Coefficient of `x^(10)` in the expansion of `(2+3x)e^(-x)` is

Options

A. `(-26)/((10)!)`
B. `(-28)/((10)!)`
C. `(-30)/((10)!)`
D. `(-32)/((10)!)`

Answer

To find the coefficient of \( x^{10} \) in the expansion of \( (2 + 3x)e^{-x} \), we can follow these steps:

### Step 1: Expand \( e^{-x} \)
The exponential function \( e^{-x} \) can be expanded using its Taylor series:
\[ e^{-x} = \sum_{n=0}^{\infty} \frac{(-x)^n}{n!} = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!} - \cdots \]
This series continues indefinitely.

### Step 2: Multiply by \( (2 + 3x) \)
Now, we need to multiply this series by \( (2 + 3x) \):
\[ (2 + 3x)e^{-x} = (2 + 3x) \left( 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!} - \cdots \right) \]

### Step 3: Identify the terms contributing to \( x^{10} \)
To find the coefficient of \( x^{10} \), we need to consider the contributions from:
1. The term \( 2 \) multiplied by the coefficient of \( x^{10} \) in \( e^{-x} \).
2. The term \( 3x \) multiplied by the coefficient of \( x^{9} \) in \( e^{-x} \).

#### Contribution from \( 2 \):
The coefficient of \( x^{10} \) in \( e^{-x} \) is given by:
\[ \text{Coefficient of } x^{10} = \frac{(-1)^{10}}{10!} = \frac{1}{10!} \]
Thus, the contribution from \( 2 \) is:
\[ 2 \cdot \frac{1}{10!} = \frac{2}{10!} \]

#### Contribution from \( 3x \):
The coefficient of \( x^{9} \) in \( e^{-x} \) is:
\[ \text{Coefficient of } x^{9} = \frac{(-1)^{9}}{9!} = -\frac{1}{9!} \]
Thus, the contribution from \( 3x \) is:
\[ 3 \cdot \left(-\frac{1}{9!}\right) = -\frac{3}{9!} \]

### Step 4: Combine the contributions
Now, we combine the contributions to find the total coefficient of \( x^{10} \):
\[ \text{Total coefficient} = \frac{2}{10!} - \frac{3}{9!} \]

### Step 5: Simplify the expression
To combine these fractions, we need a common denominator. The common denominator is \( 10! \):
\[ -\frac{3}{9!} = -\frac{3 \cdot 10}{10!} = -\frac{30}{10!} \]
Thus, we have:
\[ \text{Total coefficient} = \frac{2 - 30}{10!} = \frac{-28}{10!} \]

### Final Answer
The coefficient of \( x^{10} \) in the expansion of \( (2 + 3x)e^{-x} \) is:
\[ \boxed{-\frac{28}{10!}} \]