denominators. In this comprehensive guide, we will embark on a step-by-step journey to unravel the process of determining the LCD of two given rational expressions: $ rac{p+3}{p^2+7 p+10}$ and $ rac{p+5}{p^2+5 p+6}$. By mastering this technique, you will unlock the ability to manipulate and simplify complex algebraic expressions with ease.
The lowest common denominator is the least common multiple (LCM) of the denominators of two or more fractions. When we add or subtract fractions, they must have the same denominator. The LCD is the smallest denominator that all the fractions can share, making it easier to perform these operations. For instance, consider adding the fractions 1/4 and 1/6. The denominators are 4 and 6. The LCM of 4 and 6 is 12, so the LCD is 12. We can rewrite the fractions as 3/12 and 2/12, and then add them to get 5/12. Finding the LCD is crucial for simplifying expressions and solving equations involving fractions, as it allows us to combine terms effectively and avoid unnecessary complications.
Step 1: Factor the Denominators
The first crucial step in determining the LCD is to factor each denominator completely. Factoring allows us to identify the prime factors that constitute each denominator, which is essential for finding the least common multiple. Let's begin by factoring the first denominator, $p^2 + 7p + 10$.
Factoring $p^2 + 7p + 10$
To factor this quadratic expression, we seek two numbers that multiply to 10 and add up to 7. By careful observation, we can identify that 2 and 5 satisfy these conditions. Therefore, we can factor the expression as follows:
Now, let's move on to factoring the second denominator, $p^2 + 5p + 6$.
Factoring $p^2 + 5p + 6$
Similarly, we need to find two numbers that multiply to 6 and add up to 5. The numbers 2 and 3 fulfill these requirements. Thus, we can factor the expression as:
With both denominators now factored, we have:
\frac{p+3}{(p+2)(p+5)}$ and $\frac{p+5}{(p+2)(p+3)}
The next step involves identifying the unique factors present in both denominators.
Step 2: Identify Unique Factors
Having factored the denominators, our next task is to identify all the unique factors that appear in either denominator. This step is crucial because the LCD must include each unique factor raised to its highest power present in any of the denominators. By identifying the unique factors, we ensure that the LCD is divisible by each original denominator, which is a fundamental requirement for performing operations on rational expressions.
Looking at the factored denominators, $(p + 2)(p + 5)$ and $(p + 2)(p + 3)$, we can observe the following factors:
- (p + 2): This factor appears in both denominators.
- (p + 5): This factor appears only in the first denominator.
- (p + 3): This factor appears only in the second denominator.
Therefore, the unique factors are $(p + 2)$, $(p + 5)$, and $(p + 3)$. These factors will form the building blocks of our LCD. We must now ensure that each factor is included in the LCD with the appropriate exponent.
Step 3: Determine the Highest Power of Each Factor
After identifying the unique factors, the next critical step is to determine the highest power to which each factor appears in any of the denominators. This ensures that the LCD is indeed the least common multiple, as it includes each factor only to the extent necessary to be divisible by all original denominators. By considering the highest powers, we avoid including unnecessary factors or exponents, which would result in a common denominator, but not the lowest one.
Examining the factored denominators, we have:
Let's analyze each unique factor:
- (p + 2): This factor appears in both denominators. In the first denominator, it appears as $(p + 2)^1$, and in the second denominator, it also appears as $(p + 2)^1$. Therefore, the highest power of $(p + 2)$ is 1.
- (p + 5): This factor appears only in the first denominator as $(p + 5)^1$. So, the highest power of $(p + 5)$ is 1.
- (p + 3): This factor appears only in the second denominator as $(p + 3)^1$. Thus, the highest power of $(p + 3)$ is 1.
Now that we have determined the highest power of each unique factor, we are ready to construct the LCD.
Step 4: Construct the LCD
With the unique factors and their highest powers identified, we can now construct the LCD. The LCD is simply the product of each unique factor raised to its highest power. This ensures that the LCD is divisible by each of the original denominators, making it the least common denominator.
Recall that we identified the following unique factors and their highest powers:
- (p + 2): Highest power is 1.
- (p + 5): Highest power is 1.
- (p + 3): Highest power is 1.
Therefore, the LCD is the product of these factors raised to their respective powers:
Simplifying, we get:
Thus, the lowest common denominator of the given rational expressions is $(p + 2)(p + 5)(p + 3)$. This result allows us to confidently perform operations such as addition and subtraction on the original rational expressions.
Conclusion
In this comprehensive exploration, we have meticulously dissected the process of finding the lowest common denominator (LCD) of the rational expressions $\frac{p+3}{p^2+7 p+10}$ and $\frac{p+5}{p^2+5 p+6}$. By systematically factoring the denominators, identifying unique factors, determining the highest power of each factor, and constructing the LCD, we have arrived at the solution: $(p + 2)(p + 5)(p + 3)$.
Mastering the concept of the LCD is not merely an academic exercise; it is a fundamental skill that empowers you to manipulate and simplify complex algebraic expressions. The LCD serves as the bedrock for performing operations on rational expressions, enabling you to combine fractions, solve equations, and tackle a wide array of mathematical challenges. By diligently practicing these techniques, you will solidify your understanding and elevate your proficiency in algebra.
Therefore, the correct answer is B) $(p+5)(p+2)(p+3)$.
