# Conics

## Hyperbolas

### Graph a Hyperbola with Center at (0, 0)

The last conic section we will look at is called a hyperbola. We will see that the equation of a hyperbola looks the same as the equation of an ellipse, except it is a difference rather than a sum. While the equations of an ellipse and a hyperbola are very similar, their graphs are very different.
We define a hyperbola as all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a focus of the hyperbola.
The line through the foci, is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola.
Again our goal is to connect the geometry of a conic with algebra. Placing the hyperbola on a rectangular coordinate system gives us that opportunity. In the figure, we placed the hyperbola so the foci  are on the x-axis and the center is the origin.
The definition states the difference of the distance from the foci to a point  is constant. So  is a constant that we will call  so  We will use the distance formula to lead us to an algebraic formula for an ellipse.

To graph the hyperbola, it will be helpful to know about the intercepts. We will find the x-intercepts and y-intercepts using the formula.

The a, b values in the equation also help us find the asymptotes of the hyperbola. The asymptotes are intersecting straight lines that the branches of the graph approach but never intersect as the x, y values get larger and larger.
To find the asymptotes, we sketch a rectangle whose sides intersect the x-axis at the vertices   and intersect the y-axis at   The lines containing the diagonals of this rectangle are the asymptotes of the hyperbola. The rectangle and asymptotes are not part of the hyperbola, but they help us graph the hyperbola.
The asymptotes pass through the origin and we can evaluate their slope using the rectangle we sketched. They have equations  and
There are two equations for hyperbolas, depending whether the transverse axis is vertical or horizontal. We can tell whether the transverse axis is horizontal by looking at the equation. When the equation is in standard form, if the x2-term is positive, the transverse axis is horizontal. When the equation is in standard form, if the y2-term is positive, the transverse axis is vertical.
The second equations could be derived similarly to what we have done. We will summarize the results here.
We will use these properties to graph hyperbolas.
We summarize the steps for reference.
Sometimes the equation for a hyperbola needs to be first placed in standard form before we graph it.

### Graph a Hyperbola with Center at

Hyperbolas are not always centered at the origin. When a hyperbola is centered at  the equations changes a bit as reflected in the table.
We summarize the steps for easy reference.
Be careful as you identify the center. The standard equation has  and  with the center as
Again, sometimes we have to put the equation in standard form as our first step.

### Identify Conic Sections by their Equations

Now that we have completed our study of the conic sections, we will take a look at the different equations and recognize some ways to identify a conic by its equation. When we are given an equation to graph, it is helpful to identify the conic so we know what next steps to take.
To identify a conic from its equation, it is easier if we put the variable terms on one side of the equation and the constants on the other.

### Key Concepts

1. Hyperbola: A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant.

Each of the fixed points is called a focus of the hyperbola.
The line through the foci, is called the transverse axis.
The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola.
The midpoint of the segment joining the foci is called the center of the hyperbola.
The line perpendicular to the transverse axis that passes through the center is called the conjugate axis.
Each piece of the graph is called a branch of the hyperbola.

2. How to graph a hyperbola centered at

3. How to graph a hyperbola centered at

### Practice Makes Perfect

Graph a Hyperbola with Center at
In the following exercises, graph.
Graph a Hyperbola with Center at
In the following exercises, graph.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola
In the following exercises, identify the type of graph.
Mixed Practice
In the following exercises, graph each equation.

### 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?


# Conics

## Solve Systems of Nonlinear Equations

### Solve a System of Nonlinear Equations Using Graphing

We learned how to solve systems of linear equations with two variables by graphing, substitution and elimination. We will be using these same methods as we look at nonlinear systems of equations with two equations and two variables. A system of nonlinear equations is a system where at least one of the equations is not linear.
For example each of the following systems is a system of nonlinear equations.
Just as with systems of linear equations, a solution of a nonlinear system is an ordered pair that makes both equations true. In a nonlinear system, there may be more than one solution. We will see this as we solve a system of nonlinear equations by graphing.
When we solved systems of linear equations, the solution of the system was the point of intersection of the two lines. With systems of nonlinear equations, the graphs may be circles, parabolas or hyperbolas and there may be several points of intersection, and so several solutions. Once you identify the graphs, visualize the different ways the graphs could intersect and so how many solutions there might be.
To solve systems of nonlinear equations by graphing, we use basically the same steps as with systems of linear equations modified slightly for nonlinear equations. The steps are listed below for reference.
To identify the graph of each equation, keep in mind the characteristics of the  and  terms of each conic.

### Solve a System of Nonlinear Equations Using Substitution

The graphing method works well when the points of intersection are integers and so easy to read off the graph. But more often it is difficult to read the coordinates of the points of intersection. The substitution method is an algebraic method that will work well in many situations. It works especially well when it is easy to solve one of the equations for one of the variables.
The substitution method is very similar to the substitution method that we used for systems of linear equations. The steps are listed below for reference.
So far, each system of nonlinear equations has had at least one solution. The next example will show another option.

### Solve a System of Nonlinear Equations Using Elimination

When we studied systems of linear equations, we used the method of elimination to solve the system. We can also use elimination to solve systems of nonlinear equations. It works well when the equations have both variables squared. When using elimination, we try to make the coefficients of one variable to be opposites, so when we add the equations together, that variable is eliminated.
The elimination method is very similar to the elimination method that we used for systems of linear equations. The steps are listed for reference.
There are also four options when we consider a circle and a hyperbola.

### Use a System of Nonlinear Equations to Solve Applications

Systems of nonlinear equations can be used to model and solve many applications. We will look at an everyday geometric situation as our example.

### Key Concepts

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

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

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

### Section Exercises

### Practice Makes Perfect

Solve a System of Nonlinear Equations Using Graphing
In the following exercises, solve the system of equations by using graphing.
Solve a System of Nonlinear Equations Using Substitution
In the following exercises, solve the system of equations by using substitution.
Solve a System of Nonlinear Equations Using Elimination
In the following exercises, solve the system of equations by using elimination.
Use a System of Nonlinear Equations to Solve Applications
In the following exercises, solve the problem using a system of equations.

### 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?

### Chapter Review Exercises

### Distance and Midpoint Formulas; Circles

Use the Distance Formula
In the following exercises, find the distance between the points. Round to the nearest tenth if needed.
Use the Midpoint Formula
In the following exercises, find the midpoint of the line segments whose endpoints are given.
Write the Equation of a Circle in Standard Form
In the following exercises, write the standard form of the equation of the circle with the given information.
Graph a Circle
In the following exercises, ⓐ find the center and radius, then ⓑ graph each circle.

### Parabolas

Graph Vertical Parabolas
In the following exercises, graph each equation by using its properties.
In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.
Graph Horizontal Parabolas
In the following exercises, graph each equation by using its properties.
In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.
Solve Applications with Parabolas
In the following exercises, create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in standard form.

### Ellipses

Graph an Ellipse with Center at the Origin
In the following exercises, graph each ellipse.
Find the Equation of an Ellipse with Center at the Origin
In the following exercises, find the equation of the ellipse shown in the graph.
Graph an Ellipse with Center Not at the Origin
In the following exercises, graph each ellipse.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
Solve Applications with Ellipses
In the following exercises, write the equation of the ellipse described.

### Hyperbolas

Graph a Hyperbola with Center at
In the following exercises, graph.
Graph a Hyperbola with Center at
In the following exercises, graph.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola
In the following exercises, identify the type of graph.

