Let `alpha and beta` be the roots of `x^(2)+x+1=0`, then the equation whose roots are `alpha^(2020) and beta^(2020)` 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 solve the problem, we need to find the equation whose roots are \( \alpha^{2020} \) and \( \beta^{2020} \), where \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + x + 1 = 0 \).

### Step 1: Find the roots of the equation \( x^2 + x + 1 = 0 \)

We can use the quadratic formula to find the roots:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 1, c = 1 \).

Calculating the discriminant:
\[ b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]

Now substituting into the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{-3}}{2 \cdot 1} = \frac{-1 \pm i\sqrt{3}}{2} \]
Thus, the roots are:
\[ \alpha = \frac{-1 + i\sqrt{3}}{2}, \quad \beta = \frac{-1 - i\sqrt{3}}{2} \]

### Step 2: Express the roots in terms of complex numbers

We can recognize that these roots can be expressed in terms of the cube roots of unity:
\[ \alpha = \omega, \quad \beta = \omega^2 \]
where \( \omega = e^{2\pi i / 3} = \frac{-1 + i\sqrt{3}}{2} \) and \( \omega^2 = e^{-2\pi i / 3} = \frac{-1 - i\sqrt{3}}{2} \).

### Step 3: Calculate \( \alpha^{2020} \) and \( \beta^{2020} \)

Using the property of powers of roots of unity:
\[ \omega^3 = 1 \]
We can reduce \( 2020 \) modulo \( 3 \):
\[ 2020 \mod 3 = 1 \quad (\text{since } 2020 = 3 \times 673 + 1) \]
Thus:
\[ \alpha^{2020} = \omega^{2020} = \omega^{3 \times 673 + 1} = \omega^1 = \omega \]
\[ \beta^{2020} = (\omega^2)^{2020} = \omega^{4040} = \omega^{3 \times 1346 + 2} = \omega^2 \]

### Step 4: Form the new equation with roots \( \alpha^{2020} \) and \( \beta^{2020} \)

Since we found that \( \alpha^{2020} = \omega \) and \( \beta^{2020} = \omega^2 \), the new equation whose roots are \( \omega \) and \( \omega^2 \) can be formed using the fact that the sum and product of the roots are:
- Sum: \( \omega + \omega^2 = -1 \)
- Product: \( \omega \cdot \omega^2 = \omega^3 = 1 \)

Thus, the new equation is:
\[ x^2 - (\text{sum})x + (\text{product}) = 0 \]
This gives us:
\[ x^2 + x + 1 = 0 \]

### Final Answer

The equation whose roots are \( \alpha^{2020} \) and \( \beta^{2020} \) is:
\[ \boxed{x^2 + x + 1 = 0} \]