Dive deep into the algebra of polynomials and advanced trigonometry. Encounter objects from higher math including complex numbers, vectors, and matrices.

- Understand the relationship between zeros and factors of polynomials.
- Divide polynomials using synthetic and long division.
- Understand the relationship between the value of a polynomial at a given input and the remainder obtained when dividing the polynomial by the corresponding binomial.
- Leverage the rational roots theorem as a strategy to factor polynomials.

- Understand how the multiplicity of a root of a polynomial relates to the shape of the graph near the root.
- Sketch graphs of polynomial functions by identifying end behavior, roots, and behavior near roots.
- Sketch graphs of rational functions by identifying asymptotes.

- Leverage factoring and sign tables as a strategy to solve equations and inequalities involving polynomials and rational functions.

- Use the law of sines and the law of cosines to solve for unknown angles and sides in general triangles.
- Apply trigonometric identities to simplify trigonometric expressions.
- Solve trigonometric equations by leveraging properties of trigonometric functions and extending previous knowledge of linear and quadratic equations.

- Draw connections between algebraic and geometric properties of ellipses and hyperbolas.

- Extend prior knowledge of rectangular curves to parametric and polar curves.

- Manipulate complex numbers algebraically, visualize them geometrically in the complex plane, and connect the algebraic and geometric interpretations through Euler’s formula.

- Generalize prior intuitions about arithmetic to vectors and matrices in higher-dimensional space.
- Interpret matrices as linear transformations of points in the coordinate plane.

- Understand independent events both conceptually and quantitatively from the perspective of conditional probability.
- Define probability mass functions and apply binomial and geometric distributions in modeling contexts.
- Construct an appropriate regression model for a data set and use it to make predictions.

1.

Sequences & Series
7 topics

1.1. Finite Geometric Series

1.1.1. | The Sum of a Finite Geometric Series | |

1.1.2. | The Sum of the First N Terms of a Geometric Series | |

1.1.3. | Writing Geometric Series in Sigma Notation | |

1.1.4. | Finding the Sum of a Geometric Series Given in Sigma Notation | |

1.1.5. | Solving Geometric Series Problems Using Exponential Equations and Inequalities | |

1.1.6. | Modeling With Geometric Series | |

1.1.7. | Modeling Financial Problems Using Geometric Series |

2.

Polynomials
28 topics

2.2. Factoring Polynomials

2.2.1. | Factoring Polynomials Using the Greatest Common Factor | |

2.2.2. | Factoring Higher-Order Polynomials as a Difference of Squares | |

2.2.3. | Factoring Cubic Expressions by Grouping | |

2.2.4. | Factoring Sums and Differences of Cubes | |

2.2.5. | Factoring Biquadratic Expressions |

2.3. Dividing Polynomials

2.3.1. | Dividing Polynomials Using Synthetic Division | |

2.3.2. | Dividing Polynomials by Linear Binomials Using Long Division | |

2.3.3. | Dividing Polynomials Using Long Division | |

2.3.4. | Dividing Polynomials by Manipulating Rational Expressions | |

2.3.5. | Closure Properties of Polynomials |

2.4. Polynomial Equations

2.4.1. | Determining the Roots of Polynomials | |

2.4.2. | Solving Polynomial Equations Using the Greatest Common Factor | |

2.4.3. | Solving Cubic Equations by Grouping | |

2.4.4. | Solving Biquadratic Equations |

2.5. Polynomial Theorems

2.5.1. | The Factor Theorem | |

2.5.2. | Determining Polynomial Coefficients Using the Factor Theorem | |

2.5.3. | Factoring Cubic Polynomials Using the Factor Theorem | |

2.5.4. | Factoring Quartic Polynomials Using the Factor Theorem | |

2.5.5. | Multiplicities of the Roots of Polynomials | |

2.5.6. | Finding Multiplicities of the Roots of Quartic Polynomials by Factoring | |

2.5.7. | The Remainder Theorem | |

2.5.8. | The Rational Roots Theorem |

2.6. Graphs of Polynomials

2.6.1. | Graphing Elementary Cubic Functions | |

2.6.2. | Graphing Cubic Curves Containing Three Distinct Real Roots | |

2.6.3. | Graphing Cubic Curves Containing a Double Root | |

2.6.4. | Graphing Cubic Curves Containing One Distinct Real Root | |

2.6.5. | End Behavior of Polynomials | |

2.6.6. | Graphing General Polynomials |

3.

