# Graphs and Functions

## Graphs of Functions

### Use the Vertical Line Test

In the last section we learned how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping or an equation. We will now look at how to tell if a graph is that of a function.
An ordered pair  is a solution of a linear equation, if the equation is a true statement when the x- and y-values of the ordered pair are substituted into the equation.
The graph of a linear equation is a straight line where every point on the line is a solution of the equation and every solution of this equation is a point on this line.
In , we can see that, in graph of the equation  for every x-value there is only one y-value, as shown in the accompanying table.
A relation is a function if every element of the domain has exactly one value in the range. So the relation defined by the equation  is a function.
If we look at the graph, each vertical dashed line only intersects the line at one point. This makes sense as in a function, for every x-value there is only one y-value.
If the vertical line hit the graph twice, the x-value would be mapped to two y-values, and so the graph would not represent a function.
This leads us to the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function.

### Identify Graphs of Basic Functions

We used the equation  and its graph as we developed the vertical line test. We said that the relation defined by the equation  is a function.
We can write this as in function notation as  It still means the same thing. The graph of the function is the graph of all ordered pairs  where  So we can write the ordered pairs as  It looks different but the graph will be the same.
Compare the graph of  previously shown in  with the graph of  shown in . Nothing has changed but the notation.
As we move forward in our study, it is helpful to be familiar with the graphs of several basic functions and be able to identify them.
Through our earlier work, we are familiar with the graphs of linear equations. The process we used to decide if  is a function would apply to all linear equations. All non-vertical linear equations are functions. Vertical lines are not functions as the x-value has infinitely many y-values.
We wrote linear equations in several forms, but it will be most helpful for us here to use the slope-intercept form of the linear equation. The slope-intercept form of a linear equation is  In function notation, this linear function becomes  where m is the slope of the line and b is the y-intercept.
The domain is the set of all real numbers, and the range is also the set of all real numbers.
We will use the graphing techniques we used earlier, to graph the basic functions.
The next function whose graph we will look at is called the constant function and its equation is of the form  where b is any real number. If we replace the  with y, we get  We recognize this as the horizontal line whose y-intercept is b. The graph of the function  is also the horizontal line whose y-intercept is b.
Notice that for any real number we put in the function, the function value will be b. This tells us the range has only one value, b.
The identity function,  is a special case of the linear function. If we write it in linear function form,  we see the slope is 1 and the y-intercept is 0.
The next function we will look at is not a linear function. So the graph will not be a line. The only method we have to graph this function is point plotting. Because this is an unfamiliar function, we make sure to choose several positive and negative values as well as 0 for our x-values.
Looking at the result in , we can summarize the features of the square function. We call this graph a parabola. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.
The range is not all real numbers. Notice the graph consists of values of y never go below zero. This makes sense as the square of any number cannot be negative. So, the range of the square function is all non-negative real numbers.
The next function we will look at is also not a linear function so the graph will not be a line. Again we will use point plotting, and make sure to choose several positive and negative values as well as 0 for our x-values.
Looking at the result in , we can summarize the features of the cube function. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.
The range is all real numbers. This makes sense as the cube of any non-zero number can be positive or negative. So, the range of the cube function is all real numbers.
The next function we will look at does not square or cube the input values, but rather takes the square root of those values.
Let’s graph the function  and then summarize the features of the function. Remember, we can only take the square root of non-negative real numbers, so our domain will be the non-negative real numbers.
Our last basic function is the absolute value function,  Keep in mind that the absolute value of a number is its distance from zero. Since we never measure distance as a negative number, we will never get a negative number in the range.

### Read Information from a Graph of a Function

In the sciences and business, data is often collected and then graphed. The graph is analyzed, information is obtained from the graph and then often predictions are made from the data.
We will start by reading the domain and range of a function from its graph.
Remember the domain is the set of all the x-values in the ordered pairs in the function. To find the domain we look at the graph and find all the values of x that have a corresponding value on the graph. Follow the value x up or down vertically. If you hit the graph of the function then x is in the domain.
Remember the range is the set of all the y-values in the ordered pairs in the function. To find the range we look at the graph and find all the values of y that have a corresponding value on the graph. Follow the value y left or right horizontally. If you hit the graph of the function then y is in the range.
We are now going to read information from the graph that you may see in future math classes.

### Key Concepts

1. Vertical Line Test

2. Graph of a Function

3. Linear Function

4. Constant Function

5. Identity Function

6. Square Function

7. Cube Function

8. Square Root Function

9. Absolute Value Function

### Section Exercises

### Practice Makes Perfect

Use the Vertical Line Test
In the following exercises, determine whether each graph is the graph of a function.
Identify Graphs of Basic Functions
In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation.
Read Information from a Graph of a Function
In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation.
In the following exercises, use the graph of the function to find the indicated values.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

### Chapter Review Exercises

### Graph Linear Equations in Two Variables

Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system.
In the following exercises, determine which ordered pairs are solutions to the given equations.
Graph a Linear Equation by Plotting Points
In the following exercises, graph by plotting points.
Graph Vertical and Horizontal lines
In the following exercises, graph each equation.
In the following exercises, graph each pair of equations in the same rectangular coordinate system.
Find
In the following exercises, find the x- and y-intercepts.
In the following exercises, find the intercepts of each equation.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.

### Slope of a Line

Find the Slope of a Line
In the following exercises, find the slope of each line shown.
In the following exercises, find the slope of each line.
Use the Slope Formula to find the Slope of a Line between Two Points
In the following exercises, use the slope formula to find the slope of the line between each pair of points.
Graph a Line Given a Point and the Slope
In the following exercises, graph each line with the given point and slope.
Graph a Line Using Its Slope and Intercept
In the following exercises, identify the slope and y-intercept of each line.
In the following exercises, graph the line of each equation using its slope and y-intercept.
In the following exercises, determine the most convenient method to graph each line.
Graph and Interpret Applications of Slope-Intercept
Use Slopes to Identify Parallel and Perpendicular Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither.

