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

by ADMIN 225 views

Understanding the Problem

The given problem involves finding the numerator of the simplified sum of two fractions. The fractions are xx2+3x+2\frac{x}{x^2+3x+2} and 3x+1\frac{3}{x+1}. To find the numerator of the simplified sum, we need to first simplify the given fractions and then add them together.

Step 1: Factorize the Denominators

The first step is to factorize the denominators of the given fractions. The denominator of the first fraction is x2+3x+2x^2+3x+2, which can be factorized as (x+1)(x+2)(x+1)(x+2). The denominator of the second fraction is x+1x+1, which is already in its simplest form.

# Factorization of Denominators
## First Fraction
### Denominator: $x^2+3x+2$
#### Factorized Form: $(x+1)(x+2)$

Second Fraction


Denominator: x+1x+1


Factorized Form: x+1x+1

Step 2: Simplify the Fractions

Now that we have factorized the denominators, we can simplify the fractions. The first fraction can be simplified as x(x+1)(x+2)\frac{x}{(x+1)(x+2)}, and the second fraction remains as 3x+1\frac{3}{x+1}.

# Simplification of Fractions
## First Fraction
### Simplified Form: $\frac{x}{(x+1)(x+2)}$

Second Fraction


Simplified Form: 3x+1\frac{3}{x+1}

Step 3: Find the Common Denominator

To add the two fractions together, we need to find the common denominator. The common denominator is the product of the two denominators, which is (x+1)(x+2)(x+1)(x+2).

# Finding the Common Denominator
## Common Denominator: $(x+1)(x+2)$

Step 4: Add the Fractions

Now that we have the common denominator, we can add the fractions together. The numerator of the first fraction is xx, and the numerator of the second fraction is 33. To add the fractions, we need to multiply the numerator of the second fraction by the factor that will make its denominator equal to the common denominator.

# Adding the Fractions
## Numerator of First Fraction: $x$
## Numerator of Second Fraction: $3$
## Common Denominator: $(x+1)(x+2)$
## Factor to Multiply Second Fraction: $(x+2)$
## Numerator of Second Fraction after Multiplication: $3(x+2)$
## Numerator of Sum: $x + 3(x+2)$
## Simplified Numerator: $x + 3x + 6$
## Final Simplified Numerator: $4x + 6$

Conclusion

The numerator of the simplified sum is 4x+64x+6. This is the final answer to the problem.

Answer Options

The answer options are:

A. x+3x+3 B. 3x+63x+6 C. 4x+64x+6 D. 4x+24x+2

Q: What is the numerator of the simplified sum?

A: The numerator of the simplified sum is 4x+64x+6. This is the final answer to the problem.

Q: How do I simplify the given fractions?

A: To simplify the given fractions, you need to factorize the denominators and then simplify the fractions. The first fraction can be simplified as x(x+1)(x+2)\frac{x}{(x+1)(x+2)}, and the second fraction remains as 3x+1\frac{3}{x+1}.

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

A: The common denominator of the two fractions is (x+1)(x+2)(x+1)(x+2).

Q: How do I add the fractions together?

A: To add the fractions together, you need to multiply the numerator of the second fraction by the factor that will make its denominator equal to the common denominator. The numerator of the first fraction is xx, and the numerator of the second fraction is 33. To add the fractions, you need to multiply the numerator of the second fraction by (x+2)(x+2).

Q: What is the numerator of the sum after adding the fractions together?

A: The numerator of the sum after adding the fractions together is x+3(x+2)x + 3(x+2), which simplifies to 4x+64x+6.

Q: What are the answer options for the problem?

A: The answer options for the problem are:

A. x+3x+3 B. 3x+63x+6 C. 4x+64x+6 D. 4x+24x+2

Q: Which answer option is correct?

A: The correct answer option is C. 4x+64x+6.

Q: Why is the numerator of the simplified sum 4x+64x+6?

A: The numerator of the simplified sum is 4x+64x+6 because when you add the two fractions together, the numerator of the first fraction is xx, and the numerator of the second fraction is 3(x+2)3(x+2), which simplifies to 3x+63x+6. When you add the two numerators together, you get x+3x+6x + 3x + 6, which simplifies to 4x+64x+6.

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

A: Yes, you can use a different method to simplify the sum. One method is to find the least common multiple (LCM) of the two denominators and then rewrite the fractions with the LCM as the denominator. However, the method used in this article is a more straightforward approach.

Q: What is the importance of simplifying fractions?

A: Simplifying fractions is important because it makes it easier to add, subtract, multiply, and divide fractions. When fractions are simplified, it is easier to see the relationships between the numerators and denominators, which can make it easier to solve problems involving fractions.

Q: Can I apply the same method to simplify other fractions?

A: Yes, you can apply the same method to simplify other fractions. The method involves finding the common denominator, adding the fractions together, and simplifying the result. However, you may need to adjust the method depending on the specific fractions you are working with.