Rational Equations & Functions
29 topics

3.7. Rational Expressions

3.7.1. | Simplifying Rational Expressions Using Polynomial Factorization | |

3.7.2. | Adding and Subtracting Rational Expressions | |

3.7.3. | Adding Rational Expressions With No Common Factors in the Denominator | |

3.7.4. | Multiplying Rational Expressions | |

3.7.5. | Dividing Rational Expressions | |

3.7.6. | Closure Properties of Rational Expressions |

3.8. Rational Equations

3.8.1. | Rational Equations With Three Terms | |

3.8.2. | Advanced Rational Equations | |

3.8.3. | Further Advanced Rational Equations |

3.9. Reciprocal Functions

3.9.1. | Graphing Reciprocal Functions | |

3.9.2. | Graph Transformations of Reciprocal Functions | |

3.9.3. | Combining Graph Transformations of Reciprocal Functions | |

3.9.4. | Domain and Range of Transformed Reciprocal Functions | |

3.9.5. | Inverses of Reciprocal Functions | |

3.9.6. | Finding Intersections of Lines and Reciprocal Functions |

3.10. Rational Functions

3.10.1. | Finding Roots of Rational Functions | |

3.10.2. | Vertical Asymptotes of Rational Functions | |

3.10.3. | Locating Holes in Rational Functions | |

3.10.4. | Horizontal Asymptotes of Rational Functions | |

3.10.5. | End Behavior of Rational Functions | |

3.10.6. | Infinite Limits of Rational Functions | |

3.10.7. | Infinite Limits of Rational Functions: Advanced Cases | |

3.10.8. | The Domain and Range of a Rational Function | |

3.10.9. | Identifying a Rational Function From a Graph | |

3.10.10. | Identifying a Rational Function From a Graph Containing Holes | |

3.10.11. | Identifying the Graph of a Rational Function |

3.11. Radical Expressions

3.11.1. | Simplifying Square Root Expressions Using Polynomial Factorization | |

3.11.2. | Rationalizing Denominators of Algebraic Expressions | |

3.11.3. | Rationalizing Denominators With Two Terms |

4.

Radical Functions
9 topics

4.12. Graphs of Radical Functions

4.12.1. | Graphing the Square Root Function | |

4.12.2. | Graph Transformations of Square Root Functions | |

4.12.3. | Graphing the Cube Root Function | |

4.12.4. | Domain, Range, and Roots of Transformed Square Root Functions | |

4.12.5. | The Domain of a Transformed Radical Function | |

4.12.6. | The Range of a Transformed Radical Function | |

4.12.7. | Roots of Transformed Radical Functions | |

4.12.8. | Inverses of Radical Functions | |

4.12.9. | Finding Intersections of Lines and Radical Functions |

5.

Inequalities
18 topics

5.13. Quadratic Inequalities

5.13.1. | Solving Elementary Quadratic Inequalities | |

5.13.2. | Solving Quadratic Inequalities From Graphs | |

5.13.3. | Solving Quadratic Inequalities Using the Graphical Method | |

5.13.4. | Solving Quadratic Inequalities Using the Sign Table Method | |

5.13.5. | Solving Discriminant Problems Using Quadratic Inequalities |

5.14. Polynomial Inequalities

5.14.1. | Inequalities Involving Powers of Binomials | |

5.14.2. | Solving Polynomial Inequalities Using a Graphical Method | |

5.14.3. | Solving Polynomial Inequalities Using Special Factoring Techniques and the Graphical Method | |