### Find the Equation of a Line

Find an Equation of the Line Given the Slope and
In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.
In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.
Find an Equation of the Line Given the Slope and a Point
In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.
Find an Equation of the Line Given Two Points
In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.
Find an Equation of a Line Parallel to a Given Line
In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.
Find an Equation of a Line Perpendicular to a Given Line
In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

### Graph Linear Inequalities in Two Variables

Verify Solutions to an Inequality in Two Variables
In the following exercises, determine whether each ordered pair is a solution to the given inequality.
Recognize the Relation Between the Solutions of an Inequality and its Graph
In the following exercises, write the inequality shown by the shaded region.
Graph Linear Inequalities in Two Variables
In the following exercises, graph each linear inequality.
Solve Applications using Linear Inequalities in Two Variables

### Relations and Functions

Find the Domain and Range of a Relation
In the following exercises, for each relation, ⓐ find the domain of the relation ⓑ find the range of the relation.
In the following exercise, use the mapping of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.
In the following exercise, use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.
Determine if a Relation is a Function
In the following exercises, use the set of ordered pairs to ⓐ determine whether the relation is a function ⓑ find the domain of the relation ⓒ find the range of the relation.
In the following exercises, use the mapping to ⓐ determine whether the relation is a function ⓑ find the domain of the function ⓒ find the range of the function.
In the following exercises, determine whether each equation is a function.
Find the Value of a Function
In the following exercises, evaluate the function:
ⓐ  ⓑ  ⓒ
In the following exercises, evaluate the function.

### Graphs of Functions

Use the Vertical line Test
In the following exercises, determine whether each graph is the graph of a function.
Identify Graphs of Basic Functions
In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation.
Read Information from a Graph of a Function
In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation
In the following exercises, use the graph of the function to find the indicated values.

### Practice Test

Find the slope of each line shown.
Graph the line for each of the following equations.
Find the equation of each line. Write the equation in slope-intercept form.
Graph each linear inequality.
In the following exercises, ⓐ graph each function ⓑ state its domain and range.Write the domain and range in interval notation.


# Systems of Linear Equations

## Introduction

Climb into your car. Put on your seatbelt. Choose your destination and then…relax. That’s right. You don’t have to do anything else because you are in an autonomous car, or one that navigates its way to your destination! No cars are fully autonomous at the moment and so you theoretically still need to have your hands on the wheel. Self-driving cars may help ease traffic congestion, prevent accidents, and lower pollution. The technology is thanks to computer programmers who are developing software to control the navigation of the car. These programmers rely on their understanding of mathematics, including relationships between equations. In this chapter, you will learn how to solve systems of linear equations in different ways and use them to analyze real-world situations.


# Systems of Linear Equations

## Solve Systems of Linear Equations with Two Variables

### Determine Whether an Ordered Pair is a Solution of a System of Equations

In Solving Linear Equations, we learned how to solve linear equations with one variable. Now we will work with two or more linear equations grouped together, which is known as a system of linear equations.
In this section, we will focus our work on systems of two linear equations in two unknowns. We will solve larger systems of equations later in this chapter.
An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.
A linear equation in two variables, such as  has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.
To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs  that make both equations true. These are called the solutions of a system of equations.
To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.

### Solve a System of Linear Equations by Graphing

In this section, we will use three methods to solve a system of linear equations. The first method we’ll use is graphing.
The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we’ll find the solution to the system.
Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.
Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown.
Each time we demonstrate a new method, we will use it on the same system of linear equations. At the end of the section you’ll decide which method was the most convenient way to solve this system.
The steps to use to solve a system of linear equations by graphing are shown here.
In the next example, we’ll first re-write the equations into slope–intercept form as this will make it easy for us to quickly graph the lines.
In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we’ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.
Sometimes the equations in a system represent the same line. Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true. There are infinitely many solutions to the system.
When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same y-intercept.
The systems of equations in  and  each had two intersecting lines. Each system had one solution.
In , the equations gave coincident lines, and so the system had infinitely many solutions.
The systems in those three examples had at least one solution. A system of equations that has at least one solution is called a consistent system.
A system with parallel lines, like , has no solution. We call a system of equations like this inconsistent. It has no solution.
We also categorize the equations in a system of equations by calling the equations independent or dependent. If two equations are independent, they each have their own set of solutions. Intersecting lines and parallel lines are independent.
If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations, we get coincident lines.
Let’s sum this up by looking at the graphs of the three types of systems. See below and .
Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between  and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.

### Solve a System of Equations by Substitution

We will now solve systems of linear equations by the substitution method.
We will use the same system we used first for graphing.
We will first solve one of the equations for either x or y. We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy.
Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those!
After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.
Be very careful with the signs in the next example.

### Solve a System of Equations by Elimination

We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression.
The third method of solving systems of linear equations is called the Elimination Method. When we solved a system by substitution, we started with two equations and two variables and reduced it to one equation with one variable. This is what we’ll do with the elimination method, too, but we’ll have a different way to get there.
The Elimination Method is based on the Addition Property of Equality. The Addition Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of Equality to say that when you add equal quantities to both sides of an equation, the results are equal.
For any expressions a, b, c, and d.
To solve a system of equations by elimination, we start with both equations in standard form. Then we decide which variable will be easiest to eliminate. How do we decide? We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable.
Notice how that works when we add these two equations together:
The y’s add to zero and we have one equation with one variable.
Let’s try another one:
This time we don’t see a variable that can be immediately eliminated if we add the equations.
But if we multiply the first equation by  we will make the coefficients of x opposites. We must multiply every term on both sides of the equation by
Then rewrite the system of equations.
Now we see that the coefficients of the x terms are opposites, so x will be eliminated when we add these two equations.
Once we get an equation with just one variable, we solve it. Then we substitute that value into one of the original equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both of the original equations.
Now we’ll see how to use elimination to solve the same system of equations we solved by graphing and by substitution.
The steps are listed here for easy reference.
Now we’ll do an example where we need to multiply both equations by constants in order to make the coefficients of one variable opposites.
When the system of equations contains fractions, we will first clear the fractions by multiplying each equation by the LCD of all the fractions in the equation.
When we solved the system by graphing, we saw that not all systems of linear equations have a single ordered pair as a solution. When the two equations were really the same line, there were infinitely many solutions. We called that a consistent system. When the two equations described parallel lines, there was no solution. We called that an inconsistent system.
The same is true using substitution or elimination. If the equation at the end of substitution or elimination is a true statement, we have a consistent but dependent system and the system of equations has infinitely many solutions. If the equation at the end of substitution or elimination is a false statement, we have an inconsistent system and the system of equations has no solution.

### Choose the Most Convenient Method to Solve a System of Linear Equations

When you solve a system of linear equations in in an application, you will not be told which method to use. You will need to make that decision yourself. So you’ll want to choose the method that is easiest to do and minimizes your chance of making mistakes.

