The common roots of the equations `z^(3)+2x^(2)+2z+1=0 and z^(2017)+z^(2018)+1=0` are

Options

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

Explanation

To find the common roots of the equations \( z^3 + 2z^2 + 2z + 1 = 0 \) and \( z^{2017} + z^{2018} + 1 = 0 \), we will solve each equation step by step.

### Step 1: Solve the first equation \( z^3 + 2z^2 + 2z + 1 = 0 \)

We can rearrange the first equation as follows:

\[ z^3 + 2z^2 + 2z + 1 = 0 \]

We can factor this equation. Notice that we can group the terms:

\[ z^3 + 1 + 2z^2 + 2z = 0 \]

We can rewrite \( z^3 + 1 \) as \( (z + 1)(z^2 - z + 1) \) using the sum of cubes formula:

\[ z^3 + 1 = (z + 1)(z^2 - z + 1) \]

So we can write:

\[ (z + 1)(z^2 + z + 1) = 0 \]

This gives us two factors:

1. \( z + 1 = 0 \) which implies \( z = -1 \)
2. \( z^2 + z + 1 = 0 \)

### Step 2: Solve \( z^2 + z + 1 = 0 \)

To find the roots of \( z^2 + z + 1 = 0 \), we can use the quadratic formula:

\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1, b = 1, c = 1 \):

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

Let \( \omega = \frac{-1 + i\sqrt{3}}{2} \) and \( \omega^2 = \frac{-1 - i\sqrt{3}}{2} \).

### Step 3: Roots of the first equation

Thus, the roots of the first equation are:

- \( z = -1 \)
- \( z = \omega \)
- \( z = \omega^2 \)

### Step 4: Solve the second equation \( z^{2017} + z^{2018} + 1 = 0 \)

We can factor the second equation as follows:

\[ z^{2017} + z^{2018} + 1 = z^{2017}(1 + z) + 1 = 0 \]

This can be rearranged to:

\[ z^{2017} + z^{2017}z + 1 = 0 \]

Factoring out \( z^{2017} \):

\[ z^{2017}(z + 1) + 1 = 0 \]

This implies:

\[ z^{2017}(z + 1) = -1 \]

### Step 5: Check for common roots

We can check if \( z = \omega \) and \( z = \omega^2 \) are roots of the second equation. Since \( \omega^3 = 1 \), we can express \( z^{2017} \) and \( z^{2018} \):

\[ z^{2017} = (\omega^3)^{672} \cdot \omega = \omega \]
\[ z^{2018} = (\omega^3)^{672} \cdot \omega^2 = \omega^2 \]

Substituting back into the equation:

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

This confirms that \( \omega \) and \( \omega^2 \) are indeed roots of the second equation.

### Conclusion

The common roots of the equations \( z^3 + 2z^2 + 2z + 1 = 0 \) and \( z^{2017} + z^{2018} + 1 = 0 \) are:

\[ \omega \text{ and } \omega^2 \]