Create theorems.yaml

theorems.yaml

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+# 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.