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 can follow these steps:

### Step 1: Factor the first equation
We start with the polynomial \( z^3 + 2z^2 + 2z + 1 = 0 \). We can try to factor it by checking for rational roots.

By substituting \( z = -1 \):
\[ (-1)^3 + 2(-1)^2 + 2(-1) + 1 = -1 + 2 - 2 + 1 = 0 \]
Thus, \( z = -1 \) is a root. We can factor \( z + 1 \) out of the polynomial.

Using polynomial long division or synthetic division, we divide \( z^3 + 2z^2 + 2z + 1 \) by \( z + 1 \):
\[ z^3 + 2z^2 + 2z + 1 = (z + 1)(z^2 + z + 1) \]

### Step 2: Solve the quadratic equation
Now we need to solve \( z^2 + z + 1 = 0 \). We can use 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 \( \omega^2 = \frac{-1 - i\sqrt{3}}{2} \). The roots of the first equation are:
\[ z = -1, \quad z = \omega, \quad z = \omega^2 \]

### Step 3: Check these roots in the second equation
Now we check if these roots satisfy the second equation \( z^{2020} + z^{2018} + 1 = 0 \).

#### Check \( z = -1 \):
\[ (-1)^{2020} + (-1)^{2018} + 1 = 1 + 1 + 1 = 3 \quad (\text{not a root}) \]

#### Check \( z = \omega \):
Using the property \( \omega^3 = 1 \):
\[ \omega^{2020} = \omega^{3 \cdot 673 + 1} = \omega^1 = \omega \]
\[ \omega^{2018} = \omega^{3 \cdot 672 + 2} = \omega^2 \]
Thus,
\[ \omega + \omega^2 + 1 = 0 \quad (\text{is a root}) \]

#### Check \( z = \omega^2 \):
Similarly,
\[ (\omega^2)^{2020} = \omega^2 \]
\[ (\omega^2)^{2018} = \omega \]
Thus,
\[ \omega^2 + \omega + 1 = 0 \quad (\text{is a root}) \]

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