# Sequences, Probability, and Counting Theory

## Geometric Sequences

### Learning Objectives

1. Determine if a sequence is geometric (IA 12.3.1).
2. Find the general term (nth term) of a geometric sequence (IA 12.3.2).

### Objective 1: Determine if a sequence is geometric (IA 12.3.1)

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.

### Practice Makes Perfect

Determine if each sequence is geometric. If so, indicate the common ratio.

### Practice Makes Perfect

### Objective 2: Find the general term (nth term) of a geometric sequence (IA 12.3.2)

Let’s find the 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  . We will then look for a pattern.

### Practice Makes Perfect

Find the general term (nth term) of a geometric sequence.

Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.

### Finding Common Ratios

The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.

### Writing Terms of Geometric Sequences

Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is
  and the common ratio is
  we can find subsequent terms by multiplying
  to get
  then multiplying the result
  to get
  and so on.
The first four terms are

### Using Recursive Formulas for Geometric Sequences

A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.

### Using Explicit Formulas for Geometric Sequences

Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.
Let’s take a look at the sequence
  This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is
The graph of the sequence is shown in .

### Solving Application Problems with Geometric Sequences

In real-world scenarios involving geometric sequences, we may need to use an initial term of
  instead of
  In these problems, we can alter the explicit formula slightly by using the following formula:

### Key Equations

### Key Concepts

1. A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
2. The constant ratio between two consecutive terms is called the common ratio.
3. The common ratio can be found by dividing any term in the sequence by the previous term. See .
4. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See  and .
5. A recursive formula for a geometric sequence with common ratio
  is given by
  for
  .
6. As with any recursive formula, the initial term of the sequence must be given. See .
7. An explicit formula for a geometric sequence with common ratio
  is given by
  See .
8. In application problems, we sometimes alter the explicit formula slightly to
  See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the common ratio for the geometric sequence.
For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.
For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.
For the following exercises, write the first five terms of the geometric sequence, given any two terms.
For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.
For the following exercises, find the specified term for the geometric sequence, given the first four terms.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write a recursive formula for each geometric sequence.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write an explicit formula for each geometric sequence.
For the following exercises, find the specified term for the geometric sequence given.
For the following exercises, find the number of terms in the given finite geometric sequence.

### Graphical

For the following exercises, determine whether the graph shown represents a geometric sequence.
For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

### Extensions


# Sequences, Probability, and Counting Theory

## Series and Their Notations

### Learning Objectives

1. Use summation notation to write a sum. (IA 12.1.5)
2. Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)

### Objective 1: Use summation notation to write a sum. (IA 12.1.5)

A series is the sum of the terms of a sequence. For example,  1 + 6 + 11+ 16 + 21 + 26 + 31 is the sum of the first seven terms arithmetic sequence with general term,
We write a series by using the summation notation. In order to write that summation, we will need to find the general term of our sequence and the summation will look like:

For the series, 1 + 6 + 11 + 16  + 21 + 26 + 31 + .... the summation notation is

### Practice Makes Perfect

Use summation notation to write the sum.

### Objective 2: Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)

### Practice Makes Perfect

A parent decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section, we will learn how to answer this question. To do so, we need to consider the amount of money invested and the amount of interest earned.

### Using Summation Notation

To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series. Consider, for example, the following series.
The  of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation
 represents the partial sum.
Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma,
  to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term in the series. The number above the sigma, called the upper limit of summation, is the number used to generate the last term in a series.
If we interpret the given notation, we see that it asks us to find the sum of the terms in the series
  for
  through
  We can begin by substituting the terms for
 and listing out the terms of this series.
We can find the sum of the series by adding the terms:

### Using the Formula for Arithmetic Series

Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference,
 The sum of the terms of an arithmetic sequence is called an arithmetic series. We can write the sum of the first
 terms of an arithmetic series as:
We can also reverse the order of the terms and write the sum as
If we add these two expressions for the sum of the first
  terms of an arithmetic series, we can derive a formula for the sum of the first
  terms of any arithmetic series.
Because there are
 terms in the series, we can simplify this sum to
We divide by 2 to find the formula for the sum of the first
  terms of an arithmetic series.
Use the formula to find the sum of each arithmetic series.

### Using the Formula for Geometric Series

Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio,
  We can write the sum of the first
  terms of a geometric series as
Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first
  terms of a geometric series. We will begin by multiplying both sides of the equation by

Next, we subtract this equation from the original equation.
Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for
 divide both sides by

Use the formula to find the indicated partial sum of each geometric series.

### Using the Formula for the Sum of an Infinite Geometric Series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first
 terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is

This series can also be written in summation notation as
 where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.

### Determining Whether the Sum of an Infinite Geometric Series is Defined

If the terms of an infinite geometric sequence approach 0, the sum of an infinite geometric series can be defined.  The terms in this series approach 0:
The common ratio

 As
  gets very large, the values of
 get very small and approach 0.   Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with
 approach 0; the sum of a geometric series is defined when

Determine whether the sum of the infinite series is defined.

