# Integrated Math III

Our fully accredited Common-Core aligned Integrated Math III course builds upon the strong foundations formed in Integrated Math I and II, further developing students' knowledge and skills in algebra, geometry, trigonometry, probability, and statistics. In addition, this course introduces new mathematical objects, namely vectors, matrices, and random variables. Upon completing this course, students will have gained all the necessary tools to study calculus and other foundational college-level courses successfully.

## Content

In this course, students generalize their understanding of quadratic functions to include cubic and other polynomials. Students gain a solid experience of fundamental concepts involving these higher-level polynomials, including factorization, division, solving polynomial equations, and essential polynomial theorems and their applications.

Knowledge of inequalities and their solutions is essential when studying college-level mathematics. In this course, students build on existing knowledge of functions to solve various types of inequalities, including quadratic, polynomial, rational, and two-variable nonlinear inequalities.

In this course, students further develop the knowledge gained in Integrated Math II to carry out deep explorations of trigonometric functions. They will learn how to apply the law of sines and cosines, derive and apply trigonometric identities, solve trigonometric equations, and explore inverse trigonometric functions.

At this level, students experience their first taste of alternatives to the Cartesian coordinate system, exploring the basics of parametric and polar coordinates. They will learn how to convert to and from Cartesian coordinates and plot simple curves in these coordinate systems.

Students explore advanced mathematical objects such as vectors and matrices at this level. Students will master vector addition, scalar multiplication and learn about different types of vector products. They will learn to apply various operations to matrices and explore determinants. In addition, they will use these ideas to solve geometric problems involving length, angle, area, and volume. Students will also gain a concrete understanding of linear transformations in the plane and relate these operations to their current knowledge of transformations.

Students take their knowledge of complex numbers to new depths by exploring complex numbers in polar form, De Moivreâ€™s theorem, Eulerâ€™s theorem, and the fundamental theorem of algebra. They will also explore how operations on complex numbers can be interpreted as transformations of vectors in the complex plane.

Students complete their understanding of the four conic sections to include ellipses and hyperbolas. In addition, students achieve mastery of radical and rational functions, including sketching their graphs and describing properties.

Finally, students will explore advanced concepts in probability and statistics, including conditional probability, discrete random variables, and the normal distribution.

Upon successful completion of this course, students will have mastered the following:
• Explore finite geometric series, including computing the sum of a geometric series, and use them to model financial problems.
• Dive deep into polynomials, including factoring, dividing, solving polynomial equations, the factor, remainder, and rational root theorems, graphing polynomials, and describing their closure properties.
• Simplify and manipulate rational expressions, and solve rational equations.
• Thoroughly analyze properties of reciprocal and rational functions, including exploring their end-behavior, roots, vertical asymptotes, and holes. They will describe the behavior of rational functions using limits and describe closure properties of rational functions.
• Study radical expressions and functions, including simplifying square root expressions, rationalizing denominators, and graphing radical functions.
• Solve various inequalities, including quadratic, polynomial, non-polynomial, two-variable, and rational inequalities.
• Extend their knowledge of conic sections to include ellipses and hyperbolas. This includes describing their equations and working with properties such as foci, directrices, and eccentricity.
• Delve into trigonometry with general triangles, including the law of sines, the law of cosines, areas of general triangles, and modeling and solving problems using these methods.
• Understand inverse trigonometric functions, including their graphs, domain, and range, and evaluate expressions involving these functions.
• Master trigonometric identities, including the Pythagorean identities, cofunction identities, sum and difference, and double-angle formulas.
• Solve trigonometric equations, including describing their general solutions, cases with transformed functions, and quadratic trigonometric equations.
• Develop a solid foundation of parametric equations, including graphing curves, finding intersections, and working with parametric representations of some types of conic sections.
• Understanding polar coordinates and equations, converting between Cartesian and polar coordinates, and sketching and analyzing simple polar curves.
• Explore complex numbers in depth, including how operations with complex numbers manifest in the complex plane, the polar form of a complex number, De Moivre's theorem, Euler's formula, the roots of a complex number, the fundamental theorem of algebra, and extending polynomial identities.
• Gain proficiency in vector operations, such as addition, scalar multiplication, dot product, and cross product.
• Learn the basics of matrices, including addition, subtraction, scalar multiplication, matrix multiplication, and determinants, and apply these concepts to solve systems of equations and perform linear transformations in the plane.
• Deepen their understanding of probability and statistics, including regression models, conditional probability, discrete random variables such as the binomial and geometric distribution, and the normal distribution.
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. 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
 3.11.1. Simplifying Square Root Expressions Using Polynomial Factorization 3.11.2. Rationalizing Denominators of Algebraic Expressions 3.11.3. Rationalizing Denominators With Two Terms
4.
9 topics
 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. Properties 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
 5.13.1. 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
 9.27.1. 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
10 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 10.28.8. Parametric Equations of Parabolas Centered at (h,k) 10.28.9. Parametric Equations of Horizontal Hyperbolas 10.28.10. Parametric Equations of Vertical Hyperbolas
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. Introduction to Vectors 13.35.2. The Triangle Law for the Addition 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
26 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. Symmetry Properties of the Standard Normal Distribution 15.48.3. The Normal Distribution 15.48.4. Mean and Variance of the Normal Distribution 15.48.5. Percentage Points of the Standard Normal Distribution 15.48.6. Modeling With the Normal Distribution 15.48.7. The Empirical Rule for the Normal Distribution