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| 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: Descartesβ 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: Vietaβs Formulas (Quadratic Case) | |
| statement: For axΒ² + 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 axΒ² + bx + c = 0 are given by x = [-b Β± sqrt(bΒ² - 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, Ο, and ΟΒ² where Ο = -1/2 + sqrt(3)/2 * i. | |
| tags: [roots of unity, complex, cubic] | |
| when_to_use: To factor or solve xΒ³ + 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 β 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: iΒ² = -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. | |