5.14.4. | Solving Polynomial Inequalities Using the Sign Table Method |

5.15. Non-Polynomial Inequalities

5.15.1. | Solving Inequalities Involving Radical Functions | |

5.15.2. | Solving Inequalities Involving Exponential Functions | |

5.15.3. | Solving Inequalities Involving Logarithmic Functions | |

5.15.4. | Solving Inequalities Involving Exponential Functions and Polynomials | |

5.15.5. | Solving Inequalities Involving Positive and Negative Factors |

5.16. Two-Variable Inequalities

5.16.1. | Solving Two-Variable Nonlinear Inequalities | |

5.16.2. | Further Solving of Two-Variable Nonlinear Inequalities |

5.17. Rational Inequalities

5.17.1. | Solving Rational Inequalities | |

5.17.2. | Further Solving of Rational Inequalities |

6.

Conics
20 topics

6.18. Ellipses as Conic Sections

6.18.1. | Introduction to Ellipses | |

6.18.2. | Equations of Ellipses Centered at the Origin | |

6.18.3. | Equations of Ellipses Centered at a General Point | |

6.18.4. | Finding the Center and Axes of Ellipses by Completing the Square | |

6.18.5. | Finding Intercepts of Ellipses | |

6.18.6. | Finding Intersections of Ellipses and Lines | |

6.18.7. | Foci of Ellipses | |

6.18.8. | Vertices and Eccentricity of Ellipses | |

6.18.9. | Directrices of Ellipses | |

6.18.10. | The Area of an Ellipse |

6.19. Hyperbolas as Conic Sections

6.19.1. | Equations of Hyperbolas Centered at the Origin | |

6.19.2. | Equations of Hyperbolas Centered at a General Point | |

6.19.3. | Asymptotes of Hyperbolas Centered at the Origin | |

6.19.4. | Asymptotes of Hyperbolas Centered at a General Point | |

6.19.5. | Finding Intercepts and Intersections of Hyperbolas | |

6.19.6. | Transverse Axes of Hyperbolas | |

6.19.7. | Conjugate Axes of Hyperbolas | |

6.19.8. | Foci of Hyperbolas | |

6.19.9. | Eccentricity and Vertices of Hyperbolas | |

6.19.10. | Directrices of Hyperbolas |

7.

Trigonometric Functions
10 topics

7.20. Trigonometry with General Triangles

7.20.1. | The Law of Sines | |

7.20.2. | The Law of Cosines | |

7.20.3. | The Area of a General Triangle | |

7.20.4. | Modeling Using the Law of Sines | |

7.20.5. | Modeling Using the Law of Cosines |

7.21. The Inverse Trigonometric Functions

7.21.1. | Graphing the Inverse Sine Function | |

7.21.2. | Graphing the Inverse Cosine Function | |

7.21.3. | Graphing the Inverse Tangent Function | |

7.21.4. | Evaluating Expressions Containing Inverse Trigonometric Functions | |

7.21.5. | Further Evaluating Expressions Containing Inverse Trigonometric Functions |

8.

Trigonometric Identities
22 topics

8.22. Trigonometric Identities

8.22.1. | Simplifying Expressions Using Basic Trigonometric Identities | |

8.22.2. | Simplifying Expressions Using the Pythagorean Identity | |

8.22.3. | Alternate Forms of the Pythagorean Identity | |

8.22.4. | Simplifying Expressions Using the Secant-Tangent Identity | |

8.22.5. | Alternate Forms of the Secant-Tangent Identity | |

8.22.6. | Simplifying Trigonometric Expressions Using the Cotangent-Cosecant Identity | |

