Finding The Numerator Of A Simplified Sum A Step-by-Step Guide

What is the numerator after simplifying the sum of the fractions ${\frac{x}{x^2+3 x+2} + \frac{3}{x+1}}$?

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Introduction

In the realm of mathematics, simplifying expressions involving fractions is a fundamental skill. This article delves into the process of simplifying the sum of two algebraic fractions: ${\frac{x}{x^2+3 x+2} + \frac{3}{x+1}}$. Our primary goal is to identify the numerator of the simplified sum. This involves several steps, including factoring polynomials, finding common denominators, combining fractions, and simplifying the resulting expression. We will explore each of these steps in detail to provide a comprehensive understanding of how to solve this type of problem.

Factoring the Denominator

The first step in simplifying the given expression is to factor the denominator of the first fraction, which is ${x^2 + 3x + 2}$. Factoring this quadratic expression will help us identify common factors between the two fractions, making it easier to find a common denominator. To factor ${x^2 + 3x + 2}$, we look for two numbers that multiply to 2 and add up to 3. These numbers are 1 and 2. Therefore, we can factor the quadratic expression as follows:

${x^2 + 3x + 2 = (x+1)(x+2)}$

This factorization is crucial because it reveals that ${(x+1)}$ is a factor of the denominator, which is also the denominator of the second fraction. This observation will simplify the process of finding a common denominator.

Finding a Common Denominator

Now that we have factored the denominator of the first fraction, we can rewrite the original expression as:

${\frac{x}{(x+1)(x+2)} + \frac{3}{x+1}}$

To add these two fractions, we need a common denominator. The least common denominator (LCD) is the smallest expression that is divisible by both denominators. In this case, the denominators are ${(x+1)(x+2)}$ and ${(x+1)}$. The LCD is therefore ${(x+1)(x+2)}$. To get the second fraction to have the LCD, we multiply both the numerator and the denominator by ${(x+2)}$:

${\frac{3}{x+1} \times \frac{x+2}{x+2} = \frac{3(x+2)}{(x+1)(x+2)}}$

Now both fractions have the same denominator, allowing us to add them.

Combining the Fractions

With the common denominator in place, we can combine the two fractions by adding their numerators:

${\frac{x}{(x+1)(x+2)} + \frac{3(x+2)}{(x+1)(x+2)} = \frac{x + 3(x+2)}{(x+1)(x+2)}}$

Next, we simplify the numerator by distributing the 3 and combining like terms:

${\frac{x + 3x + 6}{(x+1)(x+2)} = \frac{4x + 6}{(x+1)(x+2)}}$

Now we have a single fraction, but we still need to simplify it further.

Simplifying the Expression

To further simplify the expression, we look for common factors in the numerator and the denominator. The numerator is ${4x + 6}$, which can be factored by taking out a common factor of 2:

${4x + 6 = 2(2x + 3)}$

So our expression becomes:

${\frac{2(2x + 3)}{(x+1)(x+2)}}$

At this point, we check if there are any common factors between the numerator and the denominator that can be canceled out. In this case, there are no common factors, so the expression is in its simplest form.

Identifying the Numerator

The final simplified expression is:

${\frac{2(2x + 3)}{(x+1)(x+2)}}$

The numerator of this simplified sum is ${2(2x + 3)}$, which can also be written as ${4x + 6}$.

Detailed Steps for Solving the Problem

To better illustrate the process, let’s break down the steps involved in solving the problem and provide detailed explanations for each.

Step 1: Factor the Denominator

The first step involves factoring the denominator of the first fraction, which is ${x^2 + 3x + 2}$. Factoring this quadratic expression is essential for finding a common denominator and simplifying the expression. We need to find two numbers that multiply to 2 and add up to 3. These numbers are 1 and 2. Therefore, we can factor the quadratic expression as:

${x^2 + 3x + 2 = (x+1)(x+2)}$

This factorization allows us to rewrite the original expression as:

${\frac{x}{(x+1)(x+2)} + \frac{3}{x+1}}$

The factored form of the denominator makes it clear that ${(x+1)}$ is a common factor, which will simplify the next step.

Step 2: Find a Common Denominator

To add the two fractions, we need a common denominator. The least common denominator (LCD) is the smallest expression that is divisible by both denominators. In this case, the denominators are ${(x+1)(x+2)}$ and ${(x+1)}$. The LCD is ${(x+1)(x+2)}$. To make the second fraction have the same denominator as the first, we multiply both the numerator and the denominator of the second fraction by ${(x+2)}$:

${\frac{3}{x+1} \times \frac{x+2}{x+2} = \frac{3(x+2)}{(x+1)(x+2)}}$

Now both fractions have the same denominator:

${\frac{x}{(x+1)(x+2)} + \frac{3(x+2)}{(x+1)(x+2)}}$

This step ensures that we can combine the fractions by adding their numerators.

