If `(x-4)/(x^(2)-5x+6)` can be expanded in the ascending powers of `x`, then the coefficient of `x^(3)` is

Options

A. `(-73)/(648)`
B. `(73)/(648)`
C. `(71)/(648)`
D. `(-71)/(648)`

Answer

To find the coefficient of \( x^3 \) in the expansion of \( \frac{x-4}{x^2 - 5x + 6} \), we will first factor the denominator and then use partial fraction decomposition.

### Step 1: Factor the denominator
The denominator \( x^2 - 5x + 6 \) can be factored as follows:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 2: Set up the partial fraction decomposition
We can express \( \frac{x-4}{(x-2)(x-3)} \) as:
\[ \frac{x-4}{(x-2)(x-3)} = \frac{A}{x-2} + \frac{B}{x-3} \]
where \( A \) and \( B \) are constants to be determined.

### Step 3: Clear the denominators
Multiplying both sides by \( (x-2)(x-3) \) gives:
\[ x - 4 = A(x - 3) + B(x - 2) \]

### Step 4: Expand the right-hand side
Expanding the right-hand side:
\[ x - 4 = Ax - 3A + Bx - 2B \]
Combining like terms:
\[ x - 4 = (A + B)x - (3A + 2B) \]

### Step 5: Set up the system of equations
By equating coefficients, we get the following system of equations:
1. \( A + B = 1 \)
2. \( -3A - 2B = -4 \)

### Step 6: Solve the system of equations
From the first equation, we can express \( B \) in terms of \( A \):
\[ B = 1 - A \]
Substituting this into the second equation:
\[ -3A - 2(1 - A) = -4 \]
Simplifying:
\[ -3A - 2 + 2A = -4 \]
\[ -A - 2 = -4 \]
\[ -A = -2 \implies A = 2 \]
Now substituting \( A \) back to find \( B \):
\[ B = 1 - 2 = -1 \]

### Step 7: Write the partial fractions
Now we can write:
\[ \frac{x-4}{(x-2)(x-3)} = \frac{2}{x-2} - \frac{1}{x-3} \]

### Step 8: Expand each term using the binomial series
We can expand each term using the binomial series:
\[ \frac{2}{x-2} = 2 \cdot \frac{1}{2} \cdot \frac{1}{1 - \frac{x}{2}} = 1 \cdot \sum_{n=0}^{\infty} \left(\frac{x}{2}\right)^n = \sum_{n=0}^{\infty} \frac{x^n}{2^n} \]
\[ \frac{-1}{x-3} = -\frac{1}{3} \cdot \frac{1}{1 - \frac{x}{3}} = -\frac{1}{3} \sum_{n=0}^{\infty} \left(\frac{x}{3}\right)^n = -\frac{1}{3} \sum_{n=0}^{\infty} \frac{x^n}{3^n} \]

### Step 9: Find the coefficient of \( x^3 \)
The coefficient of \( x^3 \) in \( \frac{2}{x-2} \) is:
\[ \frac{1}{2^3} = \frac{1}{8} \]
The coefficient of \( x^3 \) in \( -\frac{1}{x-3} \) is:
\[ -\frac{1}{3} \cdot \frac{1}{3^3} = -\frac{1}{3} \cdot \frac{1}{27} = -\frac{1}{81} \]

### Step 10: Combine the coefficients
Now, we combine the coefficients:
\[ \text{Coefficient of } x^3 = \frac{1}{8} - \frac{1}{81} \]
Finding a common denominator (which is 648):
\[ \frac{1}{8} = \frac{81}{648}, \quad -\frac{1}{81} = -\frac{8}{648} \]
Thus,
\[ \text{Coefficient of } x^3 = \frac{81}{648} - \frac{8}{648} = \frac{73}{648} \]

### Final Answer
The coefficient of \( x^3 \) is:
\[ \frac{73}{648} \]