# Integrated Math III

Our fully accredited Common Core-aligned Integrated Math III course builds upon the strong foundations formed in Integrated Math I and Integrated Math II, further developing students' knowledge and skills in polynomials, rationals, radicals, exponentials, logarithms, number systems, geometry, trigonometry and probability. Upon successful completion of this course, students will have attained the necessary knowledge and skills needed to be successful in Precalculus.

## Content

As part of this comprehensive course, students generalize their knowledge of quadratics to higher-order polynomials. This includes factoring polynomials using various methods, solving polynomial equations, and dividing polynomials. Students will also grasp fundamental polynomial theorems such as the factor and remainder theorems, learn how to plot polynomials and describe their properties.

Students will learn to solve both exponential and logarithmic equations and further develop their aptitude for modeling scenarios with exponential functions, such as calculating compound interest. Additionally, they will develop fluency in radical, rational expressions and equations, including manipulating and simplifying expressions, solving equations, and plotting graphs.

Building upon the foundations laid in Integrated Math II, students will strengthen their knowledge of sequences to include arithmetic and geometric sequences. They'll also explore properties of the real number system, and use their knowledge of quadratics as a stepping stone to investigate the fascinating world of complex numbers.

Students deepen their knowledge of geometry to include circles and parabolas in the coordinate plane. They will further develop their understanding of trigonometry to include the law of sines and cosines, graphs and properties of trigonometric functions.

Upon successful completion of this course, students will have mastered the following:

### Polynomials

• Factoring differences of squares, differences of cubes, biquadratic expressions, and using greatest common factors.
• Solving polynomial equations by factoring.
• Dividing polynomials using algebraic manipulation, synthetic division and long division.
• The factor, remainder, and rational roots theorems.
• Understanding and applying the concept of root multiplicity.
• Graphing polynomials.
• Finding the least common multiple of two polynomials
• Describing numerical relationships (e.g., generating Pythagorean triples) using polynomial identities.

• Simplifying rational and radical expressions using polynomial factorization.
• Adding, subtracting, multiplying, and dividing rational expressions.
• Graphs of reciprocal and radical functions, including performing graph transformations, describing their properties, computing inverses, and calculating intersections of reciprocal functions and lines.
• Rationalizing denominators in algebraic expressions.

### Exponentials & Logarithms

• Solving exponential and logarithmic equations, including cases with different bases, using the laws of logarithms, and the zero-product property.
• Graphing logarithmic functions.
• Computing inverses of exponential and logarithmic functions.
• Using exponential functions to model real-world situations, such as calculating compound interest and cases with continuous compounding.

### Number Systems

• Describing the hierarchical structure of the real number system and classifying numbers accordingly.
• Expressing repeating decimals as fractions.
• Understanding the properties of sums and products of rational and irrational numbers.
• Addition, subtraction, and multiplication of complex numbers.
• Understanding the cyclic property of the imaginary unit and solving related problems.
• Solving quadratic equations with complex roots.

### Sequences

• Constructing an explicit formula for the nth term of an arithmetic or geometric sequence.
• Converting between explicit and recursive formulas for arithmetic and geometric sequences.
• Solving mathematical problems related to arithmetic and geometric sequences (e.g., finding the common difference of an arithmetic sequence given two non-consecutive terms).
• Using arithmetic sequences to model situations in context.

### Conic Sections

• Finding equations of circles and parabolas in a coordinate plane.
• Determining properties of circles and parabolas by completing the square.
• Understanding the focus-directrix property of a parabola and using this to construct a parabola's equation.
• Calculating points of intersection of circles and parabolas with lines.

### Trigonometry

• Understanding the law of sines and cosines and using them to solve real-world problems.
• Calculating the area of an arbitrary triangle or quadrilateral using trigonometry.
• Graphing transformed trigonometric and reciprocal functions and describing properties such as amplitude, periodicity, and evenness/oddness.
• Finding expressions for zeros and extrema of transformed trigonometric functions.
• Using trigonometric functions to model real-world situations.

