Express `(1)/((1-x))+(1)/((1+x))+(2)/((1+x^(2)))+(4)/((1+x^(4)))` as a rational expression :

Options

A. 1
B. `(1-x^(4))/(1+x^(4))`
C. `(8)/(1-x^(8))`
D. none of these

Answer

To express the given expression

\[ \frac{1}{1-x} + \frac{1}{1+x} + \frac{2}{1+x^2} + \frac{4}{1+x^4} \]

as a rational expression, we will follow these steps:

### Step 1: Find a common denominator
The denominators are \(1-x\), \(1+x\), \(1+x^2\), and \(1+x^4\). The least common multiple (LCM) of these denominators is \((1-x)(1+x)(1+x^2)(1+x^4)\).

### Step 2: Rewrite each term with the common denominator
We will express each fraction with the common denominator:

1. For \(\frac{1}{1-x}\):
\[ \frac{1 \cdot (1+x)(1+x^2)(1+x^4)}{(1-x)(1+x)(1+x^2)(1+x^4)} \]

2. For \(\frac{1}{1+x}\):
\[ \frac{1 \cdot (1-x)(1+x^2)(1+x^4)}{(1-x)(1+x)(1+x^2)(1+x^4)} \]

3. For \(\frac{2}{1+x^2}\):
\[ \frac{2 \cdot (1-x)(1+x)(1+x^4)}{(1-x)(1+x)(1+x^2)(1+x^4)} \]

4. For \(\frac{4}{1+x^4}\):
\[ \frac{4 \cdot (1-x)(1+x)(1+x^2)}{(1-x)(1+x)(1+x^2)(1+x^4)} \]

### Step 3: Combine the numerators
Now we combine all the numerators over the common denominator:
\[ \frac{(1)(1+x)(1+x^2)(1+x^4) + (1)(1-x)(1+x^2)(1+x^4) + (2)(1-x)(1+x)(1+x^4) + (4)(1-x)(1+x)(1+x^2)}{(1-x)(1+x)(1+x^2)(1+x^4)} \]

### Step 4: Simplify the numerator
We will simplify the numerator step by step:

1. Expand each term in the numerator.
2. Combine like terms.

After performing the necessary expansions and simplifications, we will find that the numerator simplifies to \(8\).

### Step 5: Final expression
Thus, the entire expression simplifies to:
\[ \frac{8}{(1-x)(1+x)(1+x^2)(1+x^4)} \]

### Step 6: Further simplification
Notice that:
\[ (1-x)(1+x) = 1-x^2 \]
So we can rewrite the expression as:
\[ \frac{8}{1-x^4} \]

### Final Result
Thus, the final expression is:
\[ \frac{8}{1-x^8} \]