The coefficient of ` x^(4) "in" (1 - 3x + x^(2))/(e^(x))` equals

Options

A. `(12)/(15)`
B. `(24)/(25)`
C. `(25)/(24)`
D. `(15)/(12)`

Answer

To find the coefficient of \( x^4 \) in the expression \( \frac{1 - 3x + x^2}{e^x} \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \frac{1 - 3x + x^2}{e^x} \]
This can be rewritten as:
\[ (1 - 3x + x^2) \cdot e^{-x} \]

### Step 2: Expand \( e^{-x} \)
The exponential function \( e^{-x} \) can be expanded using its Taylor series:
\[ e^{-x} = 1 - \frac{x}{1!} + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!} - \cdots \]
We only need terms up to \( x^4 \):
\[ e^{-x} \approx 1 - x + \frac{x^2}{2} - \frac{x^3}{6} + \frac{x^4}{24} \]

### Step 3: Multiply the Two Expressions
Now we multiply \( (1 - 3x + x^2) \) by the expansion of \( e^{-x} \):
\[ (1 - 3x + x^2) \cdot \left(1 - x + \frac{x^2}{2} - \frac{x^3}{6} + \frac{x^4}{24}\right) \]

### Step 4: Collect Terms for \( x^4 \)
We need to find the coefficient of \( x^4 \) from the product. We will consider contributions from different combinations of terms:

1. **From \( 1 \cdot \frac{x^4}{24} \)**:
- Coefficient: \( \frac{1}{24} \)

2. **From \( -3x \cdot -\frac{x^3}{6} \)**:
- Coefficient: \( -3 \cdot -\frac{1}{6} = \frac{3}{6} = \frac{1}{2} \)

3. **From \( x^2 \cdot \frac{x^2}{2} \)**:
- Coefficient: \( 1 \cdot \frac{1}{2} = \frac{1}{2} \)

### Step 5: Sum the Coefficients
Now we sum all the contributions to find the total coefficient of \( x^4 \):
\[ \text{Total Coefficient} = \frac{1}{24} + \frac{1}{2} + \frac{1}{2} \]
Converting \( \frac{1}{2} \) to a fraction with a denominator of 24:
\[ \frac{1}{2} = \frac{12}{24} \]
Thus,
\[ \text{Total Coefficient} = \frac{1}{24} + \frac{12}{24} + \frac{12}{24} = \frac{1 + 12 + 12}{24} = \frac{25}{24} \]

### Final Answer
The coefficient of \( x^4 \) in the expression \( \frac{1 - 3x + x^2}{e^x} \) is:
\[ \frac{25}{24} \]