### Solve Systems of Nonlinear Equations

Solve a System of Nonlinear Equations Using Graphing
In the following exercises, solve the system of equations by using graphing.
Solve a System of Nonlinear Equations Using Substitution
In the following exercises, solve the system of equations by using substitution.
Solve a System of Nonlinear Equations Using Elimination
In the following exercises, solve the system of equations by using elimination.
Use a System of Nonlinear Equations to Solve Applications
In the following exercises, solve the problem using a system of equations.

### Practice Test

In the following exercises, find the distance between the points and the midpoint of the line segment with the given endpoints. Round to the nearest tenth as needed.
In the following exercises, write the standard form of the equation of the circle with the given information.
In the following exercises, ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and ⓑ graph the equation.
In the following exercises, ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, ⓑ write the equation in standard form, and ⓒ graph the equation.


# Sequences, Series and Binomial Theorem

## Introduction

A strange charge suddenly appears on your credit card. But your card is in your wallet—it’s not even lost or stolen. Sadly, you may have been a victim of cyber crime. In this day and age, most transactions take advantage of the benefit of computers in some way. Cyber crime is any type of crime that uses a computer or computer network. Thankfully, many people are working to prevent cyber crime. Sometimes known as cryptographers, these people develop complex patterns in computer codes that block access to would-be thieves as well as write codes to intercept and decode information from them so that they may be identified. In this chapter, you will explore basic sequences and series related to those used by computer programmers to prevent cyber crime.


# Sequences, Series and Binomial Theorem

## Sequences

### Write the First Few Terms of a Sequence

Let’s look at the function  and evaluate it for just the counting numbers.
If we list the function values in order as 2, 4, 6, 8, and 10, … we have a sequence. A sequence is a function whose domain is the counting numbers.
A sequence can also be seen as an ordered list of numbers and each number in the list is a term. A sequence may have an infinite number of terms or a finite number of terms. Our sequence has three dots (ellipsis) at the end which indicates the list never ends. If the domain is the set of all counting numbers, then the sequence is an infinite sequence. Its domain is all counting numbers and there is an infinite number of counting numbers.
If we limit the domain to a finite number of counting numbers, then the sequence is a finite sequence. If we use only the first four counting numbers, 1, 2, 3, 4 our sequence would be the finite sequence,
Often when working with sequences we do not want to write out all the terms. We want more compact way to show how each term is defined. When we worked with functions, we wrote  and we said the expression  was the rule that defined values in the range. While a sequence is a function, we do not use the usual function notation. Instead of writing the function as  we would write it as  The  is the nth term of the sequence, the term in the nth position where n is a value in the domain. The formula for writing the nth term of the sequence is called the general term or formula of the sequence.
When we are given the general term of the sequence, we can find the terms by replacing n with the counting numbers in order. For
To find the values of a sequence, we substitute in the counting numbers in order into the general term of the sequence.
For some sequences, the variable is an exponent.
It is not uncommon to see the expressions  or  in the general term for a sequence. If we evaluate each of these expressions for a few values, we see that this expression alternates the sign for the terms.
The terms in the next example will alternate signs as a result of the powers of

### Find a Formula for the General Term (nth Term) of a Sequence

Sometimes we have a few terms of a sequence and it would be helpful to know the general term or nth term. To find the general term, we look for patterns in the terms. Often the patterns involve multiples or powers. We also look for a pattern in the signs of the terms.

### Use Factorial Notation

Sequences often have terms that are products of consecutive integers. We indicate these products with a special notation called factorial notation. For example,, read 5 factorial, means  The exclamation point is not punctuation here; it indicates the factorial notation.
The values of  for the first 5 positive integers are shown.
When there is a fraction with factorials in the numerator and denominator, we line up the factors vertically to make our calculations easier.

### Find the Partial Sum

Sometimes in applications, rather than just list the terms, it is important for us to add the terms of a sequence. Rather than just connect the terms with plus signs, we can use summation notation.
For example,  can be written as  We read this as “the sum of a sub i from i equals one to five.” The symbol  means to add and the i is the index of summation. The 1 tells us where to start (initial value) and the 5 tells us where to end (terminal value).
When we add a finite number of terms, we call the sum a partial sum.
The index does not always have to be i we can use any letter, but i and k are commonly used. The index does not have to start with 1 either—it can start and end with any positive integer.

### Use Summation Notation to Write a Sum

In the last two examples, we went from summation notation to writing out the sum. Now we will start with a sum and change it to summation notation. This is very similar to finding the general term of a sequence. We will need to look at the terms and find a pattern. Often the patterns involve multiples or powers.
When the terms of a sum have negative coefficients, we must carefully analyze the pattern of the signs.

### Key Concepts

1. Factorial Notation
If n is a positive integer, then  is

We define  as 1, so
2. Summation Notation
The sum of the first n terms of a sequence whose nth term  is written in summation notation as:

The i is the index of summation and the 1 tells us where to start and the n tells us where to end.

### Practice Makes Perfect

Write the First Few Terms of a Sequence
In the following exercises, write the first five terms of the sequence whose general term is given.
Find a Formula for the General Term (
In the following exercises, find a general term for the sequence whose first five terms are shown.
Use Factorial Notation
In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.
Find the Partial Sum
In the following exercises, expand the partial sum and find its value.
Use Summation Notation to write a Sum
In the following exercises, write each sum using summation notation.

### 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.


# Sequences, Series and Binomial Theorem

## Arithmetic Sequences

### Determine if a Sequence is Arithmetic

The last section introduced sequences and now we will look at two specific types of sequences that each have special properties. In this section we will look at arithmetic sequences and in the next section, geometric sequences.
An arithmetic sequence is a sequence where the difference between consecutive terms is constant. The difference between consecutive terms in an arithmetic sequence,  is d, the common difference, for n greater than or equal to two.
In each of these sequences, the difference between consecutive terms is constant, and so the sequence is arithmetic.
If we know the first term,  and the common difference, d, we can list a finite number of terms of the sequence.

### Find the General Term (nth Term) of an Arithmetic Sequence

Just as we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.
Let’s write the first few terms of a sequence where the first term is  and the common difference is d. We will then look for a pattern.
As we look for a pattern we see that each term starts with .
The first term adds 0d to the , the second term adds 1d, the third term adds 2d, the fourth term adds 3d, and the fifth term adds 4d. The number of ds that were added to  is one less than the number of the term. This leads us to the following
We will use this formula in the next example to find the 15th term of a sequence.
Sometimes we do not know the first term and we must use other given information to find it before we find the requested term.
Sometimes the information given leads us to two equations in two unknowns. We then use our methods for solving systems of equations to find the values needed.

### Find the Sum of the First n Terms of an Arithmetic Sequence

As with the general sequences, it is often useful to find the sum of an arithmetic sequence. The sum,  of the first  terms of any arithmetic sequence is written as  To find the sum by merely adding all the terms can be tedious. So we can also develop a formula to find the sum of a sequence using the first and last term of the sequence.
We can develop this new formula by first writing the sum by starting with the first term,  and keep adding a d to get the next term as:
We can also reverse the order of the terms and write the sum by starting with  and keep subtracting d to get the next term as
If we add these two expressions for the sum of the first n terms of an arithmetic sequence, we can derive a formula for the sum of the first n terms of any arithmetic series.
Because there are n sums of  on the right side of the equation, we rewrite the right side as
We divide by two to solve for
This gives us a general formula for the sum of the first n terms of an arithmetic sequence.
We apply this formula in the next example where the first few terms of the sequence are given.
In the next example, we are given the general term for the sequence and are asked to find the sum of the first 50 terms.
In the next example we are given the sum in summation notation. To add all the terms would be tedious, so we extract the information needed to use the formula to find the sum of the first n terms.

