If `alpha, beta` are the roots of `x^(2) + px + q = 0, and omega` is a cube root of unity, then value of `(omega alpha + omega^(2) beta) (omega^(2) alpha + omega beta)` is

Options

A. `p^(2)`
B. 3q
C. `p^(2) - 2q`
D. `p^(2) - 3q`

Answer

To solve the problem, we need to find the value of the expression \((\omega \alpha + \omega^2 \beta)(\omega^2 \alpha + \omega \beta)\), where \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2 + px + q = 0\) and \(\omega\) is a cube root of unity.

### Step-by-Step Solution:

1. **Understanding the Roots**:
The roots of the quadratic equation \(x^2 + px + q = 0\) are given by:
\[ \alpha + \beta = -p \quad \text{(sum of roots)} \]
\[ \alpha \beta = q \quad \text{(product of roots)} \]

2. **Expanding the Expression**:
We start with the expression:
\[ (\omega \alpha + \omega^2 \beta)(\omega^2 \alpha + \omega \beta) \]
We can expand this using the distributive property:
\[ = \omega^3 \alpha^2 + \omega^2 \alpha \beta + \omega \beta \omega^2 \alpha + \omega^3 \beta^2 \]

3. **Using Properties of \(\omega\)**:
Since \(\omega\) is a cube root of unity, we know:
\[ \omega^3 = 1 \]
Therefore, we can simplify the expression:
\[ = \alpha^2 + \beta^2 + \omega^2 \alpha \beta + \omega \alpha \beta \]
\[ = \alpha^2 + \beta^2 + (\omega + \omega^2) \alpha \beta \]
Since \(\omega + \omega^2 = -1\):
\[ = \alpha^2 + \beta^2 - \alpha \beta \]

4. **Expressing \(\alpha^2 + \beta^2\)**:
We can express \(\alpha^2 + \beta^2\) in terms of \(\alpha + \beta\) and \(\alpha \beta\):
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \]
Substituting the values we have:
\[ = (-p)^2 - 2q = p^2 - 2q \]

5. **Final Expression**:
Now substituting back into our expression:
\[ = (p^2 - 2q) - q = p^2 - 3q \]

Thus, the value of the expression \((\omega \alpha + \omega^2 \beta)(\omega^2 \alpha + \omega \beta)\) is:
\[ \boxed{p^2 - 3q} \]