To write the expression
[tex]\[ \frac{7}{v^2-36} - \frac{9}{v^2-2v-48} - \frac{7v}{v^2-14v+48} \][/tex]
as a single fraction and then simplify it, let's follow these steps:
### Step 1: Factor the denominators
First, we need to factor the quadratic expressions in the denominators.
1. [tex]\( v^2 - 36 \)[/tex]
[tex]\[ v^2 - 36 = (v - 6)(v + 6) \][/tex]
2. [tex]\( v^2 - 2v - 48 \)[/tex]
To factor this, find two numbers that multiply to [tex]\(-48\)[/tex] and add up to [tex]\(-2\)[/tex]. These numbers are [tex]\(6\)[/tex] and [tex]\(-8\)[/tex].
[tex]\[ v^2 - 2v - 48 = (v - 8)(v + 6) \][/tex]
3. [tex]\( v^2 - 14v + 48 \)[/tex]
To factor this, find two numbers that multiply to [tex]\(48\)[/tex] and add up to [tex]\(-14\)[/tex]. These numbers are [tex]\(-6\)[/tex] and [tex]\(-8\)[/tex].
[tex]\[ v^2 - 14v + 48 = (v - 6)(v - 8) \][/tex]
### Step 2: Find the common denominator
The common denominator is the least common multiple (LCM) of [tex]\((v - 6)(v + 6)\)[/tex], [tex]\((v - 8)(v + 6)\)[/tex], and [tex]\((v - 6)(v - 8)\)[/tex]. The LCM is:
[tex]\[ \text{LCM} = (v - 6)(v + 6)(v - 8) \][/tex]
### Step 3: Express each fraction with the common denominator
Next, express each fraction with the common denominator:
1. For [tex]\( \frac{7}{v^2 - 36} \)[/tex]:
[tex]\[ \frac{7}{(v - 6)(v + 6)} = \frac{7(v - 8)}{(v - 6)(v + 6)(v - 8)} \][/tex]
2. For [tex]\( \frac{9}{v^2 - 2v - 48} \)[/tex]:
[tex]\[ \frac{9}{(v - 8)(v + 6)} = \frac{9(v - 6)}{(v - 6)(v + 6)(v - 8)} \][/tex]
3. For [tex]\( \frac{7v}{v^2 - 14v + 48} \)[/tex]:
[tex]\[ \frac{7v}{(v - 6)(v - 8)} = \frac{7v(v + 6)}{(v - 6)(v + 6)(v - 8)} \][/tex]
### Step 4: Combine the fractions
Combine the fractions over the common denominator:
[tex]\[ \frac{7(v - 8) - 9(v - 6) - 7v(v + 6)}{(v - 6)(v + 6)(v - 8)} \][/tex]
### Step 5: Simplify the numerator
Expand and simplify the numerator:
[tex]\[ 7(v - 8) = 7v - 56 \][/tex]
[tex]\[ 9(v - 6) = 9v - 54 \][/tex]
[tex]\[ 7v(v + 6) = 7v^2 + 42v \][/tex]
Combine these:
[tex]\[ (7v - 56) - (9v - 54) - (7v^2 + 42v) \][/tex]
[tex]\[ = 7v - 56 - 9v + 54 - 7v^2 - 42v \][/tex]
[tex]\[ = -7v^2 + (7v - 9v - 42v) + (54 - 56) \][/tex]
[tex]\[ = -7v^2 - 44v - 2 \][/tex]
### Step 6: Combine the simplified numerator and denominator
So the single fraction is:
[tex]\[ \frac{-7v^2 - 44v - 2}{(v - 6)(v + 6)(v - 8)} \][/tex]
### Step 7: Final simplification
Check if the numerator can be factored further:
[tex]\[ -7v^2 - 44v - 2 \][/tex]
This numerator does not factor nicely with integer coefficients, so the final simplified expression is:
[tex]\[ \frac{-7v^2 - 44v - 2}{(v - 6)(v + 6)(v - 8)} \][/tex]