### Key Concepts

1. General Term (
The general term of an arithmetic sequence with first term  and the common difference d is

2. Sum of the First
The sum,  of the first n terms of an arithmetic sequence, where  is the first term and  is the nth term is

### Practice Makes Perfect

Determine if a Sequence is Arithmetic
In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.
In the following exercises, write the first five terms of each sequence with the given first term and common difference.
Find the General Term (
In the following exercises, find the term described using the information provided.
In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.
Find the Sum of the First
In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.
In the following exercises, find the sum of the first 50 terms of the arithmetic sequence whose general term is given.
In the following exercises, find each sum.

### 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?


# Sequences, Series and Binomial Theorem

## Geometric Sequences and Series

### Determine if a Sequence is Geometric

We are now ready to look at the second special type of sequence, the geometric sequence.
A sequence is called a geometric sequence if the ratio between consecutive terms is always the same. The ratio between consecutive terms in a geometric sequence is r, the common ratio, where n is greater than or equal to two.
Consider these sequences.
If we know the first term,  and the common ratio, r, we can list a finite number of terms of the sequence.

### Find the General Term (nth Term) of a Geometric Sequence

Just as we found a formula for the general term of a sequence and an arithmetic sequence, we can also find a formula for the general term of a geometric sequence.
Let’s write the first few terms of the sequence where the first term is  and the common ratio is r. We will then look for a pattern.
As we look for a pattern in the five terms above, we see that each of the terms starts with
The first term,  is not multiplied by any r. In the second term, the  is multiplied by r. In the third term, the  is multiplied by r two times ( or ). In the fourth term, the  is multiplied by r three times ( or ) and in the fifth term, the  is multiplied by r four times. In each term, the number of times  is multiplied by r is one less than the number of the term. This leads us to the following
We will use this formula in the next example to find the fourteenth term of a sequence.
Sometimes we do not know the common ratio and we must use the given information to find it before we find the requested term.

### Find the Sum of the First n Terms of a Geometric Sequence

We found the sum of both general sequences and arithmetic sequence. We will now do the same for geometric sequences. The sum, of the first n terms of a geometric sequence is written as  We can write this sum by starting with the first term,  and keep multiplying by r to get the next term as:
Let’s also multiply both sides of the equation by r.
Next, we subtract these equations. We will see that when we subtract, all but the first term of the top equation and the last term of the bottom equation subtract to zero.

We apply this formula in the next example where the first few terms of the sequence are given. Notice the sum of a geometric sequence typically gets very large when the common ratio is greater than one.
In the next example, we are given the sum in summation notation. While adding all the terms might be possible, most often it is easiest to use the formula to find the sum of the first n terms.
To use the formula, we need r. We can find it by writing out the first few terms of the sequence and find their ratio. Another option is to realize that in summation notation, a sequence is written in the form  where r is the common ratio.

### Find the Sum of an Infinite Geometric Series

If we take a geometric sequence and add the terms, we have a sum that is called a geometric series. An infinite geometric series is an infinite sum whose first term is  and common ratio is r and is written
We know how to find the sum of the first n terms of a geometric series using the formula,  But how do we find the sum of an infinite sum?
Let’s look at the infinite geometric series  Each term gets larger and larger so it makes sense that the sum of the infinite number of terms gets larger. Let’s look at a few partial sums for this series. We see  and
As n gets larger and larger, the sum gets larger and larger. This is true when  and we call the series divergent. We cannot find a sum of an infinite geometric series when
Let’s look at an infinite geometric series whose common ratio is a fraction less than one,. Here the terms get smaller and smaller as n gets larger. Let’s look at a few finite sums for this series. We see  and
Notice the sum gets larger and larger but also gets closer and closer to one. When  the expression  gets smaller and smaller. In this case, we call the series convergent. As n approaches infinity, (gets infinitely large),  gets closer and closer to zero. In our sum formula, we can replace the  with zero and then we get a formula for the sum, S, for an infinite geometric series when
This formula gives us the sum of the infinite geometric sequence. Notice the S does not have the subscript n as in  as we are not adding a finite number of terms.
An interesting use of infinite geometric series is to write a repeating decimal as a fraction.

### Apply Geometric Sequences and Series in the Real World

One application of geometric sequences has to do with consumer spending. If a tax rebate is given to each household, the effect on the economy is many times the amount of the individual rebate.
We have looked at a compound interest formula where a principal, P, is invested at an interest rate, r, for t years. The new balance, A, is  when interest is compounded n times a year. This formula applies when a lump sum was invested upfront and tells us the value after a certain time period.
An annuity is an investment that is a sequence of equal periodic deposits. We will be looking at annuities that pay the interest at the time of the deposits. As we develop the formula for the value of an annuity, we are going to let  That means there is one deposit per year.
Suppose P dollars is invested at the end of each year. One year later that deposit is worth  dollars, and another year later it is worth  dollars. After t years, it will be worth  dollars.
After three years, the value of the annuity is
This a sum of the terms of a geometric sequence where the first term is P and the common ratio is  We substitute these values into the sum formula. Be careful, we have two different uses of r. The r in the sum formula is the common ratio of the sequence. In this case, that is  where r is the interest rate.
Remember our premise was that one deposit was made at the end of each year.
We can adapt this formula for n deposits made per year and the interest is compounded n times a year.

### Key Concepts

1. General Term ( The general term of a geometric sequence with first term  and the common ratio r is

2. Sum of the First  The sum,  of the n terms of a geometric sequence is

where  is the first term and r is the common ratio.
3. Infinite Geometric Series: An infinite geometric series is an infinite sum whose first term is  and common ratio is r and is written

4. Sum of an Infinite Geometric Series: For an infinite geometric series whose first term is  and common ratio r,

5. Value of an Annuity with Interest Compounded  Times a Year: For a principal, P, invested at the end of a compounding period, with an interest rate, r, which is compounded n times a year, the new balance, A, after t years, is

### Practice Makes Perfect

Determine if a Sequence is Geometric
In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.
In the following exercises, determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.
In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.
Find the General Term (
In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.
In the following exercises, find the indicated term of the given sequence. Find the general term for the sequence.
Find the Sum of the First
In the following exercises, find the sum of the first fifteen terms of each geometric sequence.
In the following exercises, find the sum of the geometric sequence.
Find the Sum of an Infinite Geometric Series
In the following exercises, find the sum of each infinite geometric series.
In the following exercises, write each repeating decimal as a fraction.
Apply Geometric Sequences and Series in the Real World
In the following exercises, solve the problem.

### 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?


# Sequences, Series and Binomial Theorem

## Binomial Theorem

### Use Pascal’s Triangle to Expand a Binomial

In our previous work, we have squared binomials either by using FOIL or by using the Binomial Squares Pattern. We can also say that we expanded
To expand  we recognize that this is  and multiply.
To find a method that is less tedious that will work for higher expansions like  we again look for patterns in some expansions.
Notice the first and last terms show only one variable. Recall that  so we could rewrite the first and last terms to include both variables. For example, we could expand  to show each term with both variables.
Generally, we don’t show the zero exponents, just as we usually write x rather than 1x.
Let’s look at an example to highlight the last three patterns.
From the patterns we identified, we see the variables in the expansion of  would be
To find the coefficients of the terms, we write our expansions again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients.
The array to the right is called Pascal’s Triangle. Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.
This triangle gives the coefficients of the terms when we expand binomials.
In the next example, we will use this triangle and the patterns we recognized to expand the binomial.
In the next example we want to expand a binomial with one variable and one constant. We need to identify the a and b to carefully apply the pattern.
In the next example, the binomial is a difference and the first term has a constant times the variable. Once we identify the a and b of the pattern, we must once again carefully apply the pattern.

