The image contains exercises about a specific polynomial $P(z)$ with a focus on its roots and properties. The instructions are in French, and I'll summarize and guide you through the steps.

Summary of the tasks:

1. Task 1: Determine two obvious roots of the polynomial $P(z)$.

   • Polynomial $P(z)$ is given as $P(z) = (1 + z)^{6n+1} - z^{6n+1} - 1$.

2. Task 2: Justify that if $\alpha$ is a root of $P$, then $\bar{\alpha}$ (the complex conjugate of $\alpha$) is also a root.

3. Task 3:

   • Define $j = \frac{-1 + i \sqrt{3}}{2}$, which represents a cube root of unity.
   • a) Compute $j^3$ and deduce something from that.
   • b) Calculate $(1 + j)^{6n+1}$.
   • c) Calculate $P(j)$ and $P\left( \frac{1}{j} \right)$.

4. Task 4: Using the previous questions, show that there exists a polynomial $T(z)$ such that: $P(z) = z(z+1)(z^2 + z + 1)T(z)$

5. Task 5: For $n = 1$, factorize $P(z)$ and determine the polynomial $Q(z)$ such that: $P(z) = z(z+1)(z^2 + z + 1)Q(z)$

Would you like to go through each step in detail, starting with the first one? Also, here are five related questions:

1. What are cube roots of unity and how do they relate to the polynomial roots?
2. How can the polynomial factorization in task 4 help in simplifying $P(z)$?
3. What are the properties of the complex number $j = \frac{-1 + i\sqrt{3}}{2}$?
4. How would the behavior of $P(z)$ change for different values of $n$?
5. What role does symmetry play in determining the roots of polynomials like $P(z)$?

Tip: When calculating roots involving complex numbers, always check if their conjugates also satisfy the polynomial equation for symmetry reasons.