Exercise 3:

Let the polynomial:

\[P(x) = x^3 + 4x^2 + x - 6.\]

1. Show that \(-2\) is a root of the polynomial \( P(x) \).
2. Deduce that \( P(x) \) is divisible by \((x + 2)\).
3. Determine the real numbers \( a, b, c \) such that:

\[P(x) = (x + 2)(ax^2 + bx + c)\]

1. Solve in \( \mathbb{R} \) the equation:

\[x^2 + 2x - 3 = 0.\]

1. Solve in \( \mathbb{R} : P(x) \geq 0 \).
2. Deduce a solution of the equation:

\[(\sqrt{x - 1})^3 + 4(\sqrt{x - 1})^2 + (\sqrt{x - 1}) - 6 = 0.\]

Exercise 4:

Let the polynomial \( P \):

\[P(x) = x^5 - 3x^4 - x^3 + 11x^2 - 12x + 4.\]

1. Show that 1 is a triple root of \( P(x) \).
2. Give the Taylor expansion of \( P(x) \) at the point 1.
3. Factorize \( P(x) \) into irreducible polynomials in \( \mathbb{R}[x] \).
4. Decompose the rational fraction \( F(x) \) defined by:

\[F(x) = \frac{x^2 - 3x + 2}{P(x)}\]

in \( \mathbb{R}(x) \).