If ` omega ` is complex cube root of unity, then the value of `(1 + 2 omega )^(-1) + (2 + omega )^(-1) - (1 + omega ) ^(-1)` =

Options

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

Answer

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

\[ (1 + 2\omega)^{-1} + (2 + \omega)^{-1} - (1 + \omega)^{-1} \]

where \(\omega\) is a complex cube root of unity. The cube roots of unity are given by:

\[ \omega = e^{2\pi i / 3} = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \]
\[ \omega^2 = e^{-2\pi i / 3} = -\frac{1}{2} - \frac{\sqrt{3}}{2} i \]
and we also know that \(1 + \omega + \omega^2 = 0\).

### Step 1: Calculate each term separately

1. **Calculate \((1 + 2\omega)^{-1}\)**:
\[ (1 + 2\omega)^{-1} = \frac{1}{1 + 2\omega} \]

2. **Calculate \((2 + \omega)^{-1}\)**:
\[ (2 + \omega)^{-1} = \frac{1}{2 + \omega} \]

3. **Calculate \((1 + \omega)^{-1}\)**:
\[ (1 + \omega)^{-1} = \frac{1}{1 + \omega} \]

### Step 2: Combine the fractions

Now we need to combine these fractions:

\[ (1 + 2\omega)^{-1} + (2 + \omega)^{-1} - (1 + \omega)^{-1} \]

We can find a common denominator for these fractions. The common denominator will be \((1 + 2\omega)(2 + \omega)(1 + \omega)\).

### Step 3: Write the expression with a common denominator

The expression becomes:

\[ \frac{(2 + \omega)(1 + \omega) + (1 + 2\omega)(1 + \omega) - (1 + 2\omega)(2 + \omega)}{(1 + 2\omega)(2 + \omega)(1 + \omega)} \]

### Step 4: Simplify the numerator

Now we simplify the numerator:

1. Expand \((2 + \omega)(1 + \omega)\):
\[ = 2 + 2\omega + \omega + \omega^2 = 2 + 3\omega + \omega^2 \]

2. Expand \((1 + 2\omega)(1 + \omega)\):
\[ = 1 + \omega + 2\omega + 2\omega^2 = 1 + 3\omega + 2\omega^2 \]

3. Expand \((1 + 2\omega)(2 + \omega)\):
\[ = 2 + \omega + 4\omega + 2\omega^2 = 2 + 5\omega + 2\omega^2 \]

### Step 5: Combine the results

Putting it all together, we have:

\[ (2 + 3\omega + \omega^2) + (1 + 3\omega + 2\omega^2) - (2 + 5\omega + 2\omega^2) \]

Combine like terms:

\[ = (2 + 1 - 2) + (3\omega + 3\omega - 5\omega) + (\omega^2 + 2\omega^2 - 2\omega^2) \]
\[ = 1 + (6\omega - 5\omega) + (1\omega^2) \]
\[ = 1 + \omega + \omega^2 \]

### Step 6: Use the property of cube roots of unity

Since \(1 + \omega + \omega^2 = 0\), we have:

\[ 1 + \omega + \omega^2 = 0 \]

Thus, the entire expression evaluates to:

\[ 0 \]

### Final Answer

The value of the expression is:

\[ \boxed{0} \]