# Polynomial and Rational Functions

## Modeling Using Variation

A pre-owned car dealer 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. As she considers the offer, she takes into account the typical price of the dealer's cars, the overall market, and how many she can reasonably expect to sell. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.

### Solving Direct Variation Problems

In the example above, Nicole’s earnings can be found by multiplying her sales by her commission. The formula 
 tells us her earnings, 
 come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive. See .
 Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called direct variation. Each variable in this type of relationship varies directly with the other.
 represents the data for Nicole’s potential earnings. We say that earnings vary directly with the sales price of the car. The formula 
 is used for direct variation. The value 
 is a nonzero constant greater than zero and is called the constant of variation. In this case, 
 and 
 


### Solving Inverse Variation Problems

Water temperature in an ocean varies inversely to the water’s depth. Between the depths of 250 feet and 500 feet, the formula 
 gives us the temperature in degrees Fahrenheit at a depth in feet below Earth’s surface. Consider the Atlantic Ocean, which covers 22% of Earth’s surface. At a certain location, at the depth of 500 feet, the temperature may be 28°F.
If we create , we observe that, as the depth increases, the water temperature decreases.
We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be inversely proportional and each term varies inversely with the other. Inversely proportional relationships are also called inverse variations.
For our example,  depicts the inverse variation. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula 
 for inverse variation in this case uses 



### Solving Problems Involving Joint Variation

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable 
  cost, varies jointly with the number of students, 
  and the distance, 



### Key Equations


### Key Concepts

1. A relationship where one quantity is a constant multiplied by another quantity is called direct variation. See .
2. Two variables that are directly proportional to one another will have a constant ratio.
3. A relationship where one quantity is a constant divided by another quantity is called inverse variation. See .
4. Two variables that are inversely proportional to one another will have a constant multiple. See .
5. In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, write an equation describing the relationship of the given variables.

### Numeric

For the following exercises, use the given information to find the unknown value.

### Technology

For the following exercises, use a calculator to graph the equation implied by the given variation.

###  Extensions

For the following exercises, use Kepler’s Law, which states that the square of the time, 
 required for a planet to orbit the Sun varies directly with the cube of the mean distance, 
 that the planet is from the Sun.

###  Real-World Applications

For the following exercises, use the given information to answer the questions.

### Chapter Review Exercises

You have reached the end of Chapter 3: Polynomial and Rational Functions. Let’s review some of the Key Terms, Concepts and Equations you have learned.

### Complex Numbers

Perform the indicated operation with complex numbers.
Solve the following equations over the complex number system.

### Quadratic Functions

For the following exercises, write the quadratic function in standard form. Then, give the vertex and axes intercepts. Finally, graph the function.
For the following problems, find the equation of the quadratic function using the given information.
Answer the following questions.

### Power Functions and Polynomial Functions

For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.
For the following exercises, determine end behavior of the polynomial function.

### Graphs of Polynomial Functions

For the following exercises, find all zeros of the polynomial function, noting multiplicities.
For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.

### Dividing Polynomials

For the following exercises, use long division to find the quotient and remainder.
For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

### Zeros of Polynomial Functions

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.
For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

### Rational Functions

For the following rational functions, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph.
For the following exercises, find the slant asymptote.

### Inverses and Radical Functions

For the following exercises, find the inverse of the function with the domain given.

### Modeling Using Variation

For the following exercises, find the unknown value.
For the following exercises, solve the application problem.

### Chapter Test

Perform the indicated operation or solve the equation.
Give the degree and leading coefficient of the following polynomial function.
Determine the end behavior of the polynomial function.
Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.
Given information about the graph of a quadratic function, find its equation.
Solve the following application problem.
Find all zeros of the following polynomial functions, noting multiplicities.
Based on the graph, determine the zeros of the function and multiplicities.
Use long division to find the quotient.
Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.
Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.
Given the following information about a polynomial function, find the function.
Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions.
For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.
Find the slant asymptote of the rational function.
Find the inverse of the function.
Find the unknown value.
Solve the following application problem.