What is the value of ` ((1 + x)/(1 - x^(4)))/(( x^(2))/( 1 + x^(2))) xx x ( 1 - x)` ?

Options

A. `(1)/(x)`
B. ` x^(2) - 1 `
C. `x 1`
D. x

Answer

To solve the expression \( \frac{(1 + x)}{(1 - x^4)} \div \left( \frac{x^2}{(1 + x^2)} \cdot x(1 - x) \right) \), we can follow these steps:

### Step 1: Rewrite the Expression
The given expression can be rewritten as:
\[ \frac{(1 + x)}{(1 - x^4)} \cdot \frac{(1 + x^2)}{(x^2 \cdot x(1 - x))} \]

### Step 2: Factor \(1 - x^4\)
Notice that \(1 - x^4\) can be factored as:
\[ 1 - x^4 = (1 - x^2)(1 + x^2) \]
Thus, we can rewrite the expression:
\[ \frac{(1 + x)}{(1 - x^2)(1 + x^2)} \cdot \frac{(1 + x^2)}{(x^3(1 - x))} \]

### Step 3: Cancel Out Common Terms
Now, we can cancel \(1 + x^2\) from the numerator and denominator:
\[ \frac{(1 + x)}{(1 - x^2)} \cdot \frac{1}{(x^3(1 - x))} \]

### Step 4: Factor \(1 - x^2\)
Next, we can factor \(1 - x^2\) as:
\[ 1 - x^2 = (1 - x)(1 + x) \]
So, we can rewrite the expression as:
\[ \frac{(1 + x)}{(1 - x)(1 + x)} \cdot \frac{1}{(x^3(1 - x))} \]

### Step 5: Cancel \(1 + x\)
Now, we can cancel \(1 + x\) from the numerator and denominator:
\[ \frac{1}{(1 - x)(x^3(1 - x))} \]

### Step 6: Simplify the Expression
This simplifies to:
\[ \frac{1}{x^3(1 - x)^2} \]

### Final Result
Thus, the value of the given expression is:
\[ \frac{1}{x^3(1 - x)^2} \]