The equation you're solving is:

\[
z^4 + z^3 + z^2 + z + 1 = 0
\]

This is a **cyclotomic equation**, which can be solved by recognizing it as a polynomial that corresponds to finding the **5th roots of unity** (complex solutions to \( z^5 = 1 \)) except \( z = 1 \).

### Step-by-step solution:

1. Rewrite the equation:

\[
z^4 + z^3 + z^2 + z + 1 = 0
\]

This is the **cyclotomic polynomial** for the 5th roots of unity, and the roots are the complex numbers \( e^{2\pi i k / 5} \) for \( k = 0, 1, 2, 3, 4 \). These are the 5th roots of unity, excluding \( z = 1 \).

### Roots:

The solutions are:

\[
z = e^{2\pi i k / 5}, \quad \text{for} \, k = 1, 2, 3, 4
\]

These correspond to the points on the unit circle in the complex plane at angles \( \frac{2\pi}{5}, \frac{4\pi}{5}, \frac{6\pi}{5}, \frac{8\pi}{5} \) radians, respectively.

In rectangular form, the roots are:

\[
z_1 = e^{2\pi i / 5} = \cos\left(\frac{2\pi}{5}\right) + i \sin\left(\frac{2\pi}{5}\right)
\]
\[
z_2 = e^{4\pi i / 5} = \cos\left(\frac{4\pi}{5}\right) + i \sin\left(\frac{4\pi}{5}\right)
\]
\[
z_3 = e^{6\pi i / 5} = \cos\left(\frac{6\pi}{5}\right) + i \sin\left(\frac{6\pi}{5}\right)
\]
\[
z_4 = e^{8\pi i / 5} = \cos\left(\frac{8\pi}{5}\right) + i \sin\left(\frac{8\pi}{5}\right)
\]

These are the four non-real 5th roots of unity.

Would you like further details on the interpretation of these solutions?

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### 5 Related Questions:
1. What is the significance of cyclotomic polynomials?
2. How do you find the nth roots of unity geometrically?
3. Why is the root \( z = 1 \) excluded in this solution?
4. Can this method be applied to other similar equations, like \( z^5 + z^4 + z^3 + z^2 + z + 1 = 0 \)?
5. How do complex roots relate to the unit circle in the complex plane?

### Tip:
When solving cyclotomic equations, visualizing the roots on the unit circle helps in understanding the symmetry and properties of the solutions.

Explore the step-by-step solution to the cyclotomic equation z^4 + z^3 + z^2 + z + 1 = 0, including the concept of the 5th roots of unity. This problem involves algebraic methods, complex numbers, and geometric interpretation on the unit circle, suitable for students in Grades 11-12 and university levels.

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