### Probability & Statistics

• Conditional probability, the law of total probability, tree diagrams, the addition law, independent and mutually exclusive events.
• The rules of sum and product, ordering objects, permutations, combinations, and calculating probabilities using combinatorics.
• Analyzing data sets by sampling, computing their mean, variance, and standard deviation.
1.
Polynomials
30 topics
1.1. Factoring Polynomials
 1.1.1. Factoring Polynomials Using the GCF 1.1.2. Factoring Higher-Order Polynomials as a Difference of Squares 1.1.3. Factoring Cubic Expressions by Grouping 1.1.4. Factoring Sums and Differences of Cubes 1.1.5. Factoring Biquadratic Expressions
1.2. Polynomial Equations
 1.2.1. Determining the Roots of Polynomials 1.2.2. Solving Polynomial Equations Using the GCF 1.2.3. Solving Cubic Equations by Grouping 1.2.4. Solving Biquadratic Equations
1.3. Dividing Polynomials
 1.3.1. Dividing Polynomials Using Synthetic Division 1.3.2. Dividing Polynomials by Linear Binomials Using Long Division 1.3.3. Dividing Polynomials Using Long Division 1.3.4. Dividing Polynomials by Manipulating Rational Expressions
1.4. Polynomial Theorems
 1.4.1. The Factor Theorem 1.4.2. Determining Polynomial Coefficients Using the Factor Theorem 1.4.3. Factoring Cubic Polynomials Using the Factor Theorem 1.4.4. Factoring Quartic Polynomials Using the Factor Theorem 1.4.5. Multiplicities of the Roots of Polynomials 1.4.6. Finding Multiplicities of the Roots of Quartic Polynomials by Factoring 1.4.7. The Remainder Theorem 1.4.8. The Rational Roots Theorem
1.5. Graphs of Polynomials
 1.5.1. Graphing Elementary Cubic Functions 1.5.2. Graphing Cubic Curves Containing Three Distinct Real Roots 1.5.3. Graphing Cubic Curves Containing a Double Root 1.5.4. Graphing Cubic Curves Containing One Distinct Real Root 1.5.5. End Behavior of Polynomials 1.5.6. Graphing General Polynomials
1.6. Properties of Polynomials
 1.6.1. The Least Common Multiple of Two Monomials 1.6.2. The Least Common Multiple of Two Polynomials 1.6.3. Describing Relationships Using Polynomial Identities
2.
24 topics
2.7. Rational Expressions
 2.7.1. Simplifying Rational Expressions Using Polynomial Factorization 2.7.2. Adding and Subtracting Rational Expressions 2.7.3. Adding Rational Expressions With No Common Factors in the Denominator 2.7.4. Multiplying Rational Expressions 2.7.5. Dividing Rational Expressions
2.8. Reciprocal Functions
 2.8.1. Graphing Reciprocal Functions 2.8.2. Graph Transformations of Reciprocal Functions 2.8.3. Combining Graph Transformations of Reciprocal Functions 2.8.4. Domain and Range of Transformed Reciprocal Functions 2.8.5. Inverses of Reciprocal Functions 2.8.6. Finding Intersections of Lines and Reciprocal Functions
 2.9.1. Simplifying Square Root Expressions Using Polynomial Factorization 2.9.2. Rationalizing Denominators of Algebraic Expressions 2.9.3. Rationalizing Denominators With Two Terms
 2.10.1. Graphing the Square Root Function 2.10.2. Graph Transformations of Square Root Functions 2.10.3. Graphing the Cube Root Function 2.10.4. Properties of Transformed Square Root Functions 2.10.5. The Domain of a Transformed Radical Function 2.10.6. The Range of a Transformed Radical Function 2.10.7. Roots of Transformed Radical Functions 2.10.8. Inverses of Radical Functions 2.10.9. Finding Intersections of Lines and Radical Functions 2.10.10. Solving Advanced Radical Equations
3.
Exponentials & Logarithms
18 topics
3.11. Exponential Equations
 3.11.1. Solving Exponential Equations Using Logarithms 3.11.2. Solving Equations Containing the Exponential Function 3.11.3. Solving Exponential Equations With Different Bases 3.11.4. Solving Exponential Equations With Different Bases Using Logarithms 3.11.5. Solving Exponential Equations Using the Zero-Product Property
3.12. Logarithmic Equations
 3.12.1. Solving Logarithmic Equations 3.12.2. Solving Logarithmic Equations Containing the Natural Logarithm 3.12.3. Solving Logarithmic Equations Using the Laws of Logarithms 3.12.4. Solving Logarithmic Equations by Combining the Laws of Logarithms 3.12.5. Solving Logarithmic Equations With Logarithms on Both Sides 3.12.6. Solving Logarithmic Equations Using the Zero-Product Property
3.13. Graphs of Logarithmic Functions
 3.13.1. Graphing Logarithmic Functions 3.13.2. Combining Graph Transformations of Logarithmic Functions 3.13.3. Properties of Transformed Logarithmic Functions 3.13.4. Inverses of Exponential and Logarithmic Functions
3.14. Modeling with Exponential Functions
 3.14.1. Modeling With Compound Interest 3.14.2. Continuously Compounded Interest 3.14.3. Converting Between Exponents
4.
Number Systems
10 topics
4.15. Real Numbers
 4.15.1. The Real Number System 4.15.2. Writing Repeating Decimals as Fractions 4.15.3. Sums and Products of Rational and Irrational Numbers
