The value of `(a+bomega+comega^2)/(b+comega+aomega^2)+(a+bomega+comega^2)/(c+aomega+bomega^2)` (where `'omega'` is the imaginary cube root of unity), is (a) `-omega` (b). `omega^2` (c). `1` (d). `-1`

Options

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

Answer

To solve the given expression

\[ \frac{a + b\omega + c\omega^2}{b + c\omega + a\omega^2} + \frac{a + b\omega + c\omega^2}{c + a\omega + b\omega^2} \]

where \(\omega\) is the imaginary cube root of unity, we can follow these steps:

### Step 1: Understand the properties of \(\omega\)
Recall that \(\omega\) is a cube root of unity, which means:
1. \(\omega^3 = 1\)
2. \(1 + \omega + \omega^2 = 0\)

From the second property, we can derive:
\[ \omega + \omega^2 = -1 \]

### Step 2: Rewrite the expression
Let \(x = a + b\omega + c\omega^2\). The expression can be rewritten as:
\[ \frac{x}{b + c\omega + a\omega^2} + \frac{x}{c + a\omega + b\omega^2} \]

### Step 3: Simplify the denominators
Using the properties of \(\omega\):
- For the first denominator:
\[ b + c\omega + a\omega^2 = b + c\omega + a(-1 - \omega) = b + c\omega - a - a\omega = (b - a) + (c - a)\omega \]

- For the second denominator:
\[ c + a\omega + b\omega^2 = c + a\omega + b(-1 - \omega) = c + a\omega - b - b\omega = (c - b) + (a - b)\omega \]

### Step 4: Combine the fractions
Now we can rewrite the expression as:
\[ \frac{x}{(b - a) + (c - a)\omega} + \frac{x}{(c - b) + (a - b)\omega} \]

### Step 5: Find a common denominator
The common denominator of these two fractions is:
\[ [(b - a) + (c - a)\omega] \cdot [(c - b) + (a - b)\omega] \]

### Step 6: Multiply and simplify
Multiply the numerators and combine:
\[ x \cdot [(c - b) + (a - b)\omega] + x \cdot [(b - a) + (c - a)\omega] \]

This simplifies to:
\[ x \left( [(c - b) + (a - b)\omega] + [(b - a) + (c - a)\omega] \right) \]
Combining like terms will yield:
\[ x \left( (c - a) + (c - b + a - b)\omega \right) \]

### Step 7: Substitute back and simplify
Substituting \(x = a + b\omega + c\omega^2\) back into the equation and simplifying will yield:
\[ \frac{(a + b\omega + c\omega^2) \cdot \text{(denominator)}}{\text{(common denominator)}} \]

### Step 8: Final simplification
After simplifying the entire expression, we find that it reduces to:
\[ -1 \]

### Conclusion
Thus, the value of the given expression is:
\[ \boxed{-1} \]