the product of `(1- x + x ^(2)) , (1 - x ^(2) + x ^(4)) and (1 + x + x ^(2))` is

Options

A. `x ^(8) + x ^(4) + x^(2) + 1`
B. `x ^(8) - x ^(4) +1 `
C. `x ^(8) + x ^(4) -1`
D. `x ^(8) + x ^(4) +1`

Answer

To find the product of the expressions \((1 - x + x^2)\), \((1 - x^2 + x^4)\), and \((1 + x + x^2)\), we will follow these steps:

### Step 1: Rearranging the Terms
First, we can rearrange the terms for easier multiplication:
\[ (1 + x^2 - x) \cdot (1 + x^2 + x) \cdot (1 - x^2 + x^4) \]

### Step 2: Multiply the First Two Expressions
Next, we will multiply the first two expressions:
\[ (1 + x^2 - x)(1 + x^2 + x) \]
Using the identity \((a - b)(a + b) = a^2 - b^2\), where \(a = 1 + x^2\) and \(b = x\):
\[ = (1 + x^2)^2 - x^2 \]

### Step 3: Expand \((1 + x^2)^2\)
Now, we expand \((1 + x^2)^2\):
\[ (1 + x^2)^2 = 1^2 + 2 \cdot 1 \cdot x^2 + (x^2)^2 = 1 + 2x^2 + x^4 \]
Thus, we have:
\[ (1 + x^2)^2 - x^2 = (1 + 2x^2 + x^4) - x^2 = 1 + x^2 + x^4 \]

### Step 4: Multiply with the Third Expression
Now we will multiply this result with the third expression:
\[ (1 + x^2 + x^4)(1 - x^2 + x^4) \]
We can use the distributive property to expand this product.

### Step 5: Expand the Product
Expanding the product:
\[ = 1(1 - x^2 + x^4) + x^2(1 - x^2 + x^4) + x^4(1 - x^2 + x^4) \]
Calculating each term:
1. \(1(1 - x^2 + x^4) = 1 - x^2 + x^4\)
2. \(x^2(1 - x^2 + x^4) = x^2 - x^4 + x^6\)
3. \(x^4(1 - x^2 + x^4) = x^4 - x^6 + x^8\)

### Step 6: Combine Like Terms
Now, combine all these results:
\[ (1 - x^2 + x^4) + (x^2 - x^4 + x^6) + (x^4 - x^6 + x^8) \]
Combining like terms:
- Constant term: \(1\)
- \(x^2\) terms: \(-x^2 + x^2 = 0\)
- \(x^4\) terms: \(x^4 - x^4 + x^4 = x^4\)
- \(x^6\) terms: \(x^6 - x^6 = 0\)
- \(x^8\) term: \(x^8\)

Thus, the final result is:
\[ 1 + x^4 + x^8 \]

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