The continued product of `(1 + x), (1 + x^2), (1 + x^4), (1 + x^8) and (1 - x) ` is :

Options

A. `(1 - x^8 + x^(16))`
B. `(x^8 + x^(16))`
C. `(1 - x^16)`
D. `(x^(16) - 1)`

Answer

To solve the problem of finding the continued product of \((1 + x), (1 + x^2), (1 + x^4), (1 + x^8)\) and \((1 - x)\), we can follow these steps:

### Step 1: Write down the product
We start with the expression:
\[ P(x) = (1 + x)(1 + x^2)(1 + x^4)(1 + x^8)(1 - x) \]

### Step 2: Combine the first two factors
Notice that we can combine \((1 + x)\) and \((1 - x)\):
\[ (1 + x)(1 - x) = 1 - x^2 \]
So now we can rewrite \(P(x)\):
\[ P(x) = (1 - x^2)(1 + x^2)(1 + x^4)(1 + x^8) \]

### Step 3: Simplify further
Next, we can combine \((1 - x^2)\) and \((1 + x^2)\):
\[ (1 - x^2)(1 + x^2) = 1 - x^4 \]
Now, we have:
\[ P(x) = (1 - x^4)(1 + x^4)(1 + x^8) \]

### Step 4: Combine the next two factors
Now, we can combine \((1 - x^4)\) and \((1 + x^4)\):
\[ (1 - x^4)(1 + x^4) = 1 - x^8 \]
Thus, we can rewrite \(P(x)\) again:
\[ P(x) = (1 - x^8)(1 + x^8) \]

### Step 5: Final combination
Finally, we combine \((1 - x^8)\) and \((1 + x^8)\):
\[ (1 - x^8)(1 + x^8) = 1 - x^{16} \]

### Conclusion
Thus, the final result of the continued product is:
\[ P(x) = 1 - x^{16} \]

### Final Answer
The continued product of \((1 + x), (1 + x^2), (1 + x^4), (1 + x^8)\) and \((1 - x)\) is:
\[ 1 - x^{16} \]

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