Step 3: Combine the Fractions

With the common denominator in place, we can combine the fractions by adding their numerators:

${\frac{x}{(x+1)(x+2)} + \frac{3(x+2)}{(x+1)(x+2)} = \frac{x + 3(x+2)}{(x+1)(x+2)}}$

Next, we simplify the numerator by distributing the 3 and combining like terms:

${\frac{x + 3x + 6}{(x+1)(x+2)} = \frac{4x + 6}{(x+1)(x+2)}}$

This step results in a single fraction with a simplified numerator.

Step 4: Simplify the Expression

To further simplify the expression, we look for common factors in the numerator and the denominator. The numerator is ${4x + 6}$, which can be factored by taking out a common factor of 2:

${4x + 6 = 2(2x + 3)}$

So our expression becomes:

${\frac{2(2x + 3)}{(x+1)(x+2)}}$

We now check if there are any common factors between the numerator and the denominator that can be canceled out. In this case, there are no common factors, so the expression is in its simplest form.

Step 5: Identify the Numerator

The final simplified expression is:

${\frac{2(2x + 3)}{(x+1)(x+2)}}$

The numerator of this simplified sum is ${2(2x + 3)}$, which can also be written as ${4x + 6}$. Therefore, the numerator of the simplified sum is ${4x + 6}$.

Common Mistakes to Avoid

When simplifying algebraic fractions, there are several common mistakes that students often make. Being aware of these mistakes can help you avoid them and ensure you arrive at the correct solution. Here are some common pitfalls to watch out for:

Mistake 1: Not Factoring Denominators Correctly

Factoring denominators is a crucial first step in simplifying algebraic fractions. A common mistake is to incorrectly factor a quadratic expression or to miss a factor altogether. For example, when factoring ${x^2 + 3x + 2}$, students might incorrectly factor it as ${(x+3)(x-1)}$ or miss the factorization entirely. Always double-check your factorization by multiplying the factors back together to ensure you get the original expression.

Mistake 2: Incorrectly Finding the Least Common Denominator (LCD)

Finding the least common denominator (LCD) is essential for adding or subtracting fractions. A common mistake is to simply multiply the denominators together without considering whether there are common factors. This can lead to a more complex expression than necessary. In our example, the denominators are ${(x+1)(x+2)}$ and ${(x+1)}$. The LCD is ${(x+1)(x+2)}$, not ${(x+1)(x+2)(x+1)}$. Always look for the smallest expression that is divisible by both denominators.

Mistake 3: Forgetting to Distribute

When combining fractions, you often need to distribute a term across multiple terms in the numerator. For example, in our problem, we had to distribute the 3 in the expression ${3(x+2)}$. Forgetting to distribute can lead to an incorrect numerator. Always ensure you distribute correctly and write out each step to avoid errors.

Mistake 4: Incorrectly Combining Like Terms

After combining the numerators, it’s important to combine like terms correctly. A common mistake is to add or subtract terms that are not like terms. For example, in the expression ${x + 3x + 6}$, only ${x}$ and ${3x}$ can be combined to get ${4x}$. The constant term 6 cannot be combined with these terms. Always ensure you are only combining terms with the same variable and exponent.

Mistake 5: Not Simplifying the Final Expression

Simplifying the final expression is crucial for arriving at the correct answer. A common mistake is to stop after combining the fractions without looking for common factors to cancel out. In our example, we factored the numerator to see if there were any common factors with the denominator. Always check for common factors and simplify the expression as much as possible.

Mistake 6: Canceling Terms Incorrectly

Students sometimes make the mistake of canceling terms that are not factors. You can only cancel factors that are multiplied, not terms that are added or subtracted. For example, in the expression ${\frac{4x + 6}{(x+1)(x+2)}}$, you cannot cancel the 6 with the 2 in ${(x+2)}$ because 6 is not a factor of the entire numerator. Always factor first and then cancel common factors.

Conclusion

In conclusion, simplifying the sum of algebraic fractions involves several key steps: factoring denominators, finding a common denominator, combining fractions, and simplifying the resulting expression. In the example ${\frac{x}{x^2+3 x+2} + \frac{3}{x+1}}$, we found that the numerator of the simplified sum is ${2(2x + 3)}$ or ${4x + 6}$. By following these steps carefully and avoiding common mistakes, you can confidently simplify algebraic fractions and solve related problems. Mastering these skills is essential for success in algebra and beyond.

By understanding each step and the underlying principles, you can tackle similar problems with confidence. Remember to practice regularly and pay attention to detail to avoid common mistakes. With a solid grasp of these concepts, you’ll be well-prepared to handle more complex algebraic manipulations.