Index
Definitions and theorems of Precalculus
The 49 definitions and 60 theorems of the theory of Precalculus, in the order of the topics. Each entry links to its box in the theory page.
Elementary logic
- Definition Proposition
- Definition Negation
- Definition Conjunction and disjunction
- Theorem De Morgan's laws
- Definition Implication
- Theorem Negation of an implication
- Definition Converse and contrapositive
- Definition Necessary condition and sufficient condition
- Definition Universal and existential quantifier
- Theorem Negation of quantifiers
- Definition Direct proof
- Definition Proof by contrapositive
- Definition Proof by contradiction
Powers and roots
- Definition Power with natural and integer exponent
- Theorem Properties of powers
- Definition N-th root and domain conditions
- Theorem Properties of roots
- Definition Power with rational exponent
- Definition Power with real exponent
- Definition Rationalization
- Theorem Comparison of powers
- Theorem Comparison of roots
Factorization, Ruffini’s rule and polynomial division
- Definition Polynomial and degree
- Theorem Identity principle for polynomials
- Theorem Square and cube of a binomial
- Theorem Difference of squares
- Theorem Sum and difference of cubes
- Theorem Division with remainder
- Theorem Remainder theorem
- Theorem Ruffini's theorem
- Definition Multiplicity of a root
- Definition Ruffini's rule
- Theorem Finding rational roots
- Definition Factoring out a common factor and factoring by grouping
- Theorem Special trinomial
- Definition Factorization strategy
Polynomial and rational equations and inequalities
- Definition First-degree equation
- Theorem Quadratic formula
- Definition Discriminant
- Definition Incomplete equations
- Theorem Sum and product of the roots
- Theorem Zero product property
- Definition Solving by factorization
- Definition Biquadratic equations
- Definition First-degree inequality
- Theorem Sign of the quadratic trinomial
- Definition Sign chart
- Theorem Sign of a factor raised to a power
- Definition Domain conditions
- Definition Rational equation
- Theorem Zero of a fraction
- Definition Rational inequalities
- Theorem Sign of a quotient
- Definition System of inequalities
Equations and inequalities with absolute value
- Definition Absolute value
- Definition Absolute value as distance
- Theorem Properties of absolute value
- Theorem Equation of the form |A| = k
- Theorem Equation of the form |A| = |B|
- Theorem Equation of the form |A| = B
- Theorem Inequality of the form |A| < k
- Theorem Inequality of the form |A| > k
- Theorem Inequalities of the form |A| < B and |A| > B
- Theorem Study of signs by regions
Irrational equations and inequalities
- Definition Domain conditions for roots of even index
- Theorem Equation of the form √A = B
- Theorem Equation of the form √A = √B
- Theorem Inequality of the form √A < B
- Theorem Inequality of the form √A > B
- Theorem Roots of odd index
Logarithmic equations and inequalities
- Definition Logarithm
- Theorem Properties of logarithms
- Theorem Change of base
- Definition Logarithmic function
- Definition Domain conditions
- Theorem Equality of logarithms with the same base
- Theorem Logarithm equal to a number
- Theorem Substitution method
- Theorem Inequalities with the same base
- Theorem Inequality with a number on the right-hand side
Exponential equations and inequalities
- Definition Exponential function
- Theorem Rules of powers
- Theorem Equality of powers with the same base
- Theorem Exponential equation with a constant right-hand side
- Theorem Substitution method
- Theorem Inequalities with the same base
Trigonometric equations and inequalities
- Definition Unit circle, sine and cosine
- Definition Tangent
- Definition Special values
- Theorem Pythagorean identity
- Theorem Related angles
- Theorem Addition and double-angle formulas
- Theorem Elementary equations
- Theorem Equality of two sines or of two cosines
- Theorem Quadratic equations in sine or cosine
- Theorem Homogeneous equations of second degree
- Theorem Auxiliary angle method
- Theorem Elementary inequalities
Analytic geometry and conic sections
- Definition Distance between two points and midpoint
- Definition Equation of a line
- Definition Slope, lines through a point, parallelism and perpendicularity
- Theorem Distance from a point to a line
- Definition Parabola y = ax² + bx + c
- Definition Circle
- Definition Completing the square
- Definition Ellipse in standard form
- Definition Hyperbola in standard form
- Theorem Recognizing a conic from its equation
- Theorem Procedure for reducing and drawing