Let I, `omega` and `omega^(2)` be the cube roots of unity. The least possible degree of a polynomial, with real coefficients having `2omega^(2), 3 + 4 omega, 3 + 4 omega^(2) ` and `5- omega - omega^(2)` as roots is -

Options

A. `4`
B. `5`
C. `6`
D. `7`

Explanation

To find the least possible degree of a polynomial with real coefficients that has the given roots \(2\omega^2\), \(3 + 4\omega\), \(3 + 4\omega^2\), and \(5 - \omega - \omega^2\), we will follow these steps:

### Step 1: Identify the roots
The roots provided are:
1. \(2\omega^2\)
2. \(3 + 4\omega\)
3. \(3 + 4\omega^2\)
4. \(5 - \omega - \omega^2\)

### Step 2: Use properties of cube roots of unity
Recall that the cube roots of unity are \(1\), \(\omega\), and \(\omega^2\), where:
\[ \omega + \omega^2 + 1 = 0 \quad \Rightarrow \quad \omega + \omega^2 = -1 \]
This means \(\omega^2 = -1 - \omega\) and \(\omega = -1 - \omega^2\).

### Step 3: Find conjugate roots
Since the polynomial must have real coefficients, the complex roots must occur in conjugate pairs.
- If \(2\omega^2\) is a root, then \(2\omega\) must also be a root because \(\omega\) and \(\omega^2\) are conjugates.
- The roots \(3 + 4\omega\) and \(3 + 4\omega^2\) are already conjugate pairs.
- The root \(5 - \omega - \omega^2\) can be simplified using the property of cube roots:
\[ 5 - \omega - \omega^2 = 5 - (-1) = 6 \]
Thus, \(6\) is also a root.

### Step 4: List all the roots
Now we have the following roots:
1. \(2\omega^2\)
2. \(2\omega\)
3. \(3 + 4\omega\)
4. \(3 + 4\omega^2\)
5. \(6\)

### Step 5: Count the distinct roots
We have identified 5 distinct roots:
- \(2\omega\)
- \(2\omega^2\)
- \(3 + 4\omega\)
- \(3 + 4\omega^2\)
- \(6\)

### Step 6: Determine the degree of the polynomial
A polynomial with \(n\) distinct roots has a degree of \(n\). Since we have 5 distinct roots, the least possible degree of the polynomial is:
\[ \text{Degree} = 5 \]

### Final Answer
The least possible degree of the polynomial with the given roots is **5**.

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