What is the coefficient of `x^(4)` in the expansion of `((1-x)/(1+x))^(2)` ?

Options

A. `-16`
B. 16
C. 8
D. `-18`

Explanation

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

### Step 1: Rewrite the Expression
We start by rewriting the expression:
\[ \left( \frac{1-x}{1+x} \right)^2 = \frac{(1-x)^2}{(1+x)^2} \]

### Step 2: Expand the Numerator and Denominator
Next, we expand both the numerator and the denominator:
- The numerator \( (1-x)^2 \) can be expanded as:
\[ (1-x)^2 = 1 - 2x + x^2 \]
- The denominator \( (1+x)^2 \) can be expanded as:
\[ (1+x)^2 = 1 + 2x + x^2 \]

### Step 3: Write the Expression in Expanded Form
Now we can write the expression as:
\[ \frac{1 - 2x + x^2}{1 + 2x + x^2} \]

### Step 4: Use the Binomial Series for the Denominator
We can express \( \frac{1}{1 + 2x + x^2} \) using the binomial series expansion. However, it is easier to rewrite it as:
\[ (1 + 2x + x^2)^{-1} = (1 + (2x + x^2))^{-1} \]
Using the binomial series expansion, we can write:
\[ (1 + u)^{-1} = 1 - u + u^2 - u^3 + \ldots \]
where \( u = 2x + x^2 \).

### Step 5: Find the Series Expansion
We need to find the first few terms of the expansion:
\[ (1 + 2x + x^2)^{-1} \approx 1 - (2x + x^2) + (2x + x^2)^2 - \ldots \]
Calculating the first few terms:
- The first term is \( 1 \).
- The second term is \( - (2x + x^2) = -2x - x^2 \).
- The third term is \( (2x + x^2)^2 = 4x^2 + 4x^3 + x^4 \).

Thus, we have:
\[ (1 + 2x + x^2)^{-1} \approx 1 - 2x - x^2 + 4x^2 + 4x^3 + x^4 \]
which simplifies to:
\[ 1 - 2x + 3x^2 + 4x^3 + x^4 \]

### Step 6: Multiply the Numerator and the Denominator
Now we multiply the numerator \( 1 - 2x + x^2 \) with the series expansion we just found:
\[ (1 - 2x + x^2)(1 - 2x + 3x^2 + 4x^3 + x^4) \]

### Step 7: Collect the Terms for \( x^4 \)
We need to collect the terms that contribute to \( x^4 \):
- From \( 1 \cdot x^4 \): contributes \( 1 \)
- From \( -2x \cdot 4x^3 \): contributes \( -8 \)
- From \( x^2 \cdot 3x^2 \): contributes \( 3 \)

Adding these contributions together:
\[ 1 - 8 + 3 = -4 \]

### Step 8: Final Coefficient
Thus, the coefficient of \( x^4 \) in the expansion of \( \left( \frac{1-x}{1+x} \right)^2 \) is:
\[ \boxed{-4} \]