If `1, omega, omega^(2)` are the cube roots of unity, then `( 1+ omega) ( 1+ omega^(2)) ( 1+ omega^(3)) ( 1+ omega + omega^(2))` is equal to `:`

Options

A. A)`-2`
B. B)`-1`
C. C)0
D. D)2

Answer

To solve the problem, we need to evaluate the expression:

\[ (1 + \omega)(1 + \omega^2)(1 + \omega^3)(1 + \omega + \omega^2) \]

where \(1, \omega, \omega^2\) are the cube roots of unity.

### Step 1: Identify the cube roots of unity
The cube roots of unity are defined as the solutions to the equation \(x^3 = 1\). The roots are:
- \(1\)
- \(\omega = e^{2\pi i / 3}\)
- \(\omega^2 = e^{4\pi i / 3}\)

### Step 2: Use properties of cube roots of unity
From the properties of cube roots of unity, we know:
- \(1 + \omega + \omega^2 = 0\)
- \(\omega^3 = 1\)

### Step 3: Evaluate \(1 + \omega^3\)
Since \(\omega^3 = 1\):
\[ 1 + \omega^3 = 1 + 1 = 2 \]

### Step 4: Evaluate \(1 + \omega + \omega^2\)
From the property we mentioned:
\[ 1 + \omega + \omega^2 = 0 \]

### Step 5: Substitute into the expression
Now we can substitute back into our expression:
\[ (1 + \omega)(1 + \omega^2)(1 + \omega^3)(1 + \omega + \omega^2) = (1 + \omega)(1 + \omega^2)(2)(0) \]

### Step 6: Simplify the expression
Since we have a multiplication by \(0\):
\[ (1 + \omega)(1 + \omega^2)(2)(0) = 0 \]

### Final Result
Thus, the value of the expression is:
\[ \boxed{0} \]

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