### Finding Sums of Infinite Series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first
 terms of a geometric series.
We will examine an infinite series with
 What happens to
  as
  increases?
The value of
  decreases rapidly. What happens for greater values of

As
  gets very large,
 gets very small. We say that, as
  increases without bound,
 approaches 0. As
  approaches 0,
 approaches 1. When this happens, the numerator approaches
  This give us a formula for the sum of an infinite geometric series.
Find the sum, if it exists.

### Solving Annuity Problems

At the beginning of the section, we looked at a problem in which a parent invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the parent invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.
We can find the value of the annuity right after the last deposit by using a geometric series with
 and
 After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.
We can find the value of the annuity after
 deposits using the formula for the sum of the first
 terms of a geometric series. In 6 years, there are 72 months, so
 We can substitute
 into the formula, and simplify to find the value of the annuity after 6 years.
After the last deposit, the parent will have a total of $4,320.44 in the account. Notice, the parent made 72 payments of $50 each for a total of
  This means that because of the annuity, the parent earned $720.44 interest in their college fund.

### Key Equations

### Key Concepts

1. The sum of the terms in a sequence is called a series.
2. A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See .
3. The sum of the terms in an arithmetic sequence is called an arithmetic series.
4. The sum of the first
 terms of an arithmetic series can be found using a formula. See  and .
5. The sum of the terms in a geometric sequence is called a geometric series.
6. The sum of the first
 terms of a geometric series can be found using a formula. See  and .
7. The sum of an infinite series exists if the series is geometric with

8. If the sum of an infinite series exists, it can be found using a formula. See , , and .
9. An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, express each description of a sum using summation notation.
For the following exercises, express each arithmetic sum using summation notation.
For the following exercises, use the formula for the sum of the first
 terms of each arithmetic sequence.
For the following exercises, express each geometric sum using summation notation.
For the following exercises, use the formula for the sum of the first
  terms of each geometric sequence, and then state the indicated sum.
For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

### Graphical

For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20.
For the following exercises, use the geometric series

### Numeric

For the following exercises, find the indicated sum.
For the following exercises, use the formula for the sum of the first  terms of an arithmetic series to find the sum.
For the following exercises, use the formula for the sum of the first
  terms of a geometric series to find the partial sum.
For the following exercises, find the sum of the infinite geometric series.
For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

### Extensions

### Real-World Applications


# Sequences, Probability, and Counting Theory

## Counting Principles

### Learning Objectives

1. Solve counting problems using the addition principle.
2. Solve counting problems using the multiplication principle.

### Objective 1: Solve counting problems using the addition principle.

In probability theory, an outcome is a possible result of an experiment or trial.

In probability theory, an event is a set of outcomes of an experiment.

Disjoint events cannot happen at the same time. In other words, they are mutually exclusive.

The addition principle applies when we are making only one selection.

### Practice Makes Perfect

Solve counting problems using the addition principle.

### Objective 2: Solve counting problems using the multiplication principle.

The Multiplication Principle applies when we are making more than one selection.

### Practice Makes Perfect

Solve counting problems using the multiplication principle.

A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. The customer can choose the order of the images and the letters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. Counting the possibilities is challenging!
We encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. We will examine this type of mathematics in this section.

### Using the Addition Principle

The company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet models and 5 supported smartphone models. The Addition Principle tells us that we can add the number of tablet options to the number of smartphone options to find the total number of options. By the Addition Principle, there are 8 total options, as we can see in .

### Using the Multiplication Principle

The Multiplication Principle applies when we are making more than one selection. Suppose we are choosing an appetizer, an entrée, and a dessert. If there are 2 appetizer options, 3 entrée options, and 2 dessert options on a fixed-price dinner menu, there are a total of 12 possible choices of one each as shown in the tree diagram in .
The possible choices are:
1. soup, chicken, cake
2. soup, chicken, pudding
3. soup, fish, cake
4. soup, fish, pudding
5. soup, steak, cake
6. soup, steak, pudding
7. salad, chicken, cake
8. salad, chicken, pudding
9. salad, fish, cake
10. salad, fish, pudding
11. salad, steak, cake
12. salad, steak, pudding
We can also find the total number of possible dinners by multiplying.
We could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.

### Finding the Number of Permutations of n Distinct Objects

The Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation.

### Finding the Number of Permutations of n Distinct Objects Using the Multiplication Principle

To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.

There are four options for the first place, so we write a 4 on the first line.

After the first place has been filled, there are three options for the second place so we write a 3 on the second line.

After the second place has been filled, there are two options for the third place so we write a 2 on the third line. Finally, we find the product.

There are 24 possible permutations of the paintings.
A family of five is having portraits taken. Use the Multiplication Principle to find the following.

### Finding the Number of Permutations of n Distinct Objects Using a Formula

For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let’s look at two common notations for permutations. If we have a set of
  objects and we want to choose
  objects from the set in order, we write
  Another way to write this is
  a notation commonly seen on computers and calculators. To calculate
  we begin by finding
  the number of ways to line up all
  objects. We then divide by
  to cancel out the
  items that we do not wish to line up.
Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is
  Using factorials, we get the same result.
There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.
Note that the formula stills works if we are choosing all
  objects and placing them in order. In that case we would be dividing by
  or
  which we said earlier is equal to 1. So the number of permutations of
  objects taken
  at a time is
  or just

A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.