### Key Concepts

1. How to solve a system of linear equations by graphing.

2. How to solve a system of equations by substitution.

3. How to solve a system of equations by elimination.

### Practice Makes Perfect

Determine Whether an Ordered Pair is a Solution of a System of Equations
In the following exercises, determine if the following points are solutions to the given system of equations.
Solve a System of Linear Equations by Graphing
In the following exercises, solve the following systems of equations by graphing.
Without graphing, determine the number of solutions and then classify the system of equations.
Solve a System of Equations by Substitution
In the following exercises, solve the systems of equations by substitution.
Solve a System of Equations by Elimination
In the following exercises, solve the systems of equations by elimination.
Choose the Most Convenient Method to Solve a System of Linear Equations
In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

### Writing Exercises

### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Systems of Linear Equations

## Solve Applications with Systems of Equations

### Solve Direct Translation Applications

Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two variables easier than setting them up with just one variable. To solve an application, we’ll first translate the words into a system of linear equations. Then we will decide the most convenient method to use, and then solve the system.
We solved number problems with one variable earlier. Let’s see how differently it works using two variables.
As you solve each application, remember to analyze which method of solving the system of equations would be most convenient.

### Solve Geometry Applications

We will now solve geometry applications using systems of linear equations. We will need to add complementary angles and supplementary angles to our list some properties of angles.
The measures of two complementary angles add to 90 degrees. The measures of two supplementary angles add to 180 degrees.
If two angles are complementary, we say that one angle is the complement of the other.
If two angles are supplementary, we say that one angle is the supplement of the other.
In the next example, we remember that the measures of supplementary angles add to 180.
Recall that the angles of a triangle add up to 180 degrees. A right triangle has one angle that is 90 degrees. What does that tell us about the other two angles? In the next example we will be finding the measures of the other two angles.
Often it is helpful when solving geometry applications to draw a picture to visualize the situation.

### Solve uniform motion applications

We used a table to organize the information in uniform motion problems when we introduced them earlier. We’ll continue using the table here. The basic equation was  where D is the distance traveled, r is the rate, and t is the time.
Our first example of a uniform motion application will be for a situation similar to some we have already seen, but now we can use two variables and two equations.
Many real-world applications of uniform motion arise because of the effects of currents—of water or air—on the actual speed of a vehicle. Cross-country airplane flights in the United States generally take longer going west than going east because of the prevailing wind currents.
Let’s take a look at a boat travelling on a river. Depending on which way the boat is going, the current of the water is either slowing it down or speeding it up.
The images below show how a river current affects the speed at which a boat is actually travelling. We’ll call the speed of the boat in still water b and the speed of the river current c.
The boat is going downstream, in the same direction as the river current. The current helps push the boat, so the boat’s actual speed is faster than its speed in still water. The actual speed at which the boat is moving is
Now, the boat is going upstream, opposite to the river current. The current is going against the boat, so the boat’s actual speed is slower than its speed in still water. The actual speed of the boat is
We’ll put some numbers to this situation in the next example.
Wind currents affect airplane speeds in the same way as water currents affect boat speeds. We’ll see this in the next example. A wind current in the same direction as the plane is flying is called a tailwind. A wind current blowing against the direction of the plane is called a headwind.

### Key Concepts

1. How To Solve Applications with Systems of Equations

### Practice Makes Perfect

Direct Translation Applications
In the following exercises, translate to a system of equations and solve.
Solve Geometry Applications
In the following exercises, translate to a system of equations and solve.
Solve Uniform Motion Applications
In the following exercises, translate to a system of equations and solve.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Systems of Linear Equations

## Solve Mixture Applications with Systems of Equations

### Solve Mixture Applications

Mixture application involve combining two or more quantities. When we solved mixture applications with coins and tickets earlier, we started by creating a table so we could organize the information. For a coin example with nickels and dimes, the table looked like this:
Using one variable meant that we had to relate the number of nickels and the number of dimes. We had to decide if we were going to let n be the number of nickels and then write the number of dimes in terms of n, or if we would let d be the number of dimes and write the number of nickels in terms of d.
Now that we know how to solve systems of equations with two variables, we’ll just let n be the number of nickels and d be the number of dimes. We’ll write one equation based on the total value column, like we did before, and the other equation will come from the number column.
For the first example, we’ll do a ticket problem where the ticket prices are in whole dollars, so we won’t need to use decimals just yet.
In the next example, we’ll solve a coin problem. Now that we know how to work with systems of two variables, naming the variables in the ‘number’ column will be easy.
Some mixture applications involve combining foods or drinks. Example situations might include combining raisins and nuts to make a trail mix or using two types of coffee beans to make a blend.
Another application of mixture problems relates to concentrated cleaning supplies, other chemicals, and mixed drinks. The concentration is given as a percent. For example, a 20% concentrated household cleanser means that 20% of the total amount is cleanser, and the rest is water. To make 35 ounces of a 20% concentration, you mix 7 ounces (20% of 35) of the cleanser with 28 ounces of water.
For these kinds of mixture problems, we’ll use “percent” instead of “value” for one of the columns in our table.

### Solve Interest Applications

The formula to model simple interest applications is  Interest, I, is the product of the principal, P, the rate, r, and the time, t. In our work here, we will calculate the interest earned in one year, so t will be 1.
We modify the column titles in the mixture table to show the formula for interest, as you’ll see in the next example.
The next example requires that we find the principal given the amount of interest earned.

### Solve applications of cost and revenue functions

Suppose a company makes and sells x units of a product. The cost to the company is the total costs to produce x units. This is the cost to manufacture for each unit times x, the number of units manufactured, plus the fixed costs.
The revenue is the money the company brings in as a result of selling x units. This is the selling price of each unit times the number of units sold.
When the costs equal the revenue we say the business has reached the break-even point.

### Key Concepts

1. Cost function: The cost function is the cost to manufacture each unit times x, the number of units manufactured, plus the fixed costs.

2. Revenue: The revenue function is the selling price of each unit times x, the number of units sold.

3. Break-even point: The break-even point is when the revenue equals the costs.

### Practice Makes Perfect

Solve Mixture Applications
In the following exercises, translate to a system of equations and solve.
Solve Interest Applications
In the following exercises, translate to a system of equations and solve.
Solve Applications of Cost and Revenue Functions

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Systems of Linear Equations

## Solve Systems of Equations with Three Variables

### Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables

In this section, we will extend our work of solving a system of linear equations. So far we have worked with systems of equations with two equations and two variables. Now we will work with systems of three equations with three variables. But first let's review what we already know about solving equations and systems involving up to two variables.
We learned earlier that the graph of a linear equation,  is a line. Each point on the line, an ordered pair  is a solution to the equation. For a system of two equations with two variables, we graph two lines. Then we can see that all the points that are solutions to each equation form a line. And, by finding what the lines have in common, we’ll find the solution to the system.
Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions
We know when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown.
Similarly, for a linear equation with three variables  every solution to the equation is an ordered triple, , that makes the equation true.
All the points that are solutions to one equation form a plane in three-dimensional space. And, by finding what the planes have in common, we’ll find the solution to the system.
When we solve a system of three linear equations represented by a graph of three planes in space, there are three possible cases.
To solve a system of three linear equations, we want to find the values of the variables that are solutions to all three equations. In other words, we are looking for the ordered triple  that makes all three equations true. These are called the solutions of the system of three linear equations with three variables.
To determine if an ordered triple is a solution to a system of three equations, we substitute the values of the variables into each equation. If the ordered triple makes all three equations true, it is a solution to the system.