4.16. Complex Numbers
 4.16.1. Imaginary Numbers 4.16.2. Quadratic Equations with Purely Imaginary Solutions 4.16.3. Complex Numbers 4.16.4. Adding and Subtracting Complex Numbers 4.16.5. Multiplying Complex Numbers 4.16.6. Solving Quadratic Equations With Complex Roots 4.16.7. The Cyclic Property of the Imaginary Unit
5.
Sequences
15 topics
5.17. Arithmetic Sequences
 5.17.1. Arithmetic Sequences 5.17.2. Recursive Formulas for Arithmetic Sequences 5.17.3. The Nth Term of an Arithmetic Sequence 5.17.4. Translating Between Explicit and Recursive Formulas for Arithmetic Sequences 5.17.5. Finding the Common Difference of an Arithmetic Sequence 5.17.6. Finding the Nth Term of an Arithmetic Sequence Given Two Terms 5.17.7. Determining Indexes of Terms in Arithmetic Sequences 5.17.8. Solving for Variables in Arithmetic Sequences 5.17.9. Modeling With Arithmetic Sequences
5.18. Geometric Sequences
 5.18.1. Introduction to Geometric Sequences 5.18.2. The Recursive Formula for a Geometric Sequence 5.18.3. The Nth Term of a Geometric Sequence 5.18.4. Translating Between Explicit and Recursive Formulas for Geometric Sequences 5.18.5. Finding the Common Ratio of a Geometric Sequence Given Two Terms 5.18.6. Determining Indexes of Terms in Geometric Sequences
6.
Conic Sections
15 topics
6.19. Circles as Conic Sections
 6.19.1. Circles in the Coordinate Plane 6.19.2. Equations of Circles Centered at the Origin 6.19.3. Equations of Circles 6.19.4. Determining Properties of Circles by Completing the Square 6.19.5. Calculating Circle Intercepts 6.19.6. Intersections of Circles with Lines
6.20. Parabolas as Conic Sections
 6.20.1. Upward and Downward Opening Parabolas 6.20.2. Left and Right Opening Parabolas 6.20.3. The Vertex of a Parabola 6.20.4. Calculating the Vertex of a Parabola by Completing the Square 6.20.5. The Focus-Directrix Property of a Parabola 6.20.6. Calculating the Focus of a Parabola 6.20.7. Calculating the Directrix of a Parabola 6.20.8. Calculating Intercepts of Parabolas 6.20.9. Intersections of Parabolas With Lines
7.
Trigonometry
30 topics
7.21. Trigonometry with General Triangles
 7.21.1. The Law of Sines 7.21.2. The Law of Cosines 7.21.3. The Area of a General Triangle 7.21.4. Modeling Using the Law of Sines 7.21.5. Modeling Using the Law of Cosines
7.22. Trigonometric Functions
 7.22.1. Graphing Sine and Cosine 7.22.2. Graphing Tangent and Cotangent 7.22.3. Graphing Secant and Cosecant 7.22.4. Describing Properties of the Sine Function 7.22.5. Describing Properties of the Cosine Function 7.22.6. Describing Properties of the Tangent Function 7.22.7. Describing Properties of the Secant Function 7.22.8. Describing Properties of the Cosecant Function 7.22.9. Describing Properties of the Cotangent Function
7.23. Graph Transformations of Trigonometric Functions
 7.23.1. Vertical Translations of Trigonometric Functions 7.23.2. Vertical Stretches of Trigonometric Functions 7.23.3. Horizontal Translations of Trigonometric Functions 7.23.4. Horizontal Stretches of Trigonometric Functions 7.23.5. Combining Graph Transformations of Sine and Cosine 7.23.6. Graph Transformations of Tangent and Cotangent 7.23.7. Combining Graph Transformations of Tangent and Cotangent 7.23.8. Combining Graph Transformations of Secant and Cosecant 7.23.9. Graphing Reflections of Trigonometric Functions 7.23.10. Graphing Reflections of Trigonometric Functions: Three or More Transformations
7.24. Properties of Transformed Trigonometric Functions
 7.24.1. Properties of Transformed Sine and Cosine Functions 7.24.2. Finding Zeros and Extrema of Transformed Sine and Cosine Functions 7.24.3. Properties of Transformed Tangent and Cotangent Functions 7.24.4. Properties of Transformed Secant and Cosecant Functions 7.24.5. Interpreting Trigonometric Models 7.24.6. Modeling With Trigonometric Functions
8.
Probability & Statistics
22 topics
8.25. Conditional Probability
 8.25.1. Conditional Probabilities From Venn Diagrams 8.25.2. Conditional Probabilities From Tables 8.25.3. The Multiplication Law for Conditional Probability 8.25.4. The Law of Total Probability 8.25.5. Tree Diagrams for Dependent Events 8.25.6. Tree Diagrams for Dependent Events: Applications 8.25.7. Independent Events 8.25.8. Tree Diagrams for Independent Events 8.25.9. The Addition Law of Probability 8.25.10. Applying the Addition Law With Event Complements 8.25.11. Mutually Exclusive Events
8.26. Combinatorics
 8.26.1. The Rule of Sum and the Rule of Product 8.26.2. Factorials 8.26.3. Factorials in Variable Expressions 8.26.4. Ordering Objects 8.26.5. Permutations 8.26.6. Combinations 8.26.7. Computing Probabilities Using Combinatorics
8.27. Analyzing Data
 8.27.1. Sampling 8.27.2. Sigma Notation 8.27.3. The Mean of a Data Set 8.27.4. Variance and Standard Deviation