### Find the Number of Combinations Using the Formula

So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with combinations. A selection of
  objects from a set of
  objects where the order does not matter can be written as
  Just as with permutations,
  can also be written as
  In this case, the general formula is as follows.
An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are
  ways to order 3 paintings. There are
  or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are not selecting 1 painting. There are 4 paintings we could choose not to select, so there are 4 ways to select 3 of the 4 paintings.

### Finding the Number of Subsets of a Set

We have looked only at combination problems in which we chose exactly
  objects. In some problems, we want to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings. Any number of toppings can be ordered. How many different pizzas are possible?
To answer this question, we need to consider pizzas with any number of toppings. There is
  way to order a pizza with no toppings. There are
  ways to order a pizza with exactly one topping. If we continue this process, we get
There are 32 possible pizzas. This result is equal to

We are presented with a sequence of choices. For each of the
  objects we have two choices: include it in the subset or not. So for the whole subset we have made
  choices, each with two options. So there are a total of
  possible resulting subsets, all the way from the empty subset, which we obtain when we say “no” each time, to the original set itself, which we obtain when we say “yes” each time.

### Finding the Number of Permutations of n Non-Distinct Objects

We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be  ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the
  permutations we counted are duplicates. The general formula for this situation is as follows.
In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are
  ways to order the stars and
  ways to order the moon.
There are 3,326,400 ways to order the sheet of stickers.

### Key Equations

### Key Concepts

1. If one event can occur in
  ways and a second event with no common outcomes can occur in
  ways, then the first or second event can occur in
  ways. See .
2. If one event can occur in
  ways and a second event can occur in
  ways after the first event has occurred, then the two events can occur in
  ways. See .
3. A permutation is an ordering of
  objects.
4. If we have a set of
  objects and we want to choose
  objects from the set in order, we write

5. Permutation problems can be solved using the Multiplication Principle or the formula for
  See  and .
6. A selection of objects where the order does not matter is a combination.
7. Given
  distinct objects, the number of ways to select
  objects from the set is
  and can be found using a formula. See .
8. A set containing
  distinct objects has
  subsets. See .
9. For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See .

### Section Exercises

### Verbal

For the following exercises, assume that there are
  ways an event
  can happen,
  ways an event
  can happen, and that
  are non-overlapping.
Answer the following questions.

### Numeric

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.
For the following exercises, compute the value of the expression.
For the following exercises, find the number of subsets in each given set.
For the following exercises, find the distinct number of arrangements.

### Extensions

### Real-World Applications


# Sequences, Probability, and Counting Theory

## Binomial Theorem

### Learning Objectives

1. Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)

### Objective 1: Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)

Pascal’s triangle helps us find the coefficients of the terms in the expansion of a binomial.

To find the coefficients of the terms, we write our expansion 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 that in each expansion the powers of a in each term decrease from n to 0, and the powers of b increase from 0 to n.

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.

To find the coefficients of the expansion of the binomial  , go to the row that has the value n as a second entry.

### Practice Makes Perfect

Use Pascal’s Triangle to expand a binomial.

A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find
  without multiplying the binomial by itself
  times.

### Identifying Binomial Coefficients

In Counting Principles, we studied combinations. In the shortcut to finding
  we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation
  instead of
  but it can be calculated in the same way. So
The combination
  is called a binomial coefficient. An example of a binomial coefficient is

### Using the Binomial Theorem

When we expand
  by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand
  we might multiply
  by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.
First, let’s examine the exponents. With each successive term, the exponent for
  decreases and the exponent for
  increases. The sum of the two exponents is
  for each term.
Next, let’s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:
These patterns lead us to the Binomial Theorem, which can be used to expand any binomial.
Another way to see the coefficients is to examine the expansion of a binomial in general form,
  to successive powers 1, 2, 3, and 4.
Can you guess the next expansion for the binomial

See , which illustrates the following:
1. There are
  terms in the expansion of

2. The degree (or sum of the exponents) for each term is

3. The powers on
  begin with
  and decrease to 0.
4. The powers on
  begin with 0 and increase to

5. The coefficients are symmetric.
To determine the expansion on
  we see
  thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of
  the pattern is as follows:
1. Introduce
  and then for each successive term reduce the exponent on
  by 1 until
  is reached.
2. Introduce
  and then increase the exponent on
  by 1 until
  is reached.

The next expansion would be
But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as Pascal's Triangle, shown in . Pascal didn't invent the triangle. The underlying principles had been developed and written about for over 1500 years, first by the Indian mathematician (and poet) Pingala in the second century BCE. Others throughout Asia and Europe worked with the concepts throughout, and the triangle was first published in its graphical form by Omar Khayyam, an Iranian mathematician and astronomer, for whom the triangle is named in Iran. French mathematician Blaise Pascal repopularized it when he republished it and used it to solve a number of probability problems.
To generate Pascal’s Triangle, we start by writing a 1. In the row below, row 2, we write two 1’s. In the 3rd row, flank the ends of the rows with 1’s, and add
  to find the middle number, 2. In the
  row, flank the ends of the row with 1’s. Each element in the triangle is the sum of the two elements immediately above it.
To see the connection between Pascal’s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form.

