# Réponse finale obtenue du bot:
response = « Here’s a theoretical framework for Algebra, covering equations and inequalities, polynomials and functions, and linear algebra and matrices:
**Equations and Inequalities**
* **Fundamental Theorem of Algebra**: Every non-constant polynomial equation has at least one complex root.
* **Rolle’s Theorem**: If a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in (a,b) such that f'(c)=0.
* **Mean Value Theorem**: If a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in (a,b) such that f(b)-f(a)=(b-a)f'(c).
* **Inequality Theory**:
+ **Triangle Inequality**: For any three numbers x, y, and z, |x+y| ≤ |x| + |y| + |z|.
+ **Cauchy-Schwarz Inequality**: For any two vectors u and v, |u · v| ≤ ||u|| ||v||.
+ **AM-GM Inequality**: For any positive real numbers a1, a2, …, an, the arithmetic mean is greater than or equal to the geometric mean: (a1+a2+…+an)/n ≥ (∏ai^(1/n))^(1/n).
**Polynomials and Functions**
* **Division Algorithm**: If f(x) and g(x) are polynomials such that g(x) ≠ 0, then there exist unique polynomials q(x) and r(x) such that f(x) = g(x)q(x)+r(x), where deg(r(x)) < deg(g(x)). * **Rational Function Theory**: For any rational function f(x)/g(x), the domain is the set of all x such that g(x) ≠ 0. * **Function Composition**: If f(x) and g(x) are functions, then (f ∘ g)(x) = f(g(x)). **Linear Algebra and Matrices** * **Vector Space Theory**: For any vector space V, the following properties hold: + Closure: The sum of two vectors in V is also in V. + Commutativity: For any vectors u and v in V, u+v=v+u. + Associativity: For any vectors u, v, and w in V, (u+v)+w=u+(v+w). + Distributivity: For any vector u in V and scalars a and b, a(u+v)=au+av. * **Linear Transformations**: A linear transformation from one vector space to another is a function that preserves the operations of addition and scalar multiplication. * **Matrices and Determinants**: + **Matrix Addition and Multiplication**: For any matrices A and B, their sum and product are defined as follows: (A+B)ij = Aj+i + Bj+i and (AB)ij = Σk=1n Akj*Bkj. + **Determinant**: The determinant of a matrix is a scalar that can be used to determine the solvability of a system of linear equations. This theoretical framework provides a foundation for understanding various concepts in Algebra, including equations and inequalities, polynomials and functions, and linear algebra and matrices. It covers fundamental results such as the Fundamental Theorem of Algebra, Rolle's Theorem, and the Mean Value Theorem, as well as important inequalities like the Triangle Inequality, Cauchy-Schwarz Inequality, and AM-GM Inequality. The framework also includes key concepts in linear algebra, such as vector spaces, linear transformations, matrices, and determinants."