Let `alpha and beta`, be the roots of the equation `x^2+x+1=0`. The equation whose roots are `alpha^19` and `beta^7` are:

Options

A. `x^(2) - x - 1 = 0`
B. `x^(2) - x + 1 = 0`
C. `x^(2) + x - 1 = 0`
D. `x^(2) + x + 1 = 0`

Answer

To solve the problem, we need to find the equation whose roots are \( \alpha^{19} \) and \( \beta^{7} \), where \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + x + 1 = 0 \).

### Step-by-Step Solution:

1. **Identify the roots \( \alpha \) and \( \beta \)**:
The roots of the equation \( x^2 + x + 1 = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 1, c = 1 \).
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2} \]
Thus, the roots are:
\[ \alpha = \frac{-1 + i\sqrt{3}}{2}, \quad \beta = \frac{-1 - i\sqrt{3}}{2} \]

2. **Express \( \alpha \) and \( \beta \) in terms of complex numbers**:
The roots can be identified as the cube roots of unity:
\[ \alpha = \omega, \quad \beta = \omega^2 \]
where \( \omega = e^{2\pi i / 3} \) and \( \omega^3 = 1 \).

3. **Calculate \( \alpha^{19} \) and \( \beta^{7} \)**:
Using the property of cube roots of unity:
\[ \alpha^{19} = \omega^{19} = \omega^{19 \mod 3} = \omega^1 = \omega \]
\[ \beta^{7} = (\omega^2)^{7} = \omega^{14} = \omega^{14 \mod 3} = \omega^2 \]

4. **Find the sum and product of the new roots**:
- The sum of the roots \( \alpha^{19} + \beta^{7} \):
\[ \alpha^{19} + \beta^{7} = \omega + \omega^2 \]
Using the property \( 1 + \omega + \omega^2 = 0 \):
\[ \omega + \omega^2 = -1 \]

- The product of the roots \( \alpha^{19} \cdot \beta^{7} \):
\[ \alpha^{19} \cdot \beta^{7} = \omega \cdot \omega^2 = \omega^3 = 1 \]

5. **Form the new quadratic equation**:
The equation whose roots are \( \alpha^{19} \) and \( \beta^{7} \) can be expressed as:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - (-1)x + 1 = 0 \implies x^2 + x + 1 = 0 \]

### Final Answer:
The equation whose roots are \( \alpha^{19} \) and \( \beta^{7} \) is:
\[ \boxed{x^2 + x + 1 = 0} \]