### Solve a System of Linear Equations with Three Variables

To solve a system of linear equations with three variables, we basically use the same techniques we used with systems that had two variables. We start with two pairs of equations and in each pair we eliminate the same variable. This will then give us a system of equations with only two variables and then we know how to solve that system!
Next, we use the values of the two variables we just found to go back to the original equation and find the third variable. We write our answer as an ordered triple and then check our results.
The steps are summarized here.
When we solve a system and end up with no variables and a false statement, we know there are no solutions and that the system is inconsistent. The next example shows a system of equations that is inconsistent.
When we solve a system and end up with no variables but a true statement, we know there are infinitely many solutions. The system is consistent with dependent equations. Our solution will show how two of the variables depend on the third.

### Solve Applications using Systems of Linear Equations with Three Variables

Applications that are modeled by a systems of equations can be solved using the same techniques we used to solve the systems. Many of the application are just extensions to three variables of the types we have solved earlier.

### Key Concepts

1. Linear Equation in Three Variables: A linear equation with three variables, where a, b, c, and d are real numbers and a, b, and c are not all 0, is of the form

Every solution to the equation is an ordered triple,  that makes the equation true.
2. How to solve a system of linear equations with three variables.

### Practice Makes Perfect

Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables
In the following exercises, determine whether the ordered triple is a solution to the system.
Solve a System of Linear Equations with Three Variables
In the following exercises, solve the system of equations.
Solve Applications using Systems of Linear Equations with Three Variables
In the following exercises, solve the given problem.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Systems of Linear Equations

## Solve Systems of Equations Using Matrices

### Write the Augmented Matrix for a System of Equations

Solving a system of equations can be a tedious operation where a simple mistake can wreak havoc on finding the solution. An alternative method which uses the basic procedures of elimination but with notation that is simpler is available. The method involves using a matrix. A matrix is a rectangular array of numbers arranged in rows and columns.
We will use a matrix to represent a system of linear equations. We write each equation in standard form and the coefficients of the variables and the constant of each equation becomes a row in the matrix. Each column then would be the coefficients of one of the variables in the system or the constants. A vertical line replaces the equal signs. We call the resulting matrix the augmented matrix for the system of equations.
Notice the first column is made up of all the coefficients of x, the second column is the all the coefficients of y, and the third column is all the constants.
It is important as we solve systems of equations using matrices to be able to go back and forth between the system and the matrix. The next example asks us to take the information in the matrix and write the system of equations.

### Use Row Operations on a Matrix

Once a system of equations is in its augmented matrix form, we will perform operations on the rows that will lead us to the solution.
To solve by elimination, it doesn’t matter which order we place the equations in the system. Similarly, in the matrix we can interchange the rows.
When we solve by elimination, we often multiply one of the equations by a constant. Since each row represents an equation, and we can multiply each side of an equation by a constant, similarly we can multiply each entry in a row by any real number except 0.
In elimination, we often add a multiple of one row to another row. In the matrix we can replace a row with its sum with a multiple of another row.
These actions are called row operations and will help us use the matrix to solve a system of equations.
Performing these operations is easy to do but all the arithmetic can result in a mistake. If we use a system to record the row operation in each step, it is much easier to go back and check our work.
We use capital letters with subscripts to represent each row. We then show the operation to the left of the new matrix. To show interchanging a row:
To multiply row 2 by :
To multiply row 2 by  and add it to row 1:
Now that we have practiced the row operations, we will look at an augmented matrix and figure out what operation we will use to reach a goal. This is exactly what we did when we did elimination. We decided what number to multiply a row by in order that a variable would be eliminated when we added the rows together.
Given this system, what would you do to eliminate x?
This next example essentially does the same thing, but to the matrix.

### Solve Systems of Equations Using Matrices

To solve a system of equations using matrices, we transform the augmented matrix into a matrix in row-echelon form using row operations. For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.
Once we get the augmented matrix into row-echelon form, we can write the equivalent system of equations and read the value of at least one variable. We then substitute this value in another equation to continue to solve for the other variables. This process is illustrated in the next example.
The steps are summarized here.
Here is a visual to show the order for getting the 1’s and 0’s in the proper position for row-echelon form.
We use the same procedure when the system of equations has three equations.
So far our work with matrices has only been with systems that are consistent and independent, which means they have exactly one solution. Let’s now look at what happens when we use a matrix for a dependent or inconsistent system.
The last system was inconsistent and so had no solutions. The next example is dependent and has infinitely many solutions.

### Key Concepts

1. Matrix: A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with m rows and n columns has order  The matrix on the left below has 2 rows and 3 columns and so it has order  We say it is a 2 by 3 matrix.

Each number in the matrix is called an element or entry in the matrix.
2. Row Operations: In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.

3. Row-Echelon Form: For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

4. How to solve a system of equations using matrices.

### Practice Makes Perfect

Write the Augmented Matrix for a System of Equations
In the following exercises, write each system of linear equations as an augmented matrix.
Write the system of equations that corresponds to the augmented matrix.
Use Row Operations on a Matrix
In the following exercises, perform the indicated operations on the augmented matrices.
Solve Systems of Equations Using Matrices
In the following exercises, solve each system of equations using a matrix.
In the following exercises, solve each system of equations using a matrix.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


# Systems of Linear Equations

## Solve Systems of Equations Using Determinants

In this section we will learn of another method to solve systems of linear equations called Cramer’s rule. Before we can begin to use the rule, we need to learn some new definitions and notation.

### Evaluate the Determinant of a  Matrix

If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant. To find the determinant of the square matrix  we first write it as  To get the real number value of the determinant we subtract the products of the diagonals, as shown.