### Evaluate a Binomial Coefficient

While Pascal’s Triangle is one method to expand a binomial, we will also look at another method. Before we get to that, we need to introduce some more factorial notation. This notation is not only used to expand binomials, but also in the study and use of probability.
To find the coefficients of the terms of expanded binomials, we will need to be able to evaluate the notation  which is called a binomial coefficient. We read  as “n choose r” or “n taken r at a time”.
In the previous example, parts (a), (b), (c) demonstrate some special properties of binomial coefficients.

### Use the Binomial Theorem to Expand a Binomial

We are now ready to use the alternate method of expanding binomials. The Binomial Theorem uses the same pattern for the variables, but uses the binomial coefficient for the coefficient of each term.
Notice that when we expanded  in the last example, using the Binomial Theorem, we got the same coefficients we would get from using Pascal’s Triangle.
The next example, the binomial is a difference. When the binomial is a difference, we must be careful in identifying the values we will use in the pattern.
Things can get messy when both terms have a coefficient and a variable.
The real beauty of the Binomial Theorem is that it gives a formula for any particular term of the expansion without having to compute the whole sum. Let’s look for a pattern in the Binomial Theorem.
Notice, that in each case the exponent on the b is one less than the number of the term. The  term is the term where the exponent of b is r. So we can use the format of the  term to find the value of a specific term.

### Key Concepts

1. Patterns in the expansion of

2. Pascal’s Triangle

3. Binomial Coefficient
: A binomial coefficient  where r and n are integers with  is defined as

We read  as “n choose r” or “n taken r at a time”.
4. Properties of Binomial Coefficients

5. Binomial Theorem: For any real numbers a, b, and positive integer n,

### Section Exercises

### Practice Makes Perfect

Use Pascal’s Triangle to Expand a Binomial
In the following exercises, expand each binomial using Pascal’s Triangle.
Evaluate a Binomial Coefficient
In the following exercises, evaluate.
Use the Binomial Theorem to Expand a Binomial
In the following exercises, expand each binomial.
In the following exercises, find the indicated term in the expansion of the binomial.
In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.

### 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

### Sequences

Write the First Few Terms of a Sequence
In the following exercises, write the first five terms of the sequence whose general term is given.
Find a Formula for the General Term (
In the following exercises, find a general term for the sequence whose first five terms are shown.
Use Factorial Notation
In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.
Find the Partial Sum
In the following exercises, expand the partial sum and find its value.
Use Summation Notation to write a Sum
In the following exercises, write each sum using summation notation.

### Arithmetic Sequences

Determine if a Sequence is Arithmetic
In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.
In the following exercises, write the first five terms of each arithmetic sequence with the given first term and common difference.
Find the General Term (
In the following exercises, find the term described using the information provided.
In the following exercises, find the indicated term and give the formula for the general term.
In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.
Find the Sum of the First
In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.
In the following exercises, find the sum of the first fifteen terms of the arithmetic sequence whose general term is given.
In the following exercises, find each sum.

### Geometric Sequences and Series

Determine if a Sequence is Geometric
In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.
In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.
Find the General Term (
In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.
In the following exercises, find the indicated term of the given sequence. Find the general term of the sequence.
Find the Sum of the First
In the following exercises, find the sum of the first fifteen terms of each geometric sequence.
In the following exercises, find the sum
Find the Sum of an Infinite Geometric Series
In the following exercises, find the sum of each infinite geometric series.
In the following exercises, write each repeating decimal as a fraction.
Apply Geometric Sequences and Series in the Real World
In the following exercises, solve the problem.

### Binomial Theorem

Use Pascal’s Triangle to Expand a Binomial
In the following exercises, expand each binomial using Pascal’s Triangle.
Evaluate a Binomial Coefficient
In the following exercises, evaluate.
Use the Binomial Theorem to Expand a Binomial
In the following exercises, expand each binomial, using the Binomial Theorem.
In the following exercises, find the indicated term in the expansion of the binomial.
In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.

### Practice Test

In the following exercises, write the first five terms of the sequence whose general term is given.
In the following exercises, determine if the sequence is arithmetic, geometric, or neither. If arithmetic, then find the common difference. If geometric, then find the common ratio.
In the following exercises, find the sum.


# Whole Numbers

## Introduction

Even though counting is first taught at a young age, mastering mathematics, which is the study of numbers, requires constant attention. If it has been a while since you have studied math, it can be helpful to review basic topics. In this chapter, we will focus on numbers used for counting as well as four arithmetic operations—addition, subtraction, multiplication, and division. We will also discuss some vocabulary that we will use throughout this book.


# Whole Numbers

## Introduction to Whole Numbers

### Identify Counting Numbers and Whole Numbers

Learning algebra is similar to learning a language. You start with a basic vocabulary and then add to it as you go along. You need to practice often until the vocabulary becomes easy to you. The more you use the vocabulary, the more familiar it becomes.
Algebra uses numbers and symbols to represent words and ideas. Let’s look at the numbers first. The most basic numbers used in algebra are those we use to count objects:  and so on. These are called the counting numbers. The notation “…” is called an ellipsis, which is another way to show “and so on”, or that the pattern continues endlessly. Counting numbers are also called natural numbers.
Counting numbers and whole numbers can be visualized on a number line as shown in .
The point labeled  is called the origin. The points are equally spaced to the right of  and labeled with the counting numbers. When a number is paired with a point, it is called the coordinate of the point.
The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the whole numbers.
We stopped at  when listing the first few counting numbers and whole numbers. We could have written more numbers if they were needed to make the patterns clear.

### Model Whole Numbers

Our number system is called a place value system because the value of a digit depends on its position, or place, in a number. The number  has a different value than the number  Even though they use the same digits, their value is different because of the different placement of the  and the
Money gives us a familiar model of place value. Suppose a wallet contains three  bills, seven  bills, and four  bills. The amounts are summarized in . How much money is in the wallet?
Find the total value of each kind of bill, and then add to find the total. The wallet contains
Base-10 blocks provide another way to model place value, as shown in . The blocks can be used to represent hundreds, tens, and ones. Notice that the tens rod is made up of  ones, and the hundreds square is made of  tens, or  ones.
 shows the number  modeled with  blocks.

### Identify the Place Value of a Digit

By looking at money and  blocks, we saw that each place in a number has a different value. A place value chart is a useful way to summarize this information. The place values are separated into groups of three, called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.
Just as with the  blocks, where the value of the tens rod is ten times the value of the ones block and the value of the hundreds square is ten times the tens rod, the value of each place in the place-value chart is ten times the value of the place to the right of it.
 shows how the number  is written in a place value chart.
1. The digit  is in the millions place. Its value is
2. The digit  is in the hundred thousands place. Its value is
3. The digit  is in the ten thousands place. Its value is
4. The digit  is in the thousands place. Its value is
5. The digit  is in the hundreds place. Its value is
6. The digit  is in the tens place. Its value is
7. The digit  is in the ones place. Its value is

### Use Place Value to Name Whole Numbers

When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period followed by the name of the period without the ‘s’ at the end. Start with the digit at the left, which has the largest place value. The commas separate the periods, so wherever there is a comma in the number, write a comma between the words. The ones period, which has the smallest place value, is not named.
So the number  is written thirty-seven million, five hundred nineteen thousand, two hundred forty-eight.
Notice that the word and is not used when naming a whole number.

### Use Place Value to Write Whole Numbers

We will now reverse the process and write a number given in words as digits.

