The document you've uploaded appears to be a math exam or worksheet in French, covering various topics such as polynomials, systems of equations, trigonometry, and more. Here are the exercises and questions it contains:

Exercise 1:

• Consider the polynomial $P(x) = x^3 + 6x^2 - x - 30$.
  1. Prove that $P(x)$ is divisible by $(x - 2)$.
  2. Determine the polynomial $Q(x)$ such that $P(x) = (x - 2) Q(x)$.
  3. Solve the equation $Q(x) = 0$.
  4. Factorize the polynomial $P(x)$.
  5. Factorize the polynomial $P(x)$ into first-degree polynomials.
  6. Determine the sign table of the polynomial $P(x)$ and deduce where $P(x) < 0$.

Exercise 2:

• Consider the function $g(x) = \frac{-2x^2 - 3x + 2}{2x + 3}$.
  1. Determine the condition for the existence of $g(x)$.
  2. Solve for $g(x) = 0$.
  3. Create the sign table for $g(x)$.
  4. Deduce the value of $g(x)$.

Exercise 3:

• Solve the system of equations $S$ using Cramer's method: -x + y = 1 \\ 2x - 3y = -4 \end{cases}$$

Exercise 4:

• In an orthonormal reference frame $(O; \overrightarrow{OI}; \overrightarrow{OJ})$, consider a trigonometric circle $C$ centered at $O$ with points $M \left( \frac{-43}{3} \right)$ and $N \left( \frac{121}{6} \right)$ on the circle.
  1. Specify the main curvilinear abscissas for points $M$ and $N$.
  2. Represent the trigonometric circle $C$ and the points $M$ and $N$.
  3. Determine the main measure $\overrightarrow{OM}$, $\overrightarrow{ON}$, and deduce the nature of the triangle $OMN$.
  4. Calculate $\cos \left( \frac{-43}{3} \right)$ and $\sin \left( \frac{121}{6} \right)$.
  5. Let $x$ be a real number, find $A(x) = \cos(23x + x) + \sin(27x + x) + \cos \left( \frac{x}{10} \right) - \sin \left( \frac{x}{8} \right)$.
  6. Prove that $A(x) = -2 \cos(x)$.
  7. Determine $\tan(x)$ and $\cos(x)$, knowing that $\sin(x) = -1$.

Exercise 5:

1. Solve the equation $2 \cos(x) - \sqrt{2} = 0$ and the inequality $2 \cos(x) - \sqrt{2} \leq 0$ in the interval $[0; 2\pi]$.
2. Solve the equation $\sin(x) = -\frac{1}{2}$ and the inequality $\sin(x) < -1$ in the interval $[0; 2\pi]$.
3. Solve the equation $\tan(x) = 1$ and the inequality $\tan(x) \leq -1$ in the interval $[-\pi; \pi]$.

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