### Evaluate the Determinant of a  Matrix

To evaluate the determinant of a  matrix, we have to be able to evaluate the minor of an entry in the determinant. The minor of an entry is the  determinant found by eliminating the row and column in the  determinant that contains the entry.
To find the minor of entry  we eliminate the row and column which contain it. So we eliminate the first row and first column. Then we write the  determinant that remains.
To find the minor of entry  we eliminate the row and column that contain it. So we eliminate the 2nd row and 2nd column. Then we write the  determinant that remains.
We are now ready to evaluate a  determinant. To do this we expand by minors, which allows us to evaluate the  determinant using  determinants—which we already know how to evaluate!
To evaluate a  determinant by expanding by minors along the first row, we use the following pattern:
Remember, to find the minor of an entry we eliminate the row and column that contains the entry.
To evaluate a  determinant we can expand by minors using any row or column. Choosing a row or column other than the first row sometimes makes the work easier.
When we expand by any row or column, we must be careful about the sign of the terms in the expansion. To determine the sign of the terms, we use the following sign pattern chart.
Notice that the sign pattern in the first row matches the signs between the terms in the expansion by the first row.
Since we can expand by any row or column, how do we decide which row or column to use? Usually we try to pick a row or column that will make our calculation easier. If the determinant contains a 0, using the row or column that contains the 0 will make the calculations easier.

### Use Cramer’s Rule to Solve Systems of Equations

Cramer’s Rule is a method of solving systems of equations using determinants. It can be derived by solving the general form of the systems of equations by elimination. Here we will demonstrate the rule for both systems of two equations with two variables and for systems of three equations with three variables.
Let’s start with the systems of two equations with two variables.
Notice that to form the determinant D, we use take the coefficients of the variables.
Notice that to form the determinant  and  we substitute the constants for the coefficients of the variable we are finding.
To solve a system of three equations with three variables with Cramer’s Rule, we basically do what we did for a system of two equations. However, we now have to solve for three variables to get the solution. The determinants are also going to be  which will make our work more interesting!
Cramer’s rule does not work when the value of the D determinant is 0, as this would mean we would be dividing by 0. But when  the system is either inconsistent or dependent.
When the value of  and  and  are all zero, the system is consistent and dependent and there are infinitely many solutions.
When the value of  and  and  are not all zero, the system is inconsistent and there is no solution.
In the next example, we will use the values of the determinants to find the solution of the system.

### Solve Applications using Determinants

An interesting application of determinants allows us to test if points are collinear. Three points   and  are collinear if and only if the determinant below is zero.
We will use this property in the next example.

### Key Concepts

1. Determinant: The determinant of any square matrix  where a, b, c, and d are real numbers, is

2. Expanding by Minors along the First Row to Evaluate a 3 × 3 Determinant: To evaluate a  determinant by expanding by minors along the first row, the following pattern:

3. Sign Pattern: When expanding by minors using a row or column, the sign of the terms in the expansion follow the following pattern.

4. Cramer’s Rule: For the system of equations  the solution  can be determined by

Notice that to form the determinant D, we use take the coefficients of the variables.
5. How to solve a system of two equations using Cramer’s rule.

### Practice Makes Perfect

Evaluate the Determinant of a 2 × 2 Matrix
In the following exercises, evaluate the determinant of each square matrix.
Evaluate the Determinant of a 3 × 3 Matrix
In the following exercises, find and then evaluate the indicated minors.
In the following exercises, evaluate each determinant by expanding by minors along the first row.
In the following exercises, evaluate each determinant by expanding by minors.
Use Cramer’s Rule to Solve Systems of Equations
In the following exercises, solve each system of equations using Cramer’s Rule.
Solve Applications Using Determinants
In the following exercises, determine whether the given points are collinear.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Systems of Linear Equations

## Graphing Systems of Linear Inequalities

### Determine whether an ordered pair is a solution of a system of linear inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.
A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.
To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs  that make both inequalities true.
To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

### Solve a System of Linear Inequalities by Graphing

The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.
Systems of linear inequalities where the boundary lines are parallel might have no solution. We’ll see this in the next example.
Some systems of linear inequalities where the boundary lines are parallel will have a solution. We’ll see this in the next example.

### Solve Applications of Systems of Inequalities

The first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so we add inequalities to the system as additional requirements.
When we use variables other than x and y to define an unknown quantity, we must change the names of the axes of the graph as well.

### Key Concepts

1. Solutions of a System of Linear Inequalities: Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true. The solution of a system of linear inequalities is shown as a shaded region in the x, y coordinate system that includes all the points whose ordered pairs make the inequalities true.
2. How to solve a system of linear inequalities by graphing.

### Section Exercises

### Practice Makes Perfect

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
In the following exercises, determine whether each ordered pair is a solution to the system.
Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.
Solve Applications of Systems of Inequalities
In the following exercises, translate to a system of inequalities and solve.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

### Chapter Review Exercises

### Solve Systems of Linear Equations with Two Variables

Determine Whether an Ordered Pair is a Solution of a System of Equations.
In the following exercises, determine if the following points are solutions to the given system of equations.
Solve a System of Linear Equations by Graphing
In the following exercises, solve the following systems of equations by graphing.
In the following exercises, without graphing determine the number of solutions and then classify the system of equations.
Solve a System of Equations by Substitution
In the following exercises, solve the systems of equations by substitution.
Solve a System of Equations by Elimination
In the following exercises, solve the systems of equations by elimination
Choose the Most Convenient Method to Solve a System of Linear Equations
In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

### Solve Applications with Systems of Equations

Solve Direct Translation Applications
In the following exercises, translate to a system of equations and solve.
Solve Geometry Applications
In the following exercises, translate to a system of equations and solve.
Solve Uniform Motion Applications
In the following exercises, translate to a system of equations and solve.

### Solve Mixture Applications with Systems of Equations

Solve Mixture Applications with Systems of Equations
For the following exercises, translate to a system of equations and solve.
Solve Interest Applications
For the following exercises, translate to a system of equations and solve.

### Solve Systems of Equations with Three Variables

Solve Systems of Equations with Three Variables
In the following exercises, determine whether the ordered triple is a solution to the system.
Solve a System of Linear Equations with Three Variables
In the following exercises, solve the system of equations.
Solve Applications using Systems of Linear Equations with Three Variables

### Solve Systems of Equations Using Matrices

Write the Augmented Matrix for a System of Equations.
Write each system of linear equations as an augmented matrix.
Write the system of equations that that corresponds to the augmented matrix.
In the following exercises, perform the indicated operations on the augmented matrices.
Solve Systems of Equations Using Matrices
In the following exercises, solve each system of equations using a matrix.

### Solve Systems of Equations Using Determinants

Evaluate the Determinant of a 2 × 2 Matrix
In the following exercise, evaluate the determinate of the square matrix.
Evaluate the Determinant of a 3 × 3 Matrix
In the following exercise, find and then evaluate the indicated minors.
In the following exercise, evaluate each determinant by expanding by minors along the first row.
In the following exercise, evaluate each determinant by expanding by minors.
Use Cramer’s Rule to Solve Systems of Equations
In the following exercises, solve each system of equations using Cramer’s rule
Solve Applications Using Determinants
In the following exercises, determine whether the given points are collinear.

