Let's go through the solution step-by-step to solve the problem and find the least common denominator (LCD) for the following fractions:
[tex]\[ \frac{2}{x^2 - 3x - 4} + \frac{3}{x^2 - 6x + 8} \][/tex]
### Step 1: Factor each denominator
1. Factoring [tex]\(x^2 - 3x - 4\)[/tex]:
[tex]\(x^2 - 3x - 4\)[/tex] can be factored as:
[tex]\[ x^2 - 3x - 4 = (x - 4)(x + 1) \][/tex]
2. Factoring [tex]\(x^2 - 6x + 8\)[/tex]:
[tex]\(x^2 - 6x + 8\)[/tex] can be factored as:
[tex]\[ x^2 - 6x + 8 = (x - 4)(x - 2) \][/tex]
So the factored forms are:
[tex]\[ x^2 - 3x - 4 = (x - 4)(x + 1) \][/tex]
[tex]\[ x^2 - 6x + 8 = (x - 4)(x - 2) \][/tex]
### Step 2: Identify the least common denominator (LCD)
To determine the LCD, we need to include all unique factors from both denominators. The factors from both expressions are:
[tex]\[ (x - 4)(x + 1) \][/tex]
[tex]\[ (x - 4)(x - 2) \][/tex]
The unique factors from these are:
[tex]\[ \{x - 4, x + 1, x - 2\} \][/tex]
Hence, the least common denominator (LCD) will be:
[tex]\[ (x - 4)(x + 1)(x - 2) \][/tex]
### Step 3: Write the complete expression with the LCD
Now that we have the LCD, we rewrite each fraction with this common denominator:
[tex]\[ \frac{2}{(x - 4)(x + 1)} + \frac{3}{(x - 4)(x - 2)} \][/tex]
First, adjust the numerators to reflect the common denominator.
1. The first fraction:
[tex]\[ \frac{2}{(x - 4)(x + 1)} = \frac{2(x - 2)}{(x - 4)(x + 1)(x - 2)} = \frac{2x - 4}{(x - 4)(x + 1)(x - 2)} \][/tex]
2. The second fraction:
[tex]\[ \frac{3}{(x - 4)(x - 2)} = \frac{3(x + 1)}{(x - 4)(x + 1)(x - 2)} = \frac{3x + 3}{(x - 4)(x + 1)(x - 2)} \][/tex]
Combine the fractions:
[tex]\[ \frac{2x - 4 + 3x + 3}{(x - 4)(x + 1)(x - 2)} \][/tex]
[tex]\[ \frac{(2x + 3x) + (-4 + 3)}{(x - 4)(x + 1)(x - 2)} \][/tex]
[tex]\[ \frac{5x - 1}{(x - 4)(x + 1)(x - 2)} \][/tex]
### Final Answer:
The sum of the numerators is [tex]\(5x - 1\)[/tex], and the least common denominator is:
[tex]\[ \frac{5x - 1}{(x - 4)(x + 1)(x - 2)} \][/tex]
Therefore, the complete expression written with the LCD is:
[tex]\[ \frac{5x - 1}{(x - 4)(x + 1)(x - 2)} \][/tex]
