Common roots of the equation `z^(3)+2z^(2)+2z+1=0` and `z^(2020)+z^(2018)+1=0`, are

Options

A. `omega,omega^(2)`
B. `1,omega,omega^(2)`
C. `-1,omega,omega^(2)`
D. `-omega,-omega^(2)`

Answer

To find the common roots of the equations \( z^3 + 2z^2 + 2z + 1 = 0 \) and \( z^{2020} + z^{2018} + 1 = 0 \), we will follow these steps:

### Step 1: Factor the first equation
We start with the first equation:
\[ z^3 + 2z^2 + 2z + 1 = 0 \]
We can rearrange it as follows:
\[ z^3 + 1 + 2z = 0 \]
This can be rewritten as:
\[ z^3 + 1 = -2z \]
Using the sum of cubes formula, we can factor \( z^3 + 1 \):
\[ z^3 + 1 = (z + 1)(z^2 - z + 1) \]
Thus, we rewrite the equation:
\[ (z + 1)(z^2 - z + 1) + 2z = 0 \]
This simplifies to:
\[ (z + 1)(z^2 - z + 1) + 2z = 0 \]

### Step 2: Finding roots of the first equation
Setting \( z + 1 = 0 \):
\[ z = -1 \]
Now, we need to find the roots of \( z^2 - z + 1 = 0 \) using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{-3}}{2} \]
This gives us:
\[ z = \frac{1 \pm i\sqrt{3}}{2} \]
Let \( \omega = \frac{1 + i\sqrt{3}}{2} \) and \( \omega^2 = \frac{1 - i\sqrt{3}}{2} \). Thus, the roots of the first equation are:
\[ z = -1, \quad z = \omega, \quad z = \omega^2 \]

### Step 3: Check common roots in the second equation
Now we check the second equation:
\[ z^{2020} + z^{2018} + 1 = 0 \]
We can factor this as:
\[ z^{2018}(z^2 + 1) + 1 = 0 \]
This can be rewritten as:
\[ z^{2018}(z^2 + 1) = -1 \]
We will check if \( z = -1 \), \( z = \omega \), and \( z = \omega^2 \) satisfy this equation.

1. **For \( z = -1 \)**:
\[ (-1)^{2020} + (-1)^{2018} + 1 = 1 + 1 + 1 = 3 \neq 0 \]
So, \( z = -1 \) is not a common root.

2. **For \( z = \omega \)**:
We know \( \omega^3 = 1 \), so:
\[ \omega^{2020} = \omega^{2020 \mod 3} = \omega^1 = \omega \]
\[ \omega^{2018} = \omega^{2018 \mod 3} = \omega^2 \]
Thus:
\[ \omega + \omega^2 + 1 = 0 \]
So, \( z = \omega \) is a common root.

3. **For \( z = \omega^2 \)**:
Similarly:
\[ (\omega^2)^{2020} = \omega^{2020 \mod 3} = \omega^2 \]
\[ (\omega^2)^{2018} = \omega^{2018 \mod 3} = \omega \]
Thus:
\[ \omega^2 + \omega + 1 = 0 \]
So, \( z = \omega^2 \) is also a common root.

### Conclusion
The common roots of the equations \( z^3 + 2z^2 + 2z + 1 = 0 \) and \( z^{2020} + z^{2018} + 1 = 0 \) are:
\[ \omega \text{ and } \omega^2 \]