### Graphing Systems of Linear Inequalities

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
In the following exercises, determine whether each ordered pair is a solution to the system.
Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.
Solve Applications of Systems of Inequalities
In the following exercises, translate to a system of inequalities and solve.

### Chapter Practice Test

In the following exercises, solve the following systems by graphing.
In the following exercises, solve each system of equations. Use either substitution or elimination.
Solve the system of equations using a matrix.
Solve using Cramer’s rule.
In the following exercises, translate to a system of equations and solve.


# Polynomials and Polynomial Functions

## Introduction

You may have coins and paper money in your wallet, but you may soon want to acquire a type of currency called bitcoins. They exist only in a digital wallet on your computer. You can use bitcoins to pay for goods at some companies, or save them as an investment. Although the future of bitcoins is uncertain, investment brokers are beginning to investigate ways to make business predictions using this digital currency. Understanding how bitcoins are created and obtained requires an understanding of a type of function known as a polynomial function. In this chapter you will investigate polynomials and polynomial functions and learn how to perform mathematical operations on them.


# Polynomials and Polynomial Functions

## Add and Subtract Polynomials

### Determine the Degree of Polynomials

We have learned that a term is a constant or the product of a constant and one or more variables. A monomial is an algebraic expression with one term. When it is of the form  where a is a constant and m is a whole number, it is called a monomial in one variable. Some examples of monomials in one variable are

     and
    . Monomials can also have more than one variable such as
   and
A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.
Here are some examples of polynomials.
Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.
The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.
A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0.
Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.
Let's start by looking at a monomial. The monomial  has two variables a and b. To find the degree we need to find the sum of the exponents. The variable a doesn't have an exponent written, but remember that means the exponent is 1. The exponent of b is 2. The sum of the exponents,  is 3 so the degree is 3.
Here are some additional examples.
Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial. Get in the habit of writing the term with the highest degree first.

### Add and Subtract Polynomials

We have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficients.
Remember that like terms must have the same variables with the same exponents.
We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.
Be careful with the signs as you distribute while subtracting the polynomials in the next example.
To subtract  from  we write it as  placing the  first.
When we add and subtract more than two polynomials, the process is the same.

### Evaluate a Polynomial Function for a Given Value

A polynomial function is a function defined by a polynomial. For example,  and  are polynomial functions, because  and  are polynomials.
In Graphs and Functions, where we first introduced functions, we learned that evaluating a function means to find the value of  for a given value of x. To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.
The polynomial functions similar to the one in the next example are used in many fields to determine the height of an object at some time after it is projected into the air. The polynomial in the next function is used specifically for dropping something from 250 ft.

### Add and Subtract Polynomial Functions

Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.

### Key Concepts

1. Monomial

2. Polynomials

3. Degree of a Polynomial

### Practice Makes Perfect

Determine the Type of Polynomials
In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.
Add and Subtract Polynomials
In the following exercises, add or subtract the monomials.
In the following exercises, add the polynomials.
In the following exercises, subtract the polynomials.
In the following exercises, subtract the polynomials.
In the following exercises, find the difference of the polynomials.
In the following exercises, add the polynomials.
In the following exercises, add or subtract the polynomials.
Evaluate a Polynomial Function for a Given Value
In the following exercises, find the function values for each polynomial function.
In the following exercises, find the height for each polynomial function.
Add and Subtract Polynomial Functions
In each example, find ⓐ (f + g)(x) ⓑ (f + g)(2) ⓒ (f − g)(x) ⓓ (f − g)(−3).

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Polynomials and Polynomial Functions

## Properties of Exponents and Scientific Notation

### Simplify Expressions Using the Properties for Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression  the exponent m tells us how many times we use the base a as a factor.
Let’s review the vocabulary for expressions with exponents.
When we combine like terms by adding and subtracting, we need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.
First, we will look at an example that leads to the Product Property.
Notice that 5 is the sum of the exponents, 2 and 3. We see  is  or
The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.
Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.
Notice, in each case the bases were the same and we subtracted exponents. We see  is  or . We see  is or  When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator--notice the numerator of 1. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, and so we need a 1 in the numerator. . This leads to the Quotient Property for Exponents.
A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like  We know for any  since any number divided by itself is 1.
The Quotient Property for Exponents shows us how to simplify  when  and when  by subtracting exponents. What if  We will simplify in two ways to lead us to the definition of the Zero Exponent Property. In general, for
We see  simplifies to  and to 1. So  Any non-zero base raised to the power of zero equals 1.
In this text, we assume any variable that we raise to the zero power is not zero.

### Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents has two forms depending on whether the exponent is larger in the numerator or the denominator. What if we just subtract exponents regardless of which is larger?
Let’s consider  We subtract the exponent in the denominator from the exponent in the numerator. We see  is  or
We can also simplify  by dividing out common factors:
This implies that  and it leads us to the definition of a negative exponent. If n is an integer and  then
Let’s now look at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

This implies  and is another form of the definition of Properties of Negative Exponents.
The negative exponent tells us we can rewrite the expression by taking the reciprocal of the base and then changing the sign of the exponent.
Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.
For example, if after simplifying an expression we end up with the expression  we will take one more step and write  The answer is considered to be in simplest form when it has only positive exponents.
Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.
This leads us to the Quotient to a Negative Power Property.
Now that we have negative exponents, we will use the Product Property with expressions that have negative exponents.
Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

Notice the 6 is the product of the exponents, 2 and 3. We see that  is  or
We multiplied the exponents. This leads to the Power Property for Exponents.
We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

Notice that each factor was raised to the power and  is
The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.
Now we will look at an example that will lead us to the Quotient to a Power Property.

Notice that the exponent applies to both the numerator and the denominator.
We see that  is
This leads to the Quotient to a Power Property for Exponents.
We now have several properties for exponents. Let’s summarize them and then we’ll do some more examples that use more than one of the properties.

