If `alpha and beta` are the roots of the equation `x^(2) + x+ 1 = 0, ` then what is the equation whose roots are `alpha^(19) and beta^(7)` ?

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 find the equation whose roots are \( \alpha^{19} \) and \( \beta^{7} \), where \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + x + 1 = 0 \), we can follow these steps:

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

### Step 2: Express \( \alpha \) and \( \beta \) in terms of cube roots of unity
We recognize that \( \alpha \) and \( \beta \) can be expressed as:
\[ \alpha = \omega, \quad \beta = \omega^2 \]
where \( \omega = e^{2\pi i / 3} \) is a primitive cube root of unity.

### Step 3: Calculate \( \alpha^{19} \) and \( \beta^{7} \)
Using the property of cube roots of unity, we find:
\[ \alpha^{19} = \omega^{19} = \omega^{18} \cdot \omega = 1 \cdot \omega = \omega \]
\[ \beta^{7} = (\omega^2)^{7} = \omega^{14} = \omega^{12} \cdot \omega^{2} = 1 \cdot \omega^{2} = \omega^{2} \]

### Step 4: Find the sum and product of the new roots
Now, we need to find the sum and product of the new roots \( \alpha^{19} \) and \( \beta^{7} \):
- Sum of the roots:
\[ \alpha^{19} + \beta^{7} = \omega + \omega^{2} \]
Using the property of cube roots of unity:
\[ \omega + \omega^{2} = -1 \]

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

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

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