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 `

Explanation

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
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, \( a = 1 \), \( b = 1 \), and \( c = 1 \):
\[ x = \frac{-1 \pm \sqrt{1^{2} - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2} \]
Thus, the roots are:
\[ \alpha = \frac{-1 + i\sqrt{3}}{2}, \quad \beta = \frac{-1 - i\sqrt{3}}{2} \]

### Step 2: Express roots in exponential form
The roots \( \alpha \) and \( \beta \) can also be expressed as the cube roots of unity:
\[ \alpha = \omega, \quad \beta = \omega^2 \]
where \( \omega = e^{2\pi i / 3} \) and \( \omega^2 = e^{-2\pi i / 3} \).

### Step 3: Calculate \( \alpha^{19} \) and \( \beta^{7} \)
Using the properties of roots of unity:
\[ \alpha^{19} = \omega^{19} = \omega^{19 \mod 3} = \omega^{1} = \omega \]
\[ \beta^{7} = \omega^{14} = \omega^{14 \mod 3} = \omega^{2} \]

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

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

### Step 5: Form the new equation
The equation with roots \( \alpha^{19} \) and \( \beta^{7} \) can be formed using the standard form:
\[ x^2 - (\text{Sum of roots})x + (\text{Product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - (-1)x + 1 = 0 \implies x^2 + x + 1 = 0 \]

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