### Round Whole Numbers

In  the U.S. Census Bureau reported the population of the state of New York as  people. It might be enough to say that the population is approximately  million. The word approximately means that  million is not the exact population, but is close to the exact value.
The process of approximating a number is called rounding. Numbers are rounded to a specific place value depending on how much accuracy is needed.  million was achieved by rounding to the millions place. Had we rounded to the one hundred thousands place, we would have  as a result. Had we rounded to the ten thousands place, we would have  as a result, and so on. The place value to which we round to depends on how we need to use the number.
Using the number line can help you visualize and understand the rounding process. Look at the number line in . Suppose we want to round the number  to the nearest ten. Is  closer to  or  on the number line?
Now consider the number  Find  in .
How do we round  to the nearest ten. Find  in .
So that everyone rounds the same way in cases like this, mathematicians have agreed to round to the higher number,  So,  rounded to the nearest ten is
Now that we have looked at this process on the number line, we can introduce a more general procedure. To round a number to a specific place, look at the number to the right of that place. If the number is less than  round down. If it is greater than or equal to  round up.
So, for example, to round  to the nearest ten, we look at the digit in the ones place.
The digit in the ones place is a  Because  is greater than or equal to  we increase the digit in the tens place by one. So the  in the tens place becomes an  Now, replace any digits to the right of the  with zeros. So,  rounds to
Let’s look again at rounding  to the nearest  Again, we look to the ones place.
The digit in the ones place is  Because  is less than  we keep the digit in the tens place the same and replace the digits to the right of it with zero. So  rounded to the nearest ten is

### Key Concepts

1. Name a whole number in words.

2. Use place value to write a whole number.

3. Round a whole number to a specific place value.

### Practice Makes Perfect

Identify Counting Numbers and Whole Numbers
In the following exercises, determine which of the following numbers are ⓐ counting numbers ⓑ whole numbers.
Model Whole Numbers
In the following exercises, use place value notation to find the value of the number modeled by the  blocks.
Identify the Place Value of a Digit
In the following exercises, find the place value of the given digits.
Use Place Value to Name Whole Numbers
In the following exercises, name each number in words.
Use Place Value to Write Whole Numbers
In the following exercises, write each number as a whole number using digits.
Round Whole Numbers
In the following exercises, round to the indicated place value.

### Everyday Math

### 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.


# Whole Numbers

## Add Whole Numbers

### Use Addition Notation

A college student has a part-time job. Last week he worked  hours on Monday and  hours on Friday. To find the total number of hours he worked last week, he added  and
The operation of addition combines numbers to get a sum. The notation we use to find the sum of  and  is:
We read this as three plus four and the result is the sum of three and four. The numbers  and  are called the addends. A math statement that includes numbers and operations is called an expression.

### Model Addition of Whole Numbers

Addition is really just counting. We will model addition with  blocks. Remember, a block represents  and a rod represents  Let’s start by modeling the addition expression we just considered,
Each addend is less than  so we can use ones blocks.
There are  blocks in all. We use an equal sign  to show the sum. A math sentence that shows that two expressions are equal is called an equation. We have shown that.
When the result is  or more ones blocks, we will exchange the  blocks for one rod.
Next we will model adding two digit numbers.

### Add Whole Numbers Without Models

Now that we have used models to add numbers, we can move on to adding without models. Before we do that, make sure you know all the one digit addition facts. You will need to use these number facts when you add larger numbers.
Imagine filling in  by adding each row number along the left side to each column number across the top. Make sure that you get each sum shown. If you have trouble, model it. It is important that you memorize any number facts you do not already know so that you can quickly and reliably use the number facts when you add larger numbers.
Did you notice what happens when you add zero to a number? The sum of any number and zero is the number itself. We call this the Identity Property of Addition. Zero is called the additive identity.
Look at the pairs of sums.

Notice that when the order of the addends is reversed, the sum does not change. This property is called the Commutative Property of Addition, which states that changing the order of the addends does not change their sum.
In the previous example, the sum of the ones and the sum of the tens were both less than  But what happens if the sum is  or more? Let’s use our  model to find out.  shows the addition of  and  again.
When we add the ones,  we get  ones. Because we have more than  ones, we can exchange  of the ones for  ten. Now we have  tens and  ones. Without using the model, we show this as a small red  above the digits in the tens place.
When the sum in a place value column is greater than  we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example,  ones for  ten or  tens for  hundred.

### Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words. Now we’ll reverse the process. We’ll translate word phrases into math notation. Some of the word phrases that indicate addition are listed in .

### Add Whole Numbers in Applications

Now that we have practiced adding whole numbers, let’s use what we’ve learned to solve real-world problems. We’ll start by outlining a plan. First, we need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we write a sentence to answer the question.
Some application problems involve shapes. For example, a person might need to know the distance around a garden to put up a fence or around a picture to frame it. The perimeter is the distance around a geometric figure. The perimeter of a figure is the sum of the lengths of its sides.

### Key Concepts

1. Addition Notation To describe addition, we can use symbols and words.

2. Identity Property of Addition

3. Commutative Property of Addition

4. Add whole numbers.

### Practice Makes Perfect

Use Addition Notation
In the following exercises, translate the following from math expressions to words.
Model Addition of Whole Numbers
In the following exercises, model the addition.
Add Whole Numbers
In the following exercises, fill in the missing values in each chart.
In the following exercises, add.
Translate Word Phrases to Math Notation
In the following exercises, translate each phrase into math notation and then simplify.
Add Whole Numbers in Applications
In the following exercises, solve the problem.
In the following exercises, find the perimeter of each figure.

### Everyday Math

### 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?


# Whole Numbers

## Subtract Whole Numbers

### Use Subtraction Notation

Suppose there are seven bananas in a bowl. Elana uses three of them to make a smoothie. How many bananas are left in the bowl? To answer the question, we subtract three from seven. When we subtract, we take one number away from another to find the difference. The notation we use to subtract  from  is
We read  as seven minus three and the result is the difference of seven and three.

### Model Subtraction of Whole Numbers

A model can help us visualize the process of subtraction much as it did with addition. Again, we will use  blocks. Remember a block represents 1 and a rod represents 10. Let’s start by modeling the subtraction expression we just considered,

### Subtract Whole Numbers

Addition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition.
We know  because  Knowing all the addition number facts will help with subtraction. Then we can check subtraction by adding. In the examples above, our subtractions can be checked by addition.
To subtract numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition. Align the digits by place value, and then subtract each column starting with the ones and then working to the left.
When we modeled subtracting  from  we exchanged  ten for  ones. When we do this without the model, we say we borrow  from the tens place and add  to the ones place.

### Translate Word Phrases to Math Notation

As with addition, word phrases can tell us to operate on two numbers using subtraction. To translate from a word phrase to math notation, we look for key words that indicate subtraction. Some of the words that indicate subtraction are listed in .

### Subtract Whole Numbers in Applications

To solve applications with subtraction, we will use the same plan that we used with addition. First, we need to determine what we are asked to find. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question, using the appropriate units.

### Key Concepts

1. Subtract whole numbers.

### Practice Makes Perfect

Use Subtraction Notation
In the following exercises, translate from math notation to words.
Model Subtraction of Whole Numbers
In the following exercises, model the subtraction.
Subtract Whole Numbers
In the following exercises, subtract and then check by adding.
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate and simplify.
Mixed Practice
In the following exercises, simplify.
In the following exercises, translate and simplify.
Subtract Whole Numbers in Applications
In the following exercises, solve.

### Everyday Math

### 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?


# Whole Numbers

## Multiply Whole Numbers

### Use Multiplication Notation

Suppose you were asked to count all these pennies shown in .
Would you count the pennies individually? Or would you count the number of pennies in each row and add that number  times.
Multiplication is a way to represent repeated addition. So instead of adding  three times, we could write a multiplication expression.
We call each number being multiplied a factor and the result the product. We read  as three times eight, and the result as the product of three and eight.
There are several symbols that represent multiplication. These include the symbol  as well as the dot, , and parentheses

### Model Multiplication of Whole Numbers

There are many ways to model multiplication. Unlike in the previous sections where we used  blocks, here we will use counters to help us understand the meaning of multiplication. A counter is any object that can be used for counting. We will use round blue counters.

