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

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: Find the roots \( \alpha \) and \( \beta \)
The roots of the quadratic 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 \( c = 1 \).

Substituting these values into the formula gives:

\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm \sqrt{3}i}{2} \]

Thus, we have:

\[ \alpha = \frac{-1 + \sqrt{3}i}{2}, \quad \beta = \frac{-1 - \sqrt{3}i}{2} \]

### Step 2: Recognize \( \alpha \) and \( \beta \) as cube roots of unity
The roots \( \alpha \) and \( \beta \) can be recognized as the cube roots of unity, denoted as \( \omega \) and \( \omega^2 \), where:

\[ \omega = \frac{-1 + \sqrt{3}i}{2}, \quad \omega^2 = \frac{-1 - \sqrt{3}i}{2} \]

These satisfy the property \( \omega^3 = 1 \) and \( \omega^2 + \omega + 1 = 0 \).

### Step 3: Calculate \( \alpha^{19} \) and \( \beta^{7} \)
Using the property of cube roots of unity:

\[ \alpha^{19} = \omega^{19} = \omega^{(19 \mod 3)} = \omega^{1} = \omega \]

\[ \beta^{7} = (\omega^2)^{7} = \omega^{14} = \omega^{(14 \mod 3)} = \omega^{2} \]

### 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 \( \beta^{7} \):

1. **Sum**:
\[ \alpha^{19} + \beta^{7} = \omega + \omega^2 \]

From the property of cube roots of unity:
\[ \omega + \omega^2 = -1 \]

2. **Product**:
\[ \alpha^{19} \cdot \beta^{7} = \omega \cdot \omega^2 = \omega^3 = 1 \]

### Step 5: Form the new quadratic equation
The standard form of a quadratic equation with roots \( r_1 \) and \( r_2 \) is given by:

\[ x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0 \]

Substituting \( r_1 = \alpha^{19} \) and \( r_2 = \beta^{7} \):

\[ x^2 - (-1)x + 1 = 0 \]

This simplifies to:

\[ x^2 + x + 1 = 0 \]

### Final Answer
The equation whose roots are \( \alpha^{19} \) and \( \beta^{7} \) is:

\[ \boxed{x^2 + x + 1 = 0} \]