8.22.7. | Simplifying Trigonometric Expressions Using Cofunction Identities |

8.23. The Sum and Difference Formulas

8.23.1. | The Sum and Difference Formulas for Sine | |

8.23.2. | The Sum and Difference Formulas for Cosine | |

8.23.3. | The Sum and Difference Formulas for Tangent | |

8.23.4. | Calculating Trigonometric Ratios Using the Sum Formula for Sine | |

8.23.5. | Calculating Trigonometric Ratios Using the Sum Formula for Cosine | |

8.23.6. | Calculating Trigonometric Ratios Using the Sum Formula for Tangent | |

8.23.7. | Writing Sums of Trigonometric Functions in Amplitude-Phase Form |

8.24. The Double-Angle Formulas

8.24.1. | The Double-Angle Formula for Sine | |

8.24.2. | Verifying Trigonometric Identities Using the Double-Angle Formula for Sine | |

8.24.3. | Using the Double-Angle Formula for Sine With the Pythagorean Theorem | |

8.24.4. | The Double-Angle Formula for Cosine | |

8.24.5. | Verifying Trigonometric Identities Using the Double-Angle Formulas for Cosine | |

8.24.6. | Finding Exact Values of Trigonometric Expressions Using the Double-Angle Formulas for Cosine | |

8.24.7. | Simplifying Expressions Using the Double-Angle Formula for Tangent | |

8.24.8. | Verifying Trigonometric Identities Using the Double-Angle Formula for Tangent |

9.

Trigonometric Equations
15 topics

9.25. Elementary Trigonometric Equations

9.25.1. | Elementary Trigonometric Equations Containing Sine | |

9.25.2. | Elementary Trigonometric Equations Containing Cosine | |

9.25.3. | Elementary Trigonometric Equations Containing Tangent | |

9.25.4. | Elementary Trigonometric Equations Containing Secant | |

9.25.5. | Elementary Trigonometric Equations Containing Cosecant | |

9.25.6. | Elementary Trigonometric Equations Containing Cotangent | |

9.25.7. | General Solutions of Elementary Trigonometric Equations |

9.26. Trigonometric Equations Containing Transformed Functions

9.26.1. | General Solutions of Trigonometric Equations With Transformed Functions | |

9.26.2. | Trigonometric Equations Containing Transformed Sine Functions | |

9.26.3. | Trigonometric Equations Containing Transformed Cosine Functions | |

9.26.4. | Trigonometric Equations Containing Transformed Tangent Functions |

9.27. Advanced Trigonometric Equations

9.27.1. | Solving Trigonometric Equations Using the Sin-Cos-Tan Identity | |

9.27.2. | Solving Trigonometric Equations Using the Zero-Product Property | |

9.27.3. | Quadratic Trigonometric Equations Containing Sine or Cosine | |

9.27.4. | Quadratic Trigonometric Equations Containing Tangent or Cotangent |

10.

Parametric Equations
7 topics

10.28. Parametric Equations

10.28.1. | Graphing Curves Defined Parametrically | |

10.28.2. | Cartesian Equations of Parametric Curves | |

10.28.3. | Finding Intercepts of Curves Defined Parametrically | |

10.28.4. | Finding Intersections of Parametric Curves and Lines | |

10.28.5. | Parametric Equations of Circles | |

10.28.6. | Parametric Equations of Ellipses | |

10.28.7. | Parametric Equations of Parabolas |

11.

Polar Equations
6 topics

11.29. Polar Coordinates

11.29.1. | Introduction to Polar Coordinates | |

11.29.2. | Converting from Polar Coordinates to Cartesian Coordinates | |

11.29.3. | Polar Equations of Circles Centered at the Origin | |

11.29.4. | Polar Equations of Radial Lines | |

11.29.5. | Polar Equations of Circles Centered on the Coordinate Axes | |

11.29.6. | Finding Intersections of Polar Curves |

12.