### Multiply Whole Numbers

In order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section.
 shows the multiplication facts. Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.
What happens when you multiply a number by zero? You can see that the product of any number and zero is zero. This is called the Multiplication Property of Zero.
What happens when you multiply a number by one? Multiplying a number by one does not change its value. We call this fact the Identity Property of Multiplication, and  is called the multiplicative identity.
Earlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum. We saw that  is the same as
Is this also true for multiplication? Let’s look at a few pairs of factors.
When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.
To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.
We start by multiplying  by
We write the  in the ones place of the product. We carry the  tens by writing  above the tens place.
Then we multiply the  by the  and add the  above the tens place to the product. So  and  Write the  in the tens place of the product.
The product is
When we multiply two numbers with a different number of digits, it’s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.
When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.
When there are three or more factors, we multiply the first two and then multiply their product by the next factor. For example:

### Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words. Now we’ll reverse the process and translate word phrases into math notation. Some of the words that indicate multiplication are given in .

### Multiply Whole Numbers in Applications

We will use the same strategy we used previously to solve applications of multiplication. First, we need to determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify to get the answer. Finally, we write a sentence to answer the question.
If we want to know the size of a wall that needs to be painted or a floor that needs to be carpeted, we will need to find its area. The area is a measure of the amount of surface that is covered by the shape. Area is measured in square units. We often use square inches, square feet, square centimeters, or square miles to measure area. A square centimeter is a square that is one centimeter (cm.) on a side. A square inch is a square that is one inch on each side, and so on.
For a rectangular figure, the area is the product of the length and the width.  shows a rectangular rug with a length of  feet and a width of  feet. Each square is  foot wide by  foot long, or  square foot. The rug is made of  squares. The area of the rug is  square feet.

### Key Concepts

1. Multiplication Property of Zero

2. Identity Property of Multiplication

3. Commutative Property of Multiplication

4. Multiply two whole numbers to find the product.

### Practice Makes Perfect

Use Multiplication Notation
In the following exercises, translate from math notation to words.
Model Multiplication of Whole Numbers
In the following exercises, model the multiplication.
Multiply Whole Numbers
In the following exercises, fill in the missing values in each chart.
In the following exercises, multiply.
Translate Word Phrases to Math Notation
In the following exercises, translate and simplify.
Mixed Practice
In the following exercises, simplify.
In the following exercises, translate and simplify.
Multiply Whole Numbers in Applications
In the following exercises, solve.

### Everyday Math

### 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?


# Whole Numbers

## Divide Whole Numbers

### Use Division Notation

So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the  cookies in  and want to package them in bags with  cookies in each bag. How many bags would we need?
You might put  cookies in first bag,  in the second bag, and so on until you run out of cookies. Doing it this way, you would fill  bags.
In other words, starting with the  cookies, you would take away, or subtract,  cookies at a time. Division is a way to represent repeated subtraction just as multiplication represents repeated addition.
Instead of subtracting  repeatedly, we can write
We read this as twelve divided by four and the result is the quotient of  and  The quotient is  because we can subtract  from  exactly  times. We call the number being divided the dividend and the number dividing it the divisor. In this case, the dividend is  and the divisor is
In the past you may have used the notation , but this division also can be written as  In each case the  is the dividend and the  is the divisor.

### Model Division of Whole Numbers

As we did with multiplication, we will model division using counters. The operation of division helps us organize items into equal groups as we start with the number of items in the dividend and subtract the number in the divisor repeatedly.

### Divide Whole Numbers

We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know  because  Knowing all the multiplication number facts is very important when doing division.
We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. In , we know  is correct because
What is the quotient when you divide a number by itself?
Dividing any number  by itself produces a quotient of  Also, any number divided by  produces a quotient of the number. These two ideas are stated in the Division Properties of One.
Suppose we have  and want to divide it among  people. How much would each person get? Each person would get  Zero divided by any number is
Now suppose that we want to divide  by  That means we would want to find a number that we multiply by  to get  This cannot happen because  times any number is  Division by zero is said to be undefined.
These two ideas make up the Division Properties of Zero.
Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away  from  Because subtracting  will never change the total, we will never get an answer. So we cannot divide a number by
When the divisor or the dividend has more than one digit, it is usually easier to use the  notation. This process is called long division. Let’s work through the process by dividing  by
We would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.
Check by multiplying the quotient times the divisor to get the dividend. Multiply  to make sure that product equals the dividend,
It does, so our answer is correct.
So far all the division problems have worked out evenly. For example, if we had  cookies and wanted to make bags of  cookies, we would have  bags. But what if there were  cookies and we wanted to make bags of  Start with the  cookies as shown in .
Try to put the cookies in groups of eight as in .
There are  groups of eight cookies, and  cookies left over. We call the  cookies that are left over the remainder and show it by writing R4 next to the  (The R stands for remainder.)
To check this division we multiply  times  to get  and then add the remainder of

### Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation for division into words. Now we’ll translate word phrases into math notation. Some of the words that indicate division are given in .

### Divide Whole Numbers in Applications

We will use the same strategy we used in previous sections to solve applications. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify it to get the answer. Finally, we write a sentence to answer the question.

### Key Concepts

1. Division Properties of One

2. Division Properties of Zero

3. Divide whole numbers.

### Section Exercises

### Practice Makes Perfect

Use Division Notation
In the following exercises, translate from math notation to words.
Model Division of Whole Numbers
In the following exercises, model the division.
Divide Whole Numbers
In the following exercises, divide. Then check by multiplying.
Mixed Practice
In the following exercises, simplify.
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate and simplify.
Divide Whole Numbers in Applications
In the following exercises, solve.
Mixed Practice
In the following exercises, solve.

### Writing Exercises

### Everyday Math

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

### Chapter Review Exercises

### Introduction to Whole Numbers

Identify Counting Numbers and Whole Numbers
In the following exercises, determine which of the following are (a) counting numbers (b) whole numbers.
Model Whole Numbers
In the following exercises, model each number using  blocks and then show its value using place value notation.
Identify the Place Value of a Digit
In the following exercises, find the place value of the given digits.
Use Place Value to Name Whole Numbers
In the following exercises, name each number in words.
Use Place Value to Write Whole Numbers
In the following exercises, write as a whole number using digits.
Round Whole Numbers
In the following exercises, round to the nearest ten.
In the following exercises, round to the nearest hundred.

### Add Whole Numbers

Use Addition Notation
In the following exercises, translate the following from math notation to words.
Model Addition of Whole Numbers
In the following exercises, model the addition.
Add Whole Numbers
In the following exercises, fill in the missing values in each chart.
In the following exercises, add.
Translate Word Phrases to Math Notation
In the following exercises, translate each phrase into math notation and then simplify.
Add Whole Numbers in Applications
In the following exercises, solve.
In the following exercises, find the perimeter of each figure.

### Subtract Whole Numbers

Use Subtraction Notation
In the following exercises, translate the following from math notation to words.
Model Subtraction of Whole Numbers
In the following exercises, model the subtraction.
Subtract Whole Numbers
In the following exercises, subtract and then check by adding.
Translate Word Phrases to Math Notation
In the following exercises, translate and simplify.
Subtract Whole Numbers in Applications
In the following exercises, solve.

