Update theorems.yaml

theorems.yaml

@@ -1,105 +1,92 @@
-# theorems.yaml
-- id: 1
-  name: Fundamental Theorem of Algebra
-  statement: Every non-zero, single-variable polynomial of degree n with complex coefficients has exactly n complex roots (counting multiplicity).
-  tags: [polynomial, complex, root]
-  use_when: Solving any polynomial equation of degree ≥ 1.
-  short_explanation: Establishes the guarantee that a degree-n polynomial will always have n roots in the complex field. Forms the basis for attempting root-finding.
-
-- id: 2
-  name: Rational Root Theorem
-  statement: If a polynomial with integer coefficients has a rational root p/q (in lowest terms), then p divides the constant term and q divides the leading coefficient.
-  tags: [polynomial, rational, root]
-  use_when: Checking for rational roots in polynomials with integer coefficients.
-  short_explanation: Helps filter candidates for rational roots. Useful before attempting full symbolic solving.
-
-- id: 3
-  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: [polynomial, complex, root]
-  use_when: When roots of a real polynomial are partially complex.
-  short_explanation: Assists in predicting complex root structure and verifying solution completeness.
-
-- id: 4
-  name: Vieta's Formula
-  statement: For a polynomial of degree n, the sums and products of its roots relate directly to its coefficients.
-  tags: [polynomial, relation, roots, coefficients]
-  use_when: Understanding relationships between roots and coefficients.
-  short_explanation: Provides equations linking roots to coefficients, e.g., sum of roots = -b/a in quadratics.
-
-- id: 5
-  name: Factor Theorem
-  statement: If f(c) = 0 for a polynomial f(x), then (x - c) is a factor of f(x).
-  tags: [polynomial, factor, root]
-  use_when: Testing whether a number is a root of a polynomial.
-  short_explanation: Direct connection between root and linear factor. Basis for manual and synthetic division.
-
-- id: 6
-  name: Remainder Theorem
-  statement: The remainder of dividing a polynomial f(x) by (x - c) is f(c).
-  tags: [polynomial, division, remainder]
-  use_when: Quickly finding remainder from division.
-  short_explanation: Useful to check non-zero remainder. Supports Factor Theorem by computing f(c).
-
-- id: 7
-  name: Difference of Cubes Factorization
-  statement: a³ ± b³ = (a ± b)(a² ∓ ab + b²)
-  tags: [polynomial, cubic, factor]
-  use_when: Factoring specific cubic polynomials like x³ + 1, x³ - 8.
-  short_explanation: Used to factor sums or differences of cubes. Often a crucial step in reducing polynomials to linear factors. Depends on algebraic identities.
-
-- id: 8
-  name: Discriminant of Quadratic Equation
-  statement: For ax² + bx + c = 0, the discriminant D = b² - 4ac determines root nature.
-  tags: [quadratic, real, complex, root]
-  use_when: Assessing whether roots are real, repeated, or complex.
-  short_explanation: D > 0 implies real and distinct roots; D = 0 repeated root; D < 0 complex roots.
-
-- id: 9
-  name: Quadratic Formula
-  statement: x = [-b ± sqrt(b² - 4ac)] / 2a solves ax² + bx + c = 0.
-  tags: [polynomial, quadratic, solve]
-  use_when: Solving general second-degree equations.
-  short_explanation: Derived from completing the square. Common tool for solving quadratics, and useful for explanation of irrational/complex roots.
-
-- id: 10
-  name: Unique Solution Condition (2x2 Systems)
-  statement: A linear system ax + by = c, dx + ey = f has a unique solution if the determinant ae - bd ≠ 0.
-  tags: [linear, determinant, unique, solution]
-  use_when: Checking whether two linear equations intersect in a single point.
-  short_explanation: Ensures system consistency and solvability. Connected to invertibility of 2x2 matrices.
-
-- id: 11
-  name: Substitution Method
-  statement: Solve one equation for one variable and substitute into the other.
-  tags: [linear, substitution]
-  use_when: When one variable is already isolated or easily isolatable.
-  short_explanation: A direct technique used to reduce a two-variable system to a single-variable equation. Often first step in manual solving.
-
-- id: 12
-  name: Elimination Method
-  statement: Combine equations to eliminate one variable by addition or subtraction.
-  tags: [linear, elimination]
-  use_when: When variable coefficients are aligned or can be made equal.
-  short_explanation: Simplifies two equations to one by removing a variable. Frequently complements substitution.
-
-- id: 13
-  name: Symmetric System Solving
-  statement: For linear systems with symmetric structure, substitution or elimination is typically efficient.
-  tags: [linear, system, substitution, elimination]
-  use_when: When both equations can be easily rearranged to isolate a variable.
-  short_explanation: Establishes the use of algebraic manipulation methods for solving 2-variable linear systems. Often used before attempting matrix methods.
-
-- id: 14
-  name: Parallel and Coincident Lines
-  statement: Two linear equations with proportional coefficients (but different constants) are parallel; same constants imply they are coincident.
-  tags: [linear, geometry, inconsistency]
-  use_when: Explaining inconsistency or infinite solutions in a system.
-  short_explanation: Helps explain geometrically when lines don’t intersect (parallel) or overlap completely (coincident).
-
-- id: 15
-  name: Gaussian Elimination (2x2)
-  statement: Row-reduction method for solving 2x2 systems using augmented matrix and elementary row operations.
-  tags: [linear, matrix, elimination]
-  use_when: For educational purposes or extensions to 3x3 systems.
-  short_explanation: Encodes the algebraic elimination steps in matrix language. Generalizable to more variables and used in numerical solvers.
+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.
+
+