Precalculus

Precalculus is a course of study in mathematics that serves as a crucial bridge between algebra and calculus, designed to equip students with the advanced functional and analytical skills required for the study of change. It involves a rigorous investigation of functions—including polynomial, rational, exponential, logarithmic, and trigonometric—exploring their properties, graphs, and transformations in depth. The curriculum also typically covers advanced trigonometry, vectors, polar coordinates, and an introduction to the concept of limits, all of which provide the essential foundation for success in calculus.

1. Foundations of Precalculus
   1. Review of Fundamental Algebra
      1. The Real Number System
         1. Natural Numbers
         2. Whole Numbers
         3. Integers
         4. Rational Numbers
         5. Irrational Numbers
         6. Properties of Real Numbers
            1. Commutative Property
            2. Associative Property
            3. Distributive Property
            4. Identity Properties
            5. Inverse Properties
         7. The Number Line and Ordering
         8. Absolute Value
         9. Set Notation and Interval Notation
      2. Algebraic Expressions and Polynomials
         1. Variables and Constants
         2. Terms, Factors, and Coefficients
         3. Like Terms and Combining Terms
         4. Operations with Polynomials
            1. Addition and Subtraction
            2. Multiplication
            3. Division by Monomials
            4. Long Division of Polynomials
         5. Degree and Leading Coefficient
         6. Standard Form of Polynomials
      3. Factoring Polynomials
         1. Factoring out the Greatest Common Factor
         2. Factoring by Grouping
         3. Factoring Trinomials
            1. Trial and Error Method
            2. AC Method
         4. Factoring Special Products
            1. Difference of Squares
            2. Perfect Square Trinomials
            3. Sum and Difference of Cubes
         5. Factoring Completely
      4. Rational Expressions
         1. Simplifying Rational Expressions
         2. Multiplying and Dividing Rational Expressions
         3. Adding and Subtracting Rational Expressions
            1. Same Denominator
            2. Different Denominators
            3. Least Common Denominator
         4. Complex Fractions
         5. Restrictions on the Domain
      5. Radicals and Rational Exponents
         1. Simplifying Radicals
         2. Operations with Radicals
            1. Addition and Subtraction
            2. Multiplication and Division
         3. Rationalizing the Denominator
            1. Monomial Denominators
            2. Binomial Denominators
         4. Converting between Radical and Exponential Form
         5. Properties of Exponents
            1. Product Rule
            2. Quotient Rule
            3. Power Rule
            4. Negative Exponents
            5. Zero Exponent
   2. Equations and Inequalities
      1. Linear Equations
         1. Solving Linear Equations in One Variable
         2. Literal Equations
         3. Applications of Linear Equations
      2. Quadratic Equations
         1. Solving Quadratic Equations by Factoring
         2. Completing the Square
         3. The Quadratic Formula
            1. Derivation of the Formula
            2. Using the Quadratic Formula
         4. Discriminant and Nature of Roots
            1. Real and Distinct Roots
            2. Real and Equal Roots
            3. Complex Roots
      3. Complex Numbers
         1. Definition and Standard Form
         2. Imaginary Unit
         3. Addition and Subtraction
         4. Multiplication and Division
         5. Complex Conjugates
         6. Absolute Value of Complex Numbers
         7. Powers of i
      4. Other Types of Equations
         1. Radical Equations
            1. Isolating the Radical
            2. Squaring Both Sides
            3. Checking for Extraneous Solutions
         2. Absolute Value Equations
            1. Isolating the Absolute Value
            2. Splitting into Cases
         3. Rational Equations
            1. Finding Common Denominators
            2. Cross Multiplication
            3. Checking for Extraneous Solutions
         4. Equations with Rational Exponents
      5. Linear Inequalities
         1. Solving Linear Inequalities in One Variable
         2. Graphing on a Number Line
         3. Compound Inequalities
            1. And Inequalities
            2. Or Inequalities
         4. Interval Notation
      6. Polynomial and Rational Inequalities
         1. Sign Chart Method
         2. Critical Points
         3. Test Intervals
         4. Boundary Points
      7. Absolute Value Inequalities
         1. Less Than Cases
         2. Greater Than Cases
         3. Graphical Interpretation
   3. The Cartesian Coordinate System
      1. The Coordinate Plane
         1. Plotting Points
         2. Quadrants and Axes
         3. Ordered Pairs
      2. Distance Formula
         1. Derivation from Pythagorean Theorem
      3. Midpoint Formula
         1. Derivation and Application
         2. Finding Endpoints
      4. Circles
         1. Standard Form of a Circle
         2. General Form of a Circle
         3. Converting between Forms
         4. Graphing Circles
         5. Finding Center and Radius