### Multiply Whole Numbers

Use Multiplication Notation
In the following exercises, translate from math notation to words.
Model Multiplication of Whole Numbers
In the following exercises, model the multiplication.
Multiply Whole Numbers
In the following exercises, fill in the missing values in each chart.
In the following exercises, multiply.
Translate Word Phrases to Math Notation
In the following exercises, translate and simplify.
Multiply Whole Numbers in Applications
In the following exercises, solve.

### Divide Whole Numbers

Use Division Notation
Translate from math notation to words.
Model Division of Whole Numbers
In the following exercises, model.
Divide Whole Numbers
In the following exercises, divide. Then check by multiplying.
Translate Word Phrases to Math Notation
In the following exercises, translate and simplify.
Divide Whole Numbers in Applications
In the following exercises, solve.

### Chapter Practice Test

Simplify.
Translate each phrase to math notation and then simplify.
In the following exercises, solve.


# The Language of Algebra

## Introduction to the Language of Algebra

You may not realize it, but you already use algebra every day. Perhaps you figure out how much to tip a server in a restaurant. Maybe you calculate the amount of change you should get when you pay for something. It could even be when you compare batting averages of your favorite players. You can describe the algebra you use in specific words, and follow an orderly process. In this chapter, you will explore the words used to describe algebra and start on your path to solving algebraic problems easily, both in class and in your everyday life.


# The Language of Algebra

## Use the Language of Algebra

### Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is  years old and Alex is  so Alex is  years older than Greg. When Greg was  Alex was  When Greg is  Alex will be  No matter what Greg’s age is, Alex’s age will always be  years more, right?
In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The  years between them always stays the same, so the age difference is the constant.
In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age  Then we could use  to represent Alex’s age. See .
Letters are used to represent variables. Letters often used for variables are
To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In Whole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.
In algebra, the cross symbol,  is not used to show multiplication because that symbol may cause confusion. Does  mean  (three times ) or  (three times )? To make it clear, use • or parentheses for multiplication.
We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.
1. The sum     means add  plus  which we write as
2. The difference     means subtract  minus  which we write as
3. The product     means multiply  times  which we can write as
4. The quotient     means divide  by  which we can write as
When two quantities have the same value, we say they are equal and connect them with an equal sign.
An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that  is greater than  it means that  is to the right of  on the number line. We use the symbols  and  for inequalities.
The expressions  can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,
When we write an inequality symbol with a line under it, such as  it means  or  We read this  is less than or equal to  Also, if we put a slash through an equal sign,  it means not equal.
We summarize the symbols of equality and inequality in .
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions.  lists three of the most commonly used grouping symbols in algebra.
Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

### Identify Expressions and Equations

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb.
In algebra, we have expressions and equations. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:
Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:

### Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify  we’d first multiply  to get  and then add the  to get  A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:
Suppose we have the expression  We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write  as  and  as  In expressions such as  the  is called the base and the  is called the exponent. The exponent tells us how many factors of the base we have to multiply.
We say  is in exponential notation and  is in expanded notation.
For powers of  and  we have special names.
 lists some examples of expressions written in exponential notation.
To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

### Simplify Expressions Using the Order of Operations

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.
For example, consider the expression:
Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.
Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. Please Excuse My Dear Aunt Sally.
It’s good that ‘My Dear’ goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, ‘Aunt Sally’ goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.
When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

### Key Concepts

1. Equality Symbol

2. Inequality

1. Exponential Notation

Order of Operations When simplifying mathematical expressions perform the operations in the following order:

### Practice Makes Perfect

Use Variables and Algebraic Symbols
In the following exercises, translate from algebraic notation to words.
Identify Expressions and Equations
In the following exercises, determine if each is an expression or an equation.
Simplify Expressions with Exponents
In the following exercises, write in exponential form.
In the following exercises, write in expanded form.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify.

### Everyday Math

### 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.


# The Language of Algebra

## Evaluate, Simplify, and Translate Expressions

### Evaluate Algebraic Expressions

In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.
To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

### Identify Terms, Coefficients, and Like Terms

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are
The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term  is  When we write  the coefficient is  since   gives the coefficients for each of the terms in the left column.
An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together.  gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.
Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?
Which of these terms are like terms?
1. The terms  and  are both constant terms.
2. The terms  and  are both terms with
3. The terms  and  both have
Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms

### Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think  would simplify to? If you thought  you would be right!
We can see why this works by writing both terms as addition problems.
Add the coefficients and keep the same variable. It doesn’t matter what  is. If you have  of something and add  more of the same thing, the result is  of them. For example,  oranges plus  oranges is  oranges. We will discuss the mathematical properties behind this later.
The expression  has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.
Now it is easier to see the like terms to be combined.
When any of the terms have negative coefficients, the procedure is the same, except that you have to subtract instead of adding to combine like terms.

### Translate Words to Algebraic Expressions

In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized in .
Look closely at these phrases using the four operations:
1. the sum of  and
2. the difference of  and
3. the product of  and
4. the quotient of  and
Each phrase tells you to operate on two numbers. Look for the words  and  to find the numbers.
How old will you be in eight years? What age is eight more years than your age now? Did you add  to your present age? Eight more than means eight added to your present age.
How old were you seven years ago? This is seven years less than your age now. You subtract  from your present age. Seven less than means seven subtracted from your present age.
Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase to an algebraic expression. We’ll need to be clear about what the expression will represent. We’ll see how to do this in the next two examples.

### Key Concepts

1. Combine like terms.

### Practice Makes Perfect

Evaluate Algebraic Expressions
In the following exercises, evaluate the expression for the given value.
Identify Terms, Coefficients, and Like Terms
In the following exercises, list the terms in the given expression.
In the following exercises, identify the coefficient of the given term.
In the following exercises, identify all sets of like terms.
Simplify Expressions by Combining Like Terms
In the following exercises, simplify the given expression by combining like terms.
Translate English Phrases into Algebraic Expressions
In the following exercises, translate the given word phrase into an algebraic expression.
In the following exercises, write an algebraic expression.

### Everyday Math

In the following exercises, use algebraic expressions to solve the problem.

### 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?


# The Language of Algebra

## Solving Equations Using the Subtraction and Addition Properties of Equality

When some people hear the word algebra, they think of solving equations. The applications of solving equations are limitless and extend to all careers and fields. In this section, we will begin solving equations. We will start by solving basic equations, and then as we proceed through the course we will build up our skills to cover many different forms of equations.

### Determine Whether a Number is a Solution of an Equation

Solving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable that make the equation a true statement. Any number that makes the equation true is called a solution of the equation. It is the answer to the puzzle!
To find the solution to an equation means to find the value of the variable that makes the equation true. Can you recognize the solution of  If you said  you’re right! We say  is a solution to the equation  because when we substitute  for  the resulting statement is true.
Since  is a true statement, we know that  is indeed a solution to the equation.
The symbol  asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign  or not-equal sign

### Model the Subtraction Property of Equality

We will use a model to help you understand how the process of solving an equation is like solving a puzzle. An envelope represents the variable – since its contents are unknown – and each counter represents one.
Suppose a desk has an imaginary line dividing it in half. We place three counters and an envelope on the left side of desk, and eight counters on the right side of the desk as in . Both sides of the desk have the same number of counters, but some counters are hidden in the envelope. Can you tell how many counters are in the envelope?
What steps are you taking in your mind to figure out how many counters are in the envelope? Perhaps you are thinking “I need to remove the  counters from the left side to get the envelope by itself. Those  counters on the left match with  on the right, so I can take them away from both sides. That leaves five counters on the right, so there must be  counters in the envelope.”  shows this process.
What algebraic equation is modeled by this situation? Each side of the desk represents an expression and the center line takes the place of the equal sign. We will call the contents of the envelope  so the number of counters on the left side of the desk is  On the right side of the desk are  counters. We are told that  is equal to  so our equation is
Let’s write algebraically the steps we took to discover how many counters were in the envelope.
Now let’s check our solution. We substitute  for  in the original equation and see if we get a true statement.
Our solution is correct. Five counters in the envelope plus three more equals eight.