### Using the Binomial Theorem to Find a Single Term

Expanding a binomial with a high exponent such as
  can be a lengthy process.
Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.
Note the pattern of coefficients in the expansion of

The second term is
  The third term is
  We can generalize this result.

### Key Equations

### Key Concepts

1.
  is called a binomial coefficient and is equal to
  See .
2. The Binomial Theorem allows us to expand binomials without multiplying. See .
3. We can find a given term of a binomial expansion without fully expanding the binomial. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, evaluate the binomial coefficient.
For the following exercises, use the Binomial Theorem to expand each binomial.
For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.
For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

### Graphical

For the following exercises, use the Binomial Theorem to expand the binomial
  Then find and graph each indicated sum on one set of axes.

### Extensions


# Sequences, Probability, and Counting Theory

## Probability

### Learning Objectives

1. Introduction to Sample Spaces and Computing Basic Probabilities.

### Objective 1: Introduction to Sample Spaces and Computing Basic Probabilities.

Many events in life are inherently uncertain: will it snow tomorrow? Am I going to get an ‘A’ in this course? None of these questions can be answered with certainty, however, we might say that some are unlikely, and others are more likely.

The probability of an event is a description of how likely it is that an event will happen. A probability is a number between 0 and 1 (that is, between 0% and 100%), where probabilities closer to 100% are very likely to occur, and probabilities closer to 0% are very unlikely to occur. A probability of 0% means the event is impossible, and a probability of 100% means the event will certainly occur.

A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. It is defined by its sample space, events within the sample space, and probabilities associated with each event.

The sample space S for a probability model is the set of all possible outcomes. For example, the sample space for rolling a dice is the set 1,2,3,4,5,6.This notation is referred to as roster notation.

An event A is a subset of the sample space S. For example, the event “Rolling an even number” is the subset 2,4,6.

To calculate the probability of an event, we divide the number of possible outcomes of the event by the number of possible outcomes of the sample space.

It is important to note that in order to use this formula, all outcomes must be equally likely to happen.

For example, the probability of rolling an even number with a standard dice is:

### Practice Makes Perfect

Residents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one in . They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line represents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name. In this section, we will investigate methods for making these types of predictions.

### Constructing Probability Models

Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment, or an activity with an observable result. The numbers on the cube are possible results, or outcomes, of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is  An event is any subset of a sample space.
The likelihood of an event is known as probability. The probability of an event
 is a number that always satisfies
  where 0 indicates an impossible event and 1 indicates a certain event. A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like .
The sum of the probabilities listed in a probability model must equal 1, or 100%.

### Computing Probabilities of Equally Likely Outcomes

Let
 be a sample space for an experiment. When investigating probability, an event is any subset of
 When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in
 Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in
 so the probability of the event is

### Computing the Probability of the Union of Two Events

We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events
  is the event that occurs if either or both events occur.
Suppose the spinner in  is spun. We want to find the probability of spinning orange or spinning a

There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is
  There are a total of 6 sections, and 2 of them have a
 So the probability of spinning a
 is
  If we added these two probabilities, we would be counting the sector that is both orange and a
 twice. To find the probability of spinning an orange or a
 we need to subtract the probability that the sector is both orange and has a

The probability of spinning orange or a
 is

### Computing the Probability of Mutually Exclusive Events

Suppose the spinner in  is spun again, but this time we are interested in the probability of spinning an orange or a
 There are no sectors that are both orange and contain a
 so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is
Notice that with mutually exclusive events, the intersection of
 and
 is the empty set. The probability of spinning an orange is
 and the probability of spinning a
 is
 We can find the probability of spinning an orange or a
 simply by adding the two probabilities.
The probability of spinning an orange or a
 is

### Using the Complement Rule to Compute Probabilities

We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event
 denoted
  is the set of outcomes in the sample space that are not in
 For example, suppose we are interested in the probability that a horse will lose a race. If event
 is the horse winning the race, then the complement of event
 is the horse losing the race.
To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.
The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is
 the probability of the horse losing the race is simply

### Computing Probability Using Counting Theory

Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems.
Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are
 ways to select 2 phones that are not defective. There are 8 phones, so there are
 ways to select 2 phones. The probability of selecting 2 phones that are not defective is:

### Key Equations

### Key Concepts

1. Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.
2. The probabilities in a probability model must sum to 1. See .
3. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. See .
4. To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See .
5. To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See .
6. The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See .
7. In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See .

