What Is The Numerator Of The Simplified Sum?$\[ \frac{x}{x^2+3x+2}+\frac{3}{x+1} \\]A. \[$x+3\$\] B. \[$3x+6\$\] C. \[$4x+6\$\] D. \[$4x+2\$\]

Introduction

When dealing with algebraic expressions, simplifying fractions is a crucial step in solving equations and inequalities. In this article, we will explore the process of simplifying a sum of two fractions and finding the numerator of the resulting expression. We will use the given expression $\frac{x}{x^2+3x+2}+\frac{3}{x+1}$ as an example and walk through the steps to simplify it.

Understanding the Expression

The given expression consists of two fractions: $\frac{x}{x^2+3x+2}$ and $\frac{3}{x+1}$. To simplify the sum, we need to find a common denominator for both fractions. The common denominator is the least common multiple (LCM) of the denominators of both fractions.

Finding the Common Denominator

To find the LCM of $x^2+3x+2$ and $x+1$, we can factor both expressions:

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

$x+1$ is already factored.

The LCM of $(x+1)(x+2)$ and $x+1$ is $(x+1)(x+2)$.

Simplifying the Expression

Now that we have the common denominator, we can rewrite both fractions with the common denominator:

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

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

Combining the Fractions

Now that both fractions have the same denominator, we can combine them by adding the numerators:

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

Simplifying the Numerator

The numerator of the simplified expression is $x+3(x+2)$. We can simplify this expression by distributing the 3:

$x+3(x+2) = x+3x+6$

Conclusion

The numerator of the simplified sum is $x+3x+6$, which can be further simplified to $4x+6$. Therefore, the correct answer is:

The final answer is C. $4x+6$

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Factor the denominators of both fractions: $x^2+3x+2 = (x+1)(x+2)$ and $x+1$ is already factored.
2. Find the LCM of the denominators: The LCM of $(x+1)(x+2)$ and $x+1$ is $(x+1)(x+2)$.
3. Rewrite both fractions with the common denominator: $\frac{x}{(x+1)(x+2)}$ and $\frac{3(x+2)}{(x+1)(x+2)}$.
4. Combine the fractions by adding the numerators: $\frac{x+3(x+2)}{(x+1)(x+2)}$.
5. Simplify the numerator: $x+3(x+2) = x+3x+6$.
6. Simplify the expression: $\frac{x+3x+6}{(x+1)(x+2)} = \frac{4x+6}{(x+1)(x+2)}$.

Final Answer

The final answer is C. $4x+6$.

Introduction

In our previous article, we explored the process of simplifying a sum of two fractions and finding the numerator of the resulting expression. We used the given expression $\frac{x}{x^2+3x+2}+\frac{3}{x+1}$ as an example and walked through the steps to simplify it. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the common denominator of the two fractions?

A: The common denominator is the least common multiple (LCM) of the denominators of both fractions. In this case, the LCM of $x^2+3x+2$ and $x+1$ is $(x+1)(x+2)$.

Q: How do I find the LCM of two expressions?

A: To find the LCM of two expressions, you can factor both expressions and then multiply the highest power of each factor that appears in either expression.

Q: What is the numerator of the simplified expression?

A: The numerator of the simplified expression is $x+3(x+2)$, which can be simplified to $4x+6$.

Q: Can I simplify the expression further?

A: Yes, the expression $\frac{4x+6}{(x+1)(x+2)}$ can be simplified further by factoring the numerator: $4x+6 = 2(2x+3)$. However, this is not necessary to find the numerator of the simplified sum.

Q: What is the final answer?

A: The final answer is C. $4x+6$.

Q: Can I use a different method to simplify the expression?

A: Yes, you can use a different method to simplify the expression, such as using a calculator or a computer algebra system. However, the steps outlined in this article provide a clear and concise method for simplifying the expression.

Q: What is the importance of simplifying expressions?

A: Simplifying expressions is an important step in solving equations and inequalities. By simplifying expressions, you can make it easier to solve for the unknown variable and find the solution to the equation or inequality.

Conclusion

In this article, we answered some frequently asked questions related to the topic of simplifying a sum of two fractions and finding the numerator of the resulting expression. We hope that this article has provided a clear and concise explanation of the process and has helped to clarify any confusion.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Factor the denominators of both fractions: $x^2+3x+2 = (x+1)(x+2)$ and $x+1$ is already factored.
2. Find the LCM of the denominators: The LCM of $(x+1)(x+2)$ and $x+1$ is $(x+1)(x+2)$.
3. Rewrite both fractions with the common denominator: $\frac{x}{(x+1)(x+2)}$ and $\frac{3(x+2)}{(x+1)(x+2)}$.
4. Combine the fractions by adding the numerators: $\frac{x+3(x+2)}{(x+1)(x+2)}$.
5. Simplify the numerator: $x+3(x+2) = x+3x+6$.
6. Simplify the expression: $\frac{x+3x+6}{(x+1)(x+2)} = \frac{4x+6}{(x+1)(x+2)}$.

Final Answer

The final answer is C. $4x+6$.

Additional Resources

• Simplifying Expressions
• Least Common Multiple
• Factoring Expressions

Related Articles

• Simplifying a Sum of Two Fractions
• Finding the Numerator of a Simplified Expression
• Solving Equations and Inequalities