1. **Stating the problem:** We are given the equation $$z^{50} - 1 = 0$$ and asked to analyze it. 2. **Understanding the equation:** This is a complex equation where $z$ is a complex number and $n=50$. The equation can be rewritten as: $$z^{50} = 1$$ 3. **Formula and important rules:** The solutions to $$z^n = 1$$ are the $n$th roots of unity, given by: $$z_k = \cos\left(\frac{2\pi k}{n}\right) + i \sin\left(\frac{2\pi k}{n}\right) = e^{i \frac{2\pi k}{n}}$$ for $$k = 0, 1, 2, \ldots, n-1$$. 4. **Applying to our problem:** For $$n=50$$, the 50th roots of unity are: $$z_k = e^{i \frac{2\pi k}{50}}$$ for $$k=0,1,2,\ldots,49$$. 5. **Interpretation:** These roots lie on the unit circle in the complex plane, equally spaced at angles of $$\frac{2\pi}{50} = \frac{\pi}{25}$$ radians. 6. **Summary:** The equation $$z^{50} - 1 = 0$$ has 50 distinct complex solutions, all on the unit circle, representing the 50th roots of unity. --- Regarding the functions $$U(t)$$ and $$P(t)$$ mentioned, no explicit formulas were provided, so we cannot sketch or analyze them here.