Question Find the numerator of the simplified expression: $\frac{x}{x^{2}+3 x+2}+\frac{3}{x+1}$ .

Studdy Solution

STEP 1

Assumptions
1. We are given two fractions to add: $\frac{x}{x^{2}+3x+2}$ and $\frac{3}{x+1}$ .
2. We need to find a common denominator to combine the fractions.
3. After finding the common denominator, we will combine the numerators.
4. We will simplify the resulting fraction if possible.
5. The goal is to find the numerator of the simplified sum.

STEP 2

First, we need to factor the denominator of the first fraction to see if it has any factors in common with the denominator of the second fraction.
$x^{2}+3x+2 = (x+1)(x+2)$

STEP 3

Now we have the factorization of the first denominator, we can see that $x+1$ is a common factor with the second denominator.

STEP 4

The least common denominator (LCD) between $(x+1)(x+2)$ and $x+1$ is $(x+1)(x+2)$ .

STEP 5

We need to express the second fraction with the common denominator. Since the denominator $x+1$ is already a factor of the LCD, we only need to multiply the numerator by the missing factor which is $(x+2)$ .
$\frac{3}{x+1} = \frac{3(x+2)}{(x+1)(x+2)}$

STEP 6

Now that both fractions have the common denominator, we can add them together.
$\frac{x}{(x+1)(x+2)} + \frac{3(x+2)}{(x+1)(x+2)}$

STEP 7

Combine the numerators over the common denominator.
$\frac{x + 3(x+2)}{(x+1)(x+2)}$

STEP 8

Distribute the 3 in the numerator to simplify the expression.
$\frac{x + 3x + 6}{(x+1)(x+2)}$

STEP 9

Combine like terms in the numerator.
$\frac{4x + 6}{(x+1)(x+2)}$

STEP 10

The numerator of the simplified sum is $4x + 6$ .
The numerator of the simplified sum is $4x + 6$ .

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