The coefficient of `x^(2)` in the expansion of `(1-x+2x^(2))(x+(1)/(x))^(10)` is

Options

A. 210
B. 714
C. 504
D. 240

Answer

To find the coefficient of \( x^2 \) in the expansion of \( (1 - x + 2x^2)(x + \frac{1}{x})^{10} \), we can follow these steps:

### Step 1: Rewrite the expression
We can rewrite the expression as:
\[ (1 - x + 2x^2) \cdot (x + \frac{1}{x})^{10} \]
The term \( (x + \frac{1}{x})^{10} \) can be simplified further.

### Step 2: Expand \( (x + \frac{1}{x})^{10} \)
Using the binomial theorem, we can expand \( (x + \frac{1}{x})^{10} \):
\[ (x + \frac{1}{x})^{10} = \sum_{k=0}^{10} \binom{10}{k} x^{10-k} \left(\frac{1}{x}\right)^k = \sum_{k=0}^{10} \binom{10}{k} x^{10 - 2k} \]
This means we will have terms of the form \( x^{10 - 2k} \).

### Step 3: Identify the required powers of \( x \)
To find the coefficient of \( x^2 \) in the product, we need to find terms in \( (1 - x + 2x^2) \) that can combine with terms from \( (x + \frac{1}{x})^{10} \) to give \( x^2 \).

### Step 4: Consider the contributions from each term
1. **From \( 1 \)**: We need \( x^2 \) from \( (x + \frac{1}{x})^{10} \). This occurs when \( 10 - 2k = 2 \) or \( k = 4 \). The coefficient is:
\[ \binom{10}{4} = 210 \]

2. **From \( -x \)**: We need \( x^3 \) from \( (x + \frac{1}{x})^{10} \). This occurs when \( 10 - 2k = 3 \) or \( k = 3 \). The coefficient is:
\[ -\binom{10}{3} = -120 \]

3. **From \( 2x^2 \)**: We need \( x^0 \) from \( (x + \frac{1}{x})^{10} \). This occurs when \( 10 - 2k = 0 \) or \( k = 5 \). The coefficient is:
\[ 2 \cdot \binom{10}{5} = 2 \cdot 252 = 504 \]

### Step 5: Combine all contributions
Now, we can combine all these contributions:
\[ \text{Total coefficient of } x^2 = 210 - 120 + 504 = 594 \]

### Step 6: Final answer
Thus, the coefficient of \( x^2 \) in the expansion of \( (1 - x + 2x^2)(x + \frac{1}{x})^{10} \) is:
\[ \boxed{594} \]