Complex Numbers
28 topics

12.30. The Complex Plane

12.30.1. | The Complex Plane | |

12.30.2. | The Magnitude of a Complex Number | |

12.30.3. | The Argument of a Complex Number | |

12.30.4. | Arithmetic in the Complex Plane | |

12.30.5. | Geometry in the Complex Plane |

12.31. Further Complex Numbers

12.31.1. | The Complex Conjugate | |

12.31.2. | Special Properties of the Complex Conjugate | |

12.31.3. | The Complex Conjugate and the Roots of a Quadratic Equation | |

12.31.4. | Dividing Complex Numbers | |

12.31.5. | Solving Equations by Equating Real and Imaginary Parts | |

12.31.6. | Extending Polynomial Identities to the Complex Numbers |

12.32. Complex Numbers in Polar Form

12.32.1. | The Polar Form of a Complex Number | |

12.32.2. | Products of Complex Numbers Expressed in Polar Form | |

12.32.3. | Quotients of Complex Numbers Expressed in Polar Form | |

12.32.4. | The CIS Notation |

12.33. De Moivre's Theorem

12.33.1. | De Moivre's Theorem | |

12.33.2. | Finding Powers of Complex Numbers Using De Moivre's Theorem | |

12.33.3. | The Power-Reducing Formulas for Sine and Cosine | |

12.33.4. | Euler's Formula | |

12.33.5. | Roots of Unity | |

12.33.6. | Properties of Roots of Unity | |

12.33.7. | Square Roots of Complex Numbers | |

12.33.8. | Higher Roots of Complex Numbers |

12.34. The Fundamental Theorem of Algebra

12.34.1. | The Fundamental Theorem of Algebra for Quadratic Equations | |

12.34.2. | The Fundamental Theorem of Algebra with Cubic Equations | |

12.34.3. | Solving Cubic Equations With Complex Roots | |

12.34.4. | The Fundamental Theorem of Algebra with Quartic Equations | |

12.34.5. | Solving Quartic Equations With Complex Roots |

13.

Vectors
29 topics

13.35. Introduction to Vectors

13.35.1. | Representing Given Information as a Vector | |

13.35.2. | The Triangle Law for the Addition and Subtraction of Two Vectors | |

13.35.3. | Calculating the Magnitude of a Vector From Given Information | |

13.35.4. | Problem Solving Using Vector Diagrams | |

13.35.5. | Parallel Vectors | |

13.35.6. | Unit Vectors | |

13.35.7. | Linear Combinations of Vectors and Their Properties | |

13.35.8. | Describing the Position Vector of a Point Using Known Vectors |

13.36. Vectors in 2D Cartesian Coordinates

13.36.1. | Two-Dimensional Vectors Expressed in Component Form | |

13.36.2. | Addition and Scalar Multiplication of Cartesian Vectors in 2D | |

13.36.3. | Calculating the Magnitude of Cartesian Vectors in 2D | |

13.36.4. | Calculating the Direction of Cartesian Vectors in 2D | |

13.36.5. | Calculating the Components of Cartesian Vectors in 2D | |

13.36.6. | Velocity and Acceleration for Plane Motion | |

13.36.7. | Calculating Displacement for Plane Motion |

13.37. Vectors in 3D Cartesian Coordinates

13.37.1. | Three-Dimensional Vectors in Component Form | |

13.37.2. | Addition and Scalar Multiplication of Cartesian Vectors in 3D | |

13.37.3. | Calculating the Magnitude of Cartesian Vectors in 3D |

13.38. The Dot Product

13.38.1. | Calculating the Dot Product Using Angle and Magnitude | |

13.38.2. | Calculating the Dot Product Using Components | |

13.38.3. | The Angle Between Two Vectors | |

13.38.4. | Calculating a Scalar Projection | |

13.38.5. | Calculating a Vector Projection |

13.39. The Cross Product

13.39.1. | Calculating the Cross Product of Two Vectors Using the Definition | |

13.39.2. | Calculating the Cross Product Using Determinants | |

13.39.3. | Finding Areas Using the Cross Product | |

13.39.4. | The Scalar Triple Product | |

13.39.5. | Volumes of Parallelepipeds | |

13.39.6. | Finding Volumes of Tetrahedrons and Pyramids Using Vector Products |

14.