### Use Scientific Notation

Working with very large or very small numbers can be awkward. Since our number system is base ten we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers 4,000 and 0.004.
Using place value, we can rewrite the numbers 4,000 and 0.004. We know that 4,000 means  and 0.004 means
If we write the 1,000 as a power of ten in exponential form, we can rewrite these numbers in this way:
When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than ten, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.
It is customary in scientific notation to use as the  multiplication sign, even though we avoid using this sign elsewhere in algebra.
If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.
In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.
The power of 10 is positive when the number is larger than 1:
The power of 10 is negative when the number is between 0 and 1:
How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.
If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.
In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.
When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

### Key Concepts

1. Exponential Notation

This is read a to the  power.
In the expression , the exponent m tells us how many times we use the base a as a factor.
2. Product Property for Exponents
If a is a real number and m and n are integers, then

To multiply with like bases, add the exponents.
3. Quotient Property for Exponents
If  is a real number,  and m and n are integers, then

4. Zero Exponent

5. Negative Exponent

6. Quotient to a Negative Exponent Property
If  are real numbers,  and  is an integer, then

7. Power Property for Exponents
If  is a real number and  are integers, then

To raise a power to a power, multiply the exponents.
8. Product to a Power Property for Exponents
If a and b are real numbers and m is a whole number, then

To raise a product to a power, raise each factor to that power.
9. Quotient to a Power Property for Exponents
If  and are real numbers,  and  is an integer, then

To raise a fraction to a power, raise the numerator and denominator to that power.
10. Summary of Exponent Properties
If a and b are real numbers, and m and n are integers, then

11. Scientific Notation
A number is expressed in scientific notation when it is of the form

12. How to convert a decimal to scientific notation.

13. How to convert scientific notation to decimal form.

### Practice Makes Perfect

Simplify Expressions Using the Properties for Exponents
In the following exercises, simplify each expression using the properties for exponents.
Use the Definition of a Negative Exponent
In the following exercises, simplify each expression.
In the following exercises, simplify each expression using the Product Property.
In the following exercises, simplify each expression using the Power Property.
In the following exercises, simplify each expression using the Product to a Power Property.
In the following exercises, simplify each expression using the Quotient to a Power Property.
In the following exercises, simplify each expression by applying several properties.

### Mixed Practice

In the following exercises, simplify each expression.
Use Scientific Notation
In the following exercises, write each number in scientific notation.
In the following exercises, convert each number to decimal form.
In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all goals?


# Polynomials and Polynomial Functions

## Multiply Polynomials

### Multiply Monomials

We are ready to perform operations on polynomials. Since monomials are algebraic expressions, we can use the properties of exponents to multiply monomials.

### Multiply a Polynomial by a Monomial

Multiplying a polynomial by a monomial is really just applying the Distributive Property.

### Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.
If you multiply binomials often enough you may notice a pattern. Notice that the first term in the result is the product of the first terms in each binomial. The second and third terms are the product of multiplying the two outer terms and then the two inner terms. And the last term results from multiplying the two last terms,
We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. We use this as another method of multiplying binomials. The word FOIL is easy to remember and ensures we find all four products.
Let’s multiply  using both methods.
We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!
When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.
Now we will do an example where we use the FOIL pattern to multiply two binomials.
The final products in the last example were trinomials because we could combine the two middle terms. This is not always the case.
The FOIL method is usually the quickest method for multiplying two binomials, but it only works for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.
Now we’ll apply this same method to multiply two binomials.
We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The methods are listed here all together, to help you remember them.

### Multiply a Polynomial by a Polynomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.
We have now seen two methods you can use to multiply a polynomial by a polynomial. After you practice each method, you’ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.

### Multiply Special Products

Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and multiplying them, there is less work to do if you learn to use a pattern. Let’s start by looking at three examples and look for a pattern.
Look at these results. Do you see any patterns?
What about the number of terms? In each example we squared a binomial and the result was a trinomial.
Now look at the first term in each result. Where did it come from?
The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!
  To get the first term of the product, square the first term.
Where did the last term come from? Look at the examples and find the pattern.
The last term is the product of the last terms, which is the square of the last term.
  To get the last term of the product, square the last term.
Finally, look at the middle term. Notice it came from adding the “outer” and the “inner” terms—which are both the same! So the middle term is double the product of the two terms of the binomial.
  To get the middle term of the product, multiply the terms and double their product.
Putting it all together:
We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.
A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a conjugate pair and is of the form
There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier. Let’s look for the pattern by using FOIL to multiply some conjugate pairs.
What do you observe about the products?
The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.
Each first term is the product of the first terms of the binomials, and since they are identical it is the square of the first term.
   To get the first term, square the first term.
The last term came from multiplying the last terms, the square of the last term.
   To get the last term, square the last term.
Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.
The product of conjugates is always of the form  This is called a difference of squares.
This leads to the pattern:
We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.

### Multiply Polynomial Functions

Just as polynomials can be multiplied, polynomial functions can also be multiplied.

### Key Concepts

1. How to use the FOIL method to multiply two binomials.

2. Multiplying Two Binomials: To multiply binomials, use the:

3. Multiplying a Polynomial by a Polynomial: To multiply a trinomial by a binomial, use the:

4. Binomial Squares Pattern
If a and b are real numbers,

5. Product of Conjugates Pattern
If  are real numbers

The product is called a difference of squares.
To multiply conjugates, square the first term, square the last term, write it as a difference of squares.
6. Comparing the Special Product Patterns

7. Multiplication of Polynomial Functions:

### Practice Makes Perfect

Multiply Monomials
In the following exercises, multiply the monomials.
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
Multiply a Binomial by a Binomial
In the following exercises, multiply the binomials using ⓐ the Distributive Property; ⓑ the FOIL method; ⓒ the Vertical Method.
In the following exercises, multiply the binomials. Use any method.
Multiply a Polynomial by a Polynomial
In the following exercises, multiply using ⓐ the Distributive Property; ⓑ the Vertical Method.
Multiply Special Products
In the following exercises, multiply. Use either method.
In the following exercises, square each binomial using the Binomial Squares Pattern.
In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.
In the following exercises, find each product.

### Mixed Practice

Multiply Polynomial Functions

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Polynomials and Polynomial Functions

## Dividing Polynomials

### Dividing Monomials

We are now familiar with all the properties of exponents and used them to multiply polynomials. Next, we’ll use these properties to divide monomials and polynomials.
Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

### Divide a Polynomial by a Monomial

Now that we know how to divide a monomial by a monomial, the next procedure is to divide a polynomial of two or more terms by a monomial.
The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition. The sum  simplifies to
Now we will do this in reverse to split a single fraction into separate fractions. For example,  can be written
This is the “reverse” of fraction addition and it states that if a, b, and c are numbers where  then  We will use this to divide polynomials by monomials.

### Divide Polynomials Using Long Division

Divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.
We check division by multiplying the quotient by the divisor.
If we did the division correctly, the product should equal the dividend.
Now we will divide a trinomial by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.
When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In the next example, we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.
Look back at the dividends in previous examples. The terms were written in descending order of degrees, and there were no missing degrees. The dividend in this example will be  It is missing an  term. We will add in  as a placeholder.
In the next example, we will divide by  As we divide, we will have to consider the constants as well as the variables.