### Section Exercises

### Verbal

### Numeric

For the following exercises, use the spinner shown in  to find the probabilities indicated.
For the following exercises, two coins are tossed.
For the following exercises, four coins are tossed.
For the following exercises, one card is drawn from a standard deck of
  cards. Find the probability of drawing the following:
For the following exercises, two dice are rolled, and the results are summed.
For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:
For the following exercises, use this scenario: a bag of M&Ms contains
  blue,
  brown,
  orange,
  yellow,
  red, and
  green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.

### Extensions

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting
  numbers from the numbers
  to
  After the player makes his selections,
  winning numbers are randomly selected from numbers
  to
  A win occurs if the player has correctly selected
  or
  of the
  winning numbers. (Round all answers to the nearest hundredth of a percent.)

### Real-World Applications

Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).United States Census Bureau. http://www.census.gov

### Chapter Review Exercises

### Sequences and Their Notation

### Arithmetic Sequences

### Geometric Sequences

### Series and Their Notation

### Counting Principles

### Binomial Theorem

### Probability

For the following exercises, assume two die are rolled.
For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)

### Practice Test

For the following exercises, use the spinner in .


# Prerequisites

## Introduction to Prerequisites

It’s a cold day in Antarctica. In fact, it’s always a cold day in Antarctica. Earth’s southernmost continent, Antarctica experiences the coldest, driest, and windiest conditions known. The coldest temperature ever recorded, over one hundred degrees below zero on the Celsius scale, was recorded by remote satellite. It is no surprise then, that no native human population can survive the harsh conditions. Only explorers and scientists brave the environment for any length of time.
Measuring and recording the characteristics of weather conditions in Antarctica requires a use of different kinds of numbers. For tens of thousands of years, humans have undertaken methods to tally, track, and record numerical information. While we don't know much about their usage, the Lebombo Bone (dated to about 35,000 BCE) and the Ishango Bone (dated to about 20,000 BCE) are among the earliest mathematical artifacts. Found in Africa, their clearly deliberate groupings of notches may have been used to track time, moon cycles, or other information. Performing calculations with them and using the results to make predictions requires an understanding of relationships among numbers. In this chapter, we will review sets of numbers and properties of operations used to manipulate numbers. This understanding will serve as prerequisite knowledge throughout our study of algebra and trigonometry.


# Prerequisites

## Real Numbers: Algebra Essentials

### Learning Objectives

1. Identify the study skills leading to success in a college level mathematics course.
2. Reflect on your past math experiences and create a plan for improvement.

### Objective 1: Identify the study skills leading to success in a college level mathematics course.

Welcome to your algebra course! This course will be challenging so now is the time to set up a plan for success. In this first chapter we will focus on important strategies for success including: math study skills, time management, note taking skills, smart test taking strategies, and the idea of a growth mindset. Each of these ideas will help you to be successful in your college level math course whether you are enrolled in a face-to-face traditional section or an online section virtual section.

Complete the following survey by checking a column for each behavior based on the frequency that you engage in the behavior during your last academic term.

### Practice Makes Perfect

Identify the study skills leading to success in a college level mathematics course.

### Objective 2: Reflect on your past math experiences and create a plan for improvement.

1.
It’s important to take the opportunity to reflect on your past experiences in math classes as you begin a new term. We can learn a lot from these reflections and thus work toward developing a strategy for improvement.
In the table below list 5 challenges you had in past math courses and list a possible solution that you could try this semester.

2.
Write your math autobiography. Tell your math story by describing your past experiences as a learner of mathematics. Share how your attitudes have changed about math over the years if they have. Perhaps include what you love, hate, dread, appreciate, fear, look forward to, or find beauty in. This will help your teacher to better understand you and your current feelings about the discipline.

3.
Share your autobiographies with your study group members. This helps to create a community in the classroom when common themes emerge.

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.
Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.
But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.
Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.
Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

### Classifying a Real Number

The numbers we use for counting, or enumerating items, are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe them in set notation as
    where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero:

The set of integers adds the opposites of the natural numbers to the set of whole numbers:
    It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.
The set of rational numbers is written as
    Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.
Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:
1. ⓐa terminating decimal:
    or
2. ⓑa repeating decimal:

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

### Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even
    but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

### Real Numbers

Given any number n, we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.
The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in .

### Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as .

### Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example,
    We can raise any number to any power. In general, the exponential notation
    means that the number or variable
    is used as a factor
    times.
In this notation,
    is read as the nth power of
   or  to the  where
    is called the base and
    is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example,
    is a mathematical expression.
To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.
Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.
The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.
Let’s take a look at the expression provided.
There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify
    as 16.
Next, perform multiplication or division, left to right.
Lastly, perform addition or subtraction, left to right.
Therefore,

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

### Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

### Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.
We can better see this relationship when using real numbers.
Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.
Again, consider an example with real numbers.
It is important to note that neither subtraction nor division is commutative. For example,
    is not the same as
    Similarly,

### Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.
Consider this example.
The associative property of addition tells us that numbers may be grouped differently without affecting the sum.
This property can be especially helpful when dealing with negative integers. Consider this example.
Are subtraction and division associative? Review these examples.
As we can see, neither subtraction nor division is associative.

### Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.
This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.
Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.
To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.
A special case of the distributive property occurs when a sum of terms is subtracted.
For example, consider the difference
    We can rewrite the difference of the two terms 12 and
    by turning the subtraction expression into addition of the opposite. So instead of subtracting
    we add the opposite.
Now, distribute
    and simplify the result.
This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

### Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.
The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.
For example, we have
    and
    There are no exceptions for these properties; they work for every real number, including 0 and 1.

### Inverse Properties

The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted by (−a), that, when added to the original number, results in the additive identity, 0.
For example, if
    the additive inverse is 8, since

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted
    that, when multiplied by the original number, results in the multiplicative identity, 1.
For example, if
    the reciprocal, denoted
    is

  because

### Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as
    or
    In the expression
    5 is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.
We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.
In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.
Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

### Formulas

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation
    has the solution of 3
   because when we substitute 3 for
    in the equation, we obtain the true statement

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area
    of a circle in terms of the radius
    of the circle:
    For any value of
    the area
    can be found by evaluating the expression

### Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

### Key Concepts

1. Rational numbers may be written as fractions or terminating or repeating decimals. See  and .
2. Determine whether a number is rational or irrational by writing it as a decimal. See .
3. The rational numbers and irrational numbers make up the set of real numbers. See . A number can be classified as natural, whole, integer, rational, or irrational. See .
4. The order of operations is used to evaluate expressions. See .
5. The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. See .
6. Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. See . They take on a numerical value when evaluated by replacing variables with constants. See , , and
7. Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression. See  and .

### Section Exercises

### Verbal

### Numeric

For the following exercises, simplify the given expression.

### Algebraic

For the following exercises, evaluate the expression using the given value of the variable.
For the following exercises, simplify the expression.

### Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.
For the following exercises, solve the given problem.
For the following exercises, consider this scenario: There is a mound of
    pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.
For the following exercise, solve the given problem.

### Technology

For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth.

### Extensions


# Prerequisites

## Exponents and Scientific Notation

### Learning Objective:

1. Plan your weekly academic schedule for the term.

### Objective 1: Plan your weekly academic schedule for the term.

1.
Most college instructors advocate studying at least 2 hours for each hour in class. With this recommendation in mind, complete the following table showing credit hours enrolled in, the study time required, and total time to be devoted to college work. Assume 2 hours of study time for each hour in class to complete this table, and after your first exam you can fine tune this estimate based on your performance.

Consider spending at least 2 hours of your study time each week at your campus (or virtual) math tutoring center or with a study group, the time will be well spent!

2.
Another way to optimize your class and study time is to have a plan for efficiency, meaning make every minute count. Below is a list of good practices, check off those you feel you could utilize this term.

3.
Creating your Semester Calendar- complete the following weekly schedule being sure to labelOptional: also include if you want a more comprehensive view of your time commitments

Term: ________________________________
Name: ________________________________________
Date: ________________________________________
Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per pixel, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number.
Using a calculator, we enter
  and press ENTER. The calculator displays 1.304596316E13. What does this mean? The “E13” portion of the result represents the exponent 13 of ten, so there are a maximum of approximately
  bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers.

### Using the Product Rule of Exponents

Consider the product
       Both terms have the same base, x, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.
The result is that

Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.
Now consider an example with real numbers.
We can always check that this is true by simplifying each exponential expression. We find that
       is 8,
       is 16, and
       is 128. The product
       equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.

### Using the Quotient Rule of Exponents

The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as
       where
       Consider the example
       Perform the division by canceling common factors.
Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.
In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.
For the time being, we must be aware of the condition
       Otherwise, the difference
       could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.

### Using the Power Rule of Exponents

Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power rule of exponents. Consider the expression
       The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.
The exponent of the answer is the product of the exponents:
       In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.
Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.

### Using the Zero Exponent Rule of Exponents

Return to the quotient rule. We made the condition that
       so that the difference
       would never be zero or negative. What would happen if
       In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. To see how this is done, let us begin with an example.
If we were to simplify the original expression using the quotient rule, we would have
If we equate the two answers, the result is
       This is true for any nonzero real number, or any variable representing a real number.
The sole exception is the expression
       This appears later in more advanced courses, but for now, we will consider the value to be undefined.

### Using the Negative Rule of Exponents

Another useful result occurs if we relax the condition that
       in the quotient rule even further. For example, can we simplify
       When
       —that is, where the difference
       is negative—we can use the negative rule of exponents to simplify the expression to its reciprocal.
Divide one exponential expression by another with a larger exponent. Use our example,

If we were to simplify the original expression using the quotient rule, we would have
Putting the answers together, we have
       This is true for any nonzero real number, or any variable representing a nonzero real number.
A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.
We have shown that the exponential expression
       is defined when
       is a natural number, 0, or the negative of a natural number. That means that
       is defined for any integer
       Also, the product and quotient rules and all of the rules we will look at soon hold for any integer

### Finding the Power of a Product

To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider
       We begin by using the associative and commutative properties of multiplication to regroup the factors.
In other words,

### Finding the Power of a Quotient

To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.
Let’s rewrite the original problem differently and look at the result.
It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.