Matrices
37 topics

14.40. Introduction to Matrices

14.40.1. | Introduction to Matrices | |

14.40.2. | Index Notation for Matrices | |

14.40.3. | Adding and Subtracting Matrices | |

14.40.4. | Properties of Matrix Addition | |

14.40.5. | Scalar Multiplication of Matrices | |

14.40.6. | Zero, Square, Diagonal and Identity Matrices | |

14.40.7. | The Transpose of a Matrix |

14.41. Matrix Multiplication

14.41.1. | Multiplying a Matrix by a Column Vector | |

14.41.2. | Multiplying Square Matrices | |

14.41.3. | Conformability for Matrix Multiplication | |

14.41.4. | Multiplying Matrices | |

14.41.5. | Powers of Matrices | |

14.41.6. | Multiplying a Matrix by the Identity Matrix | |

14.41.7. | Properties of Matrix Multiplication | |

14.41.8. | Representing 2x2 Systems of Equations Using a Matrix Product | |

14.41.9. | Representing 3x3 Systems of Equations Using a Matrix Product |

14.42. Determinants

14.42.1. | The Determinant of a 2x2 Matrix | |

14.42.2. | The Geometric Interpretation of the 2x2 Determinant | |

14.42.3. | The Minors of a 3x3 Matrix | |

14.42.4. | The Determinant of a 3x3 Matrix |

14.43. The Inverse of a Matrix

14.43.1. | Introduction to the Inverse of a Matrix | |

14.43.2. | Inverses of 2x2 Matrices | |

14.43.3. | Calculating the Inverse of a 3x3 Matrix Using the Cofactor Method | |

14.43.4. | Solving 2x2 Systems of Equations Using Inverse Matrices | |

14.43.5. | Solving 3x3 Systems of Equations Using Inverse Matrices |

14.44. Linear Transformations

14.44.1. | Introduction to Linear Transformations | |

14.44.2. | The Standard Matrix of a Linear Transformation | |

14.44.3. | Linear Transformations of Points and Lines in the Plane | |

14.44.4. | Linear Transformations of Objects in the Plane | |

14.44.5. | Dilations and Reflections as Linear Transformations | |

14.44.6. | Shear and Stretch as Linear Transformations | |

14.44.7. | Right-Angle Rotations as Linear Transformations | |

14.44.8. | Rotations as Linear Transformations | |

14.44.9. | Combining Linear Transformations Using 2x2 Matrices | |

14.44.10. | Inverting Linear Transformations | |

14.44.11. | Area Scale Factors of Linear Transformations | |

14.44.12. | Singular Linear Transformations in the Plane |

15.

Statistics & Probability
25 topics

15.45. Regression

15.45.1. | Selecting a Regression Model | |

15.45.2. | Quadratic Regression | |

15.45.3. | Semi-Log Scatter Plots | |

15.45.4. | Exponential Regression |

15.46. Conditional Probability

15.46.1. | Conditional Probabilities From Venn Diagrams | |

15.46.2. | Conditional Probabilities From Tables | |

15.46.3. | The Multiplication Law for Conditional Probability | |

15.46.4. | The Law of Total Probability | |

15.46.5. | Tree Diagrams for Dependent Events | |

15.46.6. | Tree Diagrams for Dependent Events: Applications | |

15.46.7. | Independent Events | |

15.46.8. | Tree Diagrams for Independent Events |

15.47. Discrete Random Variables

15.47.1. | Probability Mass Functions of Discrete Random Variables | |

15.47.2. | Cumulative Distribution Functions for Discrete Random Variables | |

15.47.3. | Expected Values of Discrete Random Variables | |

15.47.4. | The Binomial Distribution | |

15.47.5. | Modeling With the Binomial Distribution | |

15.47.6. | The Geometric Distribution | |

15.47.7. | Modeling With the Geometric Distribution |

15.48. The Normal Distribution

15.48.1. | The Standard Normal Distribution | |

15.48.2. | The Normal Distribution | |

15.48.3. | Mean and Variance of the Normal Distribution | |

15.48.4. | Percentage Points of the Standard Normal Distribution | |

15.48.5. | Modeling With the Normal Distribution | |

15.48.6. | The Empirical Rule for the Normal Distribution |