9: Additional Topics for Ch 4, 6, 7 and 8

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9.1: Inverses and Radical Functions
In this section, we will explore the inverses of polynomial and rational functions and in particular the radical functions we encounter in the process.
- 9.1.1: Inverses and Radical Functions (Exercises)
9.2: Modeling Using Variation
A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.
- 9.2E: Modeling Using Variation (Exercises)
9.3: Partial Fractions
Decompose a ratio of polynomials by writing the partial fractions. Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations. The decomposition with repeated linear factors must account for the factors of the denominator in increasing powers. The decomposition with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor.
- 9.3.1: Partial Fractions (Exercises)
9.4: Matrices and Matrix Operations
To solve a systems of equations, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters.
- 9.4E: Matrices and Matrix Operations (Exercises)
9.5: Solving Systems with Inverses
A matrix that has a multiplicative inverse is called an invertible matrix. Only a square matrix may have a multiplicative inverse, as reversibility is a requirement. Not all square matrices have an inverse. We will look at two methods for finding the inverse of a 2×2 matrix and a third method that can be used on both 2×2 and 3×3 matrices.
- 9.5E: Solving Systems with Inverses (Exercises)
9.6: Solving Systems with Cramer's Rule
In this section, we will study two more strategies for solving systems of equations. A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Cramer’s Rule to solve a system of equations in two & three variables.
- 9.6E: Solving Systems with Cramer's Rule (Exercises)
9.7: Rotation of Axes
In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus at the pole, and a line, the directrix, which is perpendicular to the polar axis.
- 9.7E: Rotation of Axes (Exercises)
9.8: Conic Sections in Polar Coordinates
In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus at the pole, and a line, the directrix, which is perpendicular to the polar axis.
- 9.8E: Conic Sections in Polar Coordinates (Exercises)
9.9: Counting Principles
We encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this counting the possibilities.
- 9.9E: Counting Principles (Exercises)
9.10: Binomial Theorem
A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find $(x+y)^n$ without multiplying the binomial by itself $n$ times.
- 9.10E: Binomial Theorem (Exercises)
9.11: Probability
Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain. The probabilities in a probability model must sum to 1. See Example. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment.
- 9.11E: Probability (Exercises)

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