### Simplifying Exponential Expressions

Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

### Using Scientific Notation

Recall at the beginning of the section that we found the number
  when describing bits of information in digital images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an electron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers such as these?
A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places n that you moved the decimal point. Multiply the decimal number by 10 raised to a power of n. If you moved the decimal left as in a very large number,
     is positive. If you moved the decimal right as in a very small number,
     is negative.
For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.
We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.
Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.
Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

### Converting from Scientific to Standard Notation

To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal
     places to the right if
     is positive or
     places to the left if
     is negative and add zeros as needed. Remember, if
     is positive, the absolute value of the number is greater than 1, and if
     is negative, the absolute value of the number is less than one.

### Using Scientific Notation in Applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around
  molecules of water and 1 L of water holds about
  average drops. Therefore, there are approximately
  atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!
When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product
  The answer is not in proper scientific notation because 35 is greater than 10. Consider 35 as
  That adds a ten to the exponent of the answer.

### Key Equations

### Key Concepts

1. Products of exponential expressions with the same base can be simplified by adding exponents. See .
2. Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See .
3. Powers of exponential expressions with the same base can be simplified by multiplying exponents. See .
4. An expression with exponent zero is defined as 1. See .
5. An expression with a negative exponent is defined as a reciprocal. See  and .
6. The power of a product of factors is the same as the product of the powers of the same factors. See .
7. The power of a quotient of factors is the same as the quotient of the powers of the same factors. See .
8. The rules for exponential expressions can be combined to simplify more complicated expressions. See .
9. Scientific notation uses powers of 10 to simplify very large or very small numbers. See  and .
10. Scientific notation may be used to simplify calculations with very large or very small numbers. See  and .

### Section Exercises

### Verbal

### Numeric

For the following exercises, simplify the given expression. Write answers with positive exponents.
For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.
For the following exercises, express the decimal in scientific notation.
For the following exercises, convert each number in scientific notation to standard notation.

### Algebraic

For the following exercises, simplify the given expression. Write answers with positive exponents.

### Real-World Applications

### Technology

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.

### Extensions

For the following exercises, simplify the given expression. Write answers with positive exponents.


# Prerequisites

## Radicals and Rational Exponents

### Learning Objective:

1. Investigate the discipline called learning science and the idea of a knowledge space.

### Objective 1: Investigate the discipline called learning science and the idea of a knowledge space.

The brain is a complex organ. It is the control center for our bodies, while the mind is where thinking and learning take place. In an attempt to understand the processes that occur in learning, researchers study a collection of disciplines called learning sciences. This interdisciplinary field includes study of psychological, sociological, anthropological, and computational approaches to learning.

In this skill sheet we will investigate the mathematics of mastery and knowledge spaces. A knowledge space includes the possible states of knowledge of a human learner. The theory of knowledge space was introduced in 1985 by mathematical psychologists Jean-Paul Doignon and Jean-Claude Falmagne and has since been studied by many researchers.
Doignon, J.-P.; Falmagne, J.-Cl. (1985), "Spaces for the assessment of knowledge", International Journal of Man-Machine Studies.

### Practice Makes Perfect

Investigation:  There are 32 student-learning outcomes (SLO’s) in a typical College Algebra course. These are topics a student needs to master to show proficiency in College Algebra. Let’s begin by looking at just a few of these skills.  Let’s assign the variables, A, B, C, and D to the following topics. We will name the set containing each of these 4 topics, Q.

1. A = Graph the basic functions listed in the library of functions.
2. B = Find the domain of a function defined by an equation.
3. C = Create a new function through composition of functions.
4. D = Find linear functions that model data sets.
Using roster notation Q = {A, B, C, D}.
A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in , and use the Pythagorean Theorem.
Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.

### Evaluating Square Roots

When the square root of a number is squared, the result is the original number. Since
  the square root of

 is

 The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.
In general terms, if

 is a positive real number, then the square root of

 is a number that, when multiplied by itself, gives

 The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals

 The square root obtained using a calculator is the principal square root.
The principal square root of

 is written as

 The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression.

### Using the Product Rule to Simplify Square Roots

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite

 as

 We can also use the product rule to express the product of multiple radical expressions as a single radical expression.

### Using the Quotient Rule to Simplify Square Roots

Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite

 as

### Adding and Subtracting Square Roots

We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of

 and

 is

 However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression

 can be written with a

 in the radicand, as
  so

### Rationalizing Denominators

When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator.
We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.
For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is
  multiply by

For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is
  then the conjugate is

### Using Rational Roots

Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.

### Understanding nth Roots

Suppose we know that

 We want to find what number raised to the 3rd power is equal to 8. Since
  we say that 2 is the cube root of 8.
The nth root of

 is a number that, when raised to the nth power, gives

 For example,

 is the 5th root of

 because

 If

 is a real number with at least one nth root, then the principal  of

 is the number with the same sign as

 that, when raised to the nth power, equals

The principal nth root of

 is written as
  where

 is a positive integer greater than or equal to 2. In the radical expression,

 is called the index of the radical.

