theorems:
  - name: Fundamental Theorem of Algebra
    statement: Every non-zero polynomial of degree n with complex coefficients has exactly n complex roots, counted with multiplicity.
    tags: [roots, complex, algebra]
    when_to_use: When identifying the total number of complex roots of a polynomial.
    short_explanation: Guarantees that a polynomial of degree n has n roots in the complex number system.

  - name: Rational Root Theorem
    statement: Any rational root of a polynomial with integer coefficients is a factor of the constant term divided by a factor of the leading coefficient.
    tags: [rational, integer, divisibility]
    when_to_use: To test possible rational roots of polynomials with integer coefficients.
    short_explanation: Helps guess rational roots based on coefficients; useful before trying numerical methods.

  - name: Complex Conjugate Root Theorem
    statement: If a polynomial has real coefficients and a complex root a + bi, then its conjugate a - bi is also a root.
    tags: [complex, conjugate, real]
    when_to_use: After finding one complex root in real polynomials.
    short_explanation: Ensures non-real roots appear in conjugate pairs when coefficients are real.

  - name: Remainder Theorem
    statement: The remainder of f(x) divided by (x - c) is f(c).
    tags: [evaluation, factor, testing]
    when_to_use: When checking if (x - c) is a factor of a polynomial.
    short_explanation: Allows fast testing of values as roots by plugging into the polynomial.

  - name: Factor Theorem
    statement: (x - c) is a factor of f(x) if and only if f(c) = 0.
    tags: [roots, factors]
    when_to_use: After evaluating f(c) and getting 0.
    short_explanation: Links remainder zero directly to factorization.

  - name: Descartes’ Rule of Signs
    statement: The number of positive real roots is equal to the number of sign changes or less by an even number.
    tags: [signs, real, counting]
    when_to_use: To estimate number of positive or negative real roots.
    short_explanation: Gives an upper bound on number of real roots based on coefficient signs.

  - name: Vieta’s Formulas (Quadratic Case)
    statement: For ax² + bx + c = 0, sum of roots is -b/a and product is c/a.
    tags: [roots, coefficients, relationships]
    when_to_use: When relating roots to coefficients or vice versa.
    short_explanation: Encodes root relationships algebraically, useful for reverse-engineering equations.

  - name: Quadratic Formula
    statement: The solutions to ax² + bx + c = 0 are given by x = [-b ± sqrt(b² - 4ac)] / (2a).
    tags: [quadratic, formula, solution]
    when_to_use: To directly solve any quadratic equation.
    short_explanation: Universal formula for solving second-degree equations, gives real or complex roots.

  - name: Cube Root of Unity Theorem
    statement: The cube roots of unity are 1, ω, and ω² where ω = -1/2 + sqrt(3)/2 * i.
    tags: [roots of unity, complex, cubic]
    when_to_use: To factor or solve x³ + 1 = 0 or similar.
    short_explanation: Provides structure for solving special cubics using symmetric roots.

  - name: Unique Solution Condition (2x2 Systems)
    statement: A linear system ax + by = c, dx + ey = f has a unique solution if ae - bd ≠ 0.
    tags: [linear, determinant, solution condition]
    when_to_use: To check if a system of two equations in two variables has a unique solution.
    short_explanation: The determinant must be non-zero for a unique solution to exist.

  - name: Elimination Method
    statement: Linear combinations of two equations can eliminate a variable to solve the system.
    tags: [linear, elimination]
    when_to_use: To reduce a 2-variable system to one equation.
    short_explanation: Combines equations strategically to remove variables and simplify.

  - name: Substitution Method
    statement: Solve one equation for a variable and substitute into the other.
    tags: [substitution, linear]
    when_to_use: When one variable is easy to isolate.
    short_explanation: Reduces a system to a single-variable equation by replacement.

  - name: Gauss Elimination (Conceptual)
    statement: Any system of linear equations can be reduced using row operations to echelon form.
    tags: [system, reduction, matrix]
    when_to_use: For solving or analyzing larger systems or performing algorithmic solutions.
    short_explanation: Encodes the algebraic elimination steps in matrix language. Useful for generalization.

  - name: Imaginary Unit Identity
    statement: i² = -1 defines the imaginary unit.
    tags: [complex, imaginary, identity]
    when_to_use: When solving quadratics with negative discriminant.
    short_explanation: Enables extension of square roots to negative numbers, yielding complex solutions.

  - name: Root Multiplicity
    statement: If (x - c)^k divides the polynomial but (x - c)^(k+1) does not, then c is a root of multiplicity k.
    tags: [multiplicity, roots, factor]
    when_to_use: To analyze repeated roots.
    short_explanation: Explains why some roots repeat and how they affect the shape of the graph.