### Solve Equations Using the Subtraction Property of Equality

Our puzzle has given us an idea of what we need to do to solve an equation. The goal is to isolate the variable by itself on one side of the equations. In the previous examples, we used the Subtraction Property of Equality, which states that when we subtract the same quantity from both sides of an equation, we still have equality.
Think about twin brothers Andy and Bobby. They are  years old. How old was Andy  years ago? He was  years less than  so his age was  or  What about Bobby’s age  years ago? Of course, he was  also. Their ages are equal now, and subtracting the same quantity from both of them resulted in equal ages  years ago.

### Solve Equations Using the Addition Property of Equality

In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to “undo” the addition in order to isolate the variable.
But suppose we have an equation with a number subtracted from the variable, such as  We want to isolate the variable, so to “undo” the subtraction we will add the number to both sides.
We use the Addition Property of Equality, which says we can add the same number to both sides of the equation without changing the equality. Notice how it mirrors the Subtraction Property of Equality.
Remember the  twins, Andy and Bobby? In ten years, Andy’s age will still equal Bobby’s age. They will both be
We can add the same number to both sides and still keep the equality.

### Translate Word Phrases to Algebraic Equations

Remember, an equation has an equal sign between two algebraic expressions. So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation. We look for clue words that mean equals. Some words that translate to the equal sign are:
1. is equal to
2. is the same as
3. is
4. gives
5. was
6. will be
It may be helpful to put a box around the equals word(s) in the sentence to help you focus separately on each phrase. Then translate each phrase into an expression, and write them on each side of the equal sign.
We will practice translating word sentences into algebraic equations. Some of the sentences will be basic number facts with no variables to solve for. Some sentences will translate into equations with variables. The focus right now is just to translate the words into algebra.

### Translate to an Equation and Solve

Now let’s practice translating sentences into algebraic equations and then solving them. We will solve the equations by using the Subtraction and Addition Properties of Equality.

### Key Concepts

1. Determine whether a number is a solution to an equation.

           If it is not true, the number is not a solution.

2. Subtraction Property of Equality

3. Solve an equation using the Subtraction Property of Equality.

4.
Addition Property of Equality

5. Solve an equation using the Addition Property of Equality.

### Practice Makes Perfect

Determine Whether a Number is a Solution of an Equation
In the following exercises, determine whether each given value is a solution to the equation.
Model the Subtraction Property of Equality
In the following exercises, write the equation modeled by the envelopes and counters and then solve using the subtraction property of equality.
Solve Equations using the Subtraction Property of Equality
In the following exercises, solve each equation using the subtraction property of equality.
Solve Equations using the Addition Property of Equality
In the following exercises, solve each equation using the addition property of equality.
Translate Word Phrase to Algebraic Equations
In the following exercises, translate the given sentence into an algebraic equation.
Translate to an Equation and Solve
In the following exercises, translate the given sentence into an algebraic equation and then solve it.

### Everyday Math

### 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?


# The Language of Algebra

## Find Multiples and Factors

### Identify Multiples of Numbers

Annie is counting the shoes in her closet. The shoes are matched in pairs, so she doesn’t have to count each one. She counts by twos:  She has  shoes in her closet.
The numbers  are called multiples of  Multiples of  can be written as the product of a counting number and  The first six multiples of  are given below.
A multiple of a number is the product of the number and a counting number. So a multiple of  would be the product of a counting number and  Below are the first six multiples of
We can find the multiples of any number by continuing this process.  shows the multiples of  through  for the first twelve counting numbers.
Recognizing the patterns for multiples of  will be helpful to you as you continue in this course.
 shows the counting numbers from  to  Multiples of  are highlighted. Do you notice a pattern?
The last digit of each highlighted number in  is either  This is true for the product of  and any counting number. So, to tell if any number is a multiple of  look at the last digit. If it is  then the number is a multiple of
Now let’s look at multiples of   highlights all of the multiples of  between  and  What do you notice about the multiples of
All multiples of  end with either  or  Just like we identify multiples of  by looking at the last digit, we can identify multiples of  by looking at the last digit.
 highlights the multiples of  between  and  All multiples of  all end with a zero.
 highlights multiples of  The pattern for multiples of  is not as obvious as the patterns for multiples of
Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit. The pattern for multiples of  is based on the sum of the digits. If the sum of the digits of a number is a multiple of  then the number itself is a multiple of  See .
Consider the number  The digits are  and  and their sum is  Since  is a multiple of  we know that  is also a multiple of
Look back at the charts where you highlighted the multiples of  of  and of  Notice that the multiples of  are the numbers that are multiples of both  and  That is because  Likewise, since  the multiples of  are the numbers that are multiples of both  and

### Use Common Divisibility Tests

Another way to say that  is a multiple of  is to say that  is divisible by  In fact,  is  so  is  Notice in  that  is not a multiple  When we divided  by  we did not get a counting number, so  is not divisible by
Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests.  summarizes divisibility tests for some of the counting numbers between one and ten.

### Find All the Factors of a Number

There are often several ways to talk about the same idea. So
far, we’ve seen that if  is a multiple of  we can say that  is divisible by  We know that  is the product of  and  so we can say  is a multiple of  and  is a multiple of  We can also say  is divisible by  and by  Another way to talk about this is to say that  and  are factors of  When we write  we can say that we have factored
In algebra, it can be useful to determine all of the factors of a number. This is called factoring a number, and it can help us solve many kinds of problems.
For example, suppose a choreographer is planning a dance for a ballet recital. There are  dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage.
In how many ways can the dancers be put into groups of equal size? Answering this question is the same as identifying the factors of   summarizes the different ways that the choreographer can arrange the dancers.
What patterns do you see in ? Did you notice that the number of groups times the number of dancers per group is always  This makes sense, since there are always  dancers.
You may notice another pattern if you look carefully at the first two columns. These two columns contain the exact same set of numbers—but in reverse order. They are mirrors of one another, and in fact, both columns list all of the factors of  which are:
We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with  If the quotient is also a counting number, then the divisor and the quotient are factors of the number. We can stop when the quotient becomes smaller than the divisor.

### Identify Prime and Composite Numbers

Some numbers, like  have many factors. Other numbers, such as  have only two factors:  and the number. A number with only two factors is called a prime number. A number with more than two factors is called a composite number. The number  is neither prime nor composite. It has only one factor, itself.
 lists the counting numbers from  through  along with their factors. The highlighted numbers are prime, since each has only two factors.
The prime numbers less than  are  There are many larger prime numbers too. In order to determine whether a number is prime or composite, we need to see if the number has any factors other than  and itself. To do this, we can test each of the smaller prime numbers in order to see if it is a factor of the number. If none of the prime numbers are factors, then that number is also prime.

### Key Concepts

1. Factors   If , then   and   are factors of , and  is the product of   and .
2. Find all the factors of a counting number.

3. Determine if a number is prime.

### Practice Makes Perfect

Identify Multiples of Numbers
In the following exercises, list all the multiples less than  for the given number.
Use Common Divisibility Tests
In the following exercises, use the divisibility tests to determine whether each number is divisible by
Find All the Factors of a Number
In the following exercises, find all the factors of the given number.
Identify Prime and Composite Numbers
In the following exercises, determine if the given number is prime or composite.

### Everyday Math

### 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?