### Divide Polynomials using Synthetic Division

As we have mentioned before, mathematicians like to find patterns to make their work easier. Since long division can be tedious, let’s look back at the long division we did in  and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.
Synthetic division basically just removes unnecessary repeated variables and numbers. Here all the  and  are removed. as well as the  and  as they are opposite the term above.
The first row of the synthetic division is the coefficients of the dividend. The  is the opposite of the 5 in the divisor.
The second row of the synthetic division are the numbers shown in red in the division problem.
The third row of the synthetic division are the numbers shown in blue in the division problem.
Notice the quotient and remainder are shown in the third row.
The following example will explain the process.
In the next example, we will do all the steps together.

### Divide Polynomial Functions

Just as polynomials can be divided, polynomial functions can also be divided.

### Use the Remainder and Factor Theorem

Let’s look at the division problems we have just worked that ended up with a remainder. They are summarized in the chart below. If we take the dividend from each division problem and use it to define a function, we get the functions shown in the chart. When the divisor is written as  the value of the function at  is the same as the remainder from the division problem.
To see this more generally, we realize we can check a division problem by multiplying the quotient times the divisor and add the remainder. In function notation we could say, to get the dividend  we multiply the quotient,  times the divisor,  and add the remainder, r.

This leads us to the Remainder Theorem.
When we divided  by  in  the result was  To check our work, we multiply  by  to get .
Written this way, we can see that  and  are factors of  When we did the division, the remainder was zero.
Whenever a divisor,  divides a polynomial function,  and resulting in a remainder of zero, we say  is a factor of
The reverse is also true. If  is a factor of  then  will divide the polynomial function resulting in a remainder of zero.
We will state this in the Factor Theorem.

### Key Concepts

1. Division of a Polynomial by a Monomial

2. Division of Polynomial Functions

3. Remainder Theorem

4. Factor Theorem: For any polynomial function

### Section Exercises

### Practice Makes Perfect

Divide Monomials
In the following exercises, divide the monomials.
Divide a Polynomial by a Monomial
In the following exercises, divide each polynomial by the monomial.
Divide Polynomials using Long Division
In the following exercises, divide each polynomial by the binomial.
Divide Polynomials using Synthetic Division
In the following exercises, use synthetic Division to find the quotient and remainder.
Divide Polynomial Functions
In the following exercises, divide.
Use the Remainder and Factor Theorem
In the following exercises, use the Remainder Theorem to find the remainder.
In the following exercises, use the Factor Theorem to determine if  is a factor of the polynomial function.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

### Chapter Review Exercises

### Add and Subtract Polynomials

Types of Polynomials
In the following exercises, determine the type of polynomial.
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
In the following exercises, simplify.
Evaluate a Polynomial Function for a Given Value of the Variable
In the following exercises, find the function values for each polynomial function.
Add and Subtract Polynomial Functions
In the following exercises, find ⓐ (f + g)(x) ⓑ (f + g)(3) ⓒ (f − g)(x) ⓓ (f − g)(−2)

### Properties of Exponents and Scientific Notation

Simplify Expressions Using the Properties for Exponents
In the following exercises, simplify each expression using the properties for exponents.
Use the Definition of a Negative Exponent
In the following exercises, simplify each expression.
In the following exercises, simplify each expression using the Product Property.
In the following exercises, simplify each expression using the Power Property.
In the following exercises, simplify each expression using the Product to a Power Property.
In the following exercises, simplify each expression using the Quotient to a Power Property.
In the following exercises, simplify each expression by applying several properties.
In the following exercises, write each number in scientific notation.
In the following exercises, convert each number to decimal form.
In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

### Multiply Polynomials

Multiply Monomials
In the following exercises, multiply the monomials.
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
Multiply a Binomial by a Binomial
In the following exercises, multiply the binomials using:
ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.
In the following exercises, multiply the binomials. Use any method.
Multiply a Polynomial by a Polynomial
In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.
In the following exercises, multiply. Use either method.
Multiply Special Products
In the following exercises, square each binomial using the Binomial Squares Pattern.
In the following exercises, multiply each pair of conjugates using the Product of Conjugates.

### Divide Monomials

Divide Monomials
In the following exercises, divide the monomials.
Divide a Polynomial by a Monomial
In the following exercises, divide each polynomial by the monomial
Divide Polynomials using Long Division
In the following exercises, divide each polynomial by the binomial.
Divide Polynomials using Synthetic Division
In the following exercises, use synthetic Division to find the quotient and remainder.
Divide Polynomial Functions
In the following exercises, divide.
Use the Remainder and Factor Theorem
In the following exercises, use the Remainder Theorem to find the remainder.
In the following exercises, use the Factor Theorem to determine if  is a factor of the polynomial function.

### Chapter Practice Test

In the following exercises, simplify and write your answer in exponential notation.


# Factoring

## Introduction to Factoring

An epidemic of a disease has broken out. Where did it start? How is it spreading? What can be done to control it? Answers to these and other questions can be found by scientists known as epidemiologists. They collect data and analyze it to study disease and consider possible control measures. Because diseases can spread at alarming rates, these scientists must use their knowledge of mathematics involving factoring. In this chapter, you will learn how to factor and apply factoring to real-life situations.


# Factoring

## Greatest Common Factor and Factor by Grouping

### Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will reverse this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.
We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.
We summarize the steps we use to find the greatest common factor.
The next example will show us the steps to find the greatest common factor of three expressions.

### Factor the Greatest Common Factor from a Polynomial

It is sometimes useful to represent a number as a product of factors, for example, 12 as  or  In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as  and end with its factors,  To do this we apply the Distributive Property “in reverse.”
We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”
So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!
When the leading coefficient is negative, we factor the negative out as part of the GCF.
So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.

### Factor by Grouping

Sometimes there is no common factor of all the terms of a polynomial. When there are four terms we separate the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts. Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.

### Key Concepts

1. How to find the greatest common factor (GCF) of two expressions.

2. Distributive Property: If a, b, and c are real numbers, then

The form on the left is used to multiply. The form on the right is used to factor.
3. How to factor the greatest common factor from a polynomial.

4. Factor as a Noun and a Verb: We use “factor” as both a noun and a verb.

5. How to factor by grouping.

### Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions
In the following exercises, find the greatest common factor.
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.
Factor by Grouping
In the following exercises, factor by grouping.
Mixed Practice
In the following exercises, factor.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!
…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.