### Using Rational Exponents

Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index

 is even, then

 cannot be negative.
We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.
All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

### Key Concepts

1. The principal square root of a number

 is the nonnegative number that when multiplied by itself equals

 See .
2. If

 and

 are nonnegative, the square root of the product

 is equal to the product of the square roots of

 and

 See  and .
3. If

 and

 are nonnegative, the square root of the quotient

 is equal to the quotient of the square roots of

 and

 See  and .
4. We can add and subtract radical expressions if they have the same radicand and the same index. See  and .
5. Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See  and .
6. The principal nth root of

 is the number with the same sign as

 that when raised to the nth power equals

 These roots have the same properties as square roots. See .
7. Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See  and .
8. The properties of exponents apply to rational exponents. See .

### Section Exercises

### Verbal

### Numeric

For the following exercises, simplify each expression.

### Algebraic

For the following exercises, simplify each expression.

### Real-World Applications

### Extensions

For the following exercises, simplify each expression.


# Prerequisites

## Polynomials

### Learning Objectives

1. Distinguish between a fixed and a growth mindset, and how these ideas may help in learning.

### Objective 1: Distinguish between a fixed and a growth mindset, and how these ideas may help in learning.

Stanford University psychologist and researcher, Carol Dweck, PH.D., published a book in 2006 called "Mindset, The New Psychology of Success", which changed how many people think about their talents and abilities. Based on decades of research Dr. Dweck outlined two mindsets and their influence on our learning.

Dr. Dweck’s research found that people who believe that their abilities could change through learning and practice (growth mindset) more readily accepted learning challenges and persisted through these challenges.
While individuals who believe that knowledge and abilities come from natural talent and cannot be changed (fixed mindset) more often become discouraged by failure and do not persist.
Her research shows that if we believe we can learn and master something new, this belief greatly improves our ability to learn.

1.
Read through the following illustration based on Dr. Dweck’s work.

2.
It’s important to note that we as individuals do not have a strict fixed or growth mindset at all times. We can lean one way or another in certain situations or when working in different disciplines or areas. For example, a person who often plays video games may feel they can learn any new game that is released and be confident in these abilities, but at the same time avoid sports and are fixed on the idea that they will never excel at physical activities.

In terms of learning new skills in mathematics, which mindset, growth or fixed, best describes your beliefs as of today? Explain.

3.
Identify each of the following statements as coming from a student with a fixed mindset or with a growth mindset.

4.
Mindsets can be changed. As Dr. Dweck would say “You have a choice. Mindsets are just beliefs. They are powerful beliefs, but they are something in your mind and you can change your mind.”
Think about what you would like to achieve in your classes this term and how a growth mindset can help you reach these goals. Write three goals for yourself below.

Maahi is building a little free library (a small house-shaped book repository), whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which people can take and donate books. Maahi wants to find the area of the front of the library so that they can purchase the correct amount of paint. Using the measurements of the front of the house, shown in , we can create an expression that combines several variable terms, allowing us to solve this problem and others like it.
First find the area of the square in square feet.
Then find the area of the triangle in square feet.
Next find the area of the rectangular door in square feet.
The area of the front of the library can be found by adding the areas of the square and the triangle, and then subtracting the area of the rectangle. When we do this, we get
  or
  ft2.
In this section, we will examine expressions such as this one, which combine several variable terms.

### Identifying the Degree and Leading Coefficient of Polynomials

The formula just found is an example of a polynomial, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as
  is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product
  such as
  is a term of a polynomial. If a term does not contain a variable, it is called a constant.
A polynomial containing only one term, such as
  is called a monomial. A polynomial containing two terms, such as
  is called a binomial. A polynomial containing three terms, such as
  is called a trinomial.
We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form.

### Adding and Subtracting Polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example,
  and
  are like terms, and can be added to get
  but
  and
  are not like terms, and therefore cannot be added.

### Multiplying Polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

### Multiplying Polynomials Using the Distributive Property

To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the
  in
  to obtain the equivalent expression
  When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

### Using FOIL to Multiply Binomials

A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the first terms, the outer terms, the inner terms, and then the last terms of each binomial.
The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.

### Perfect Square Trinomials

Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let’s look at a few perfect square trinomials to familiarize ourselves with the form.
Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.

### Difference of Squares

Another special product is called the difference of squares, which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let’s see what happens when we multiply
  using the FOIL method.
The middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, let’s look at a few examples.
Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.

### Performing Operations with Polynomials of Several Variables

We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:

### Key Equations

### Key Concepts

1. A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See .
2. We can add and subtract polynomials by combining like terms. See  and .
3. To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. See .
4. FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See .
5. Perfect square trinomials and difference of squares are special products. See  and .
6.  Follow the same rules to work with polynomials containing several variables. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, identify the degree of the polynomial.
For the following exercises, find the sum or difference.
For the following exercises, find the product.
For the following exercises, expand the binomial.
For the following exercises, multiply the binomials.
For the following exercises, multiply the polynomials.

### Real-World Applications

### Extensions

For the following exercises, perform the given operations.