# Polynomials

## Add and Subtract Polynomials

### Identify Polynomials, Monomials, Binomials and Trinomials

You have learned that a term is a constant or the product of a constant and one or more variables. When it is of the form , where  is a constant and  is a whole number, it is called a monomial. Some examples of monomial are .
A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.
Here are some examples of polynomials.

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

### Determine the Degree of Polynomials

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.
A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0—it has no variable.
Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.
A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

### Add and Subtract Monomials

You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.
Remember that like terms must have the same variables with the same exponents.

### Add and Subtract Polynomials

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

### Evaluate a Polynomial for a Given Value

We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.

### Key Concepts

1. Monomials

2. Polynomials

3. Degree of a Polynomial

### Practice Makes Perfect

Identify Polynomials, Monomials, Binomials, and Trinomials
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.
Determine the Degree of Polynomials
In the following exercises, determine the degree of each polynomial.
Add and Subtract Monomials
In the following exercises, add or subtract the monomials.
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
Evaluate a Polynomial for a Given Value
In the following exercises, evaluate each polynomial for the given value.

### Everyday Math

### Writing Exercises

### Self Check

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


# Polynomials

## Use Multiplication Properties of Exponents

### Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example,  means to multiply 2 by itself 4 times, so  means .
Let’s review the vocabulary for expressions with exponents.
Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.
Notice the similarities and differences in ⓐ and ⓑ ! Why are the answers different? As we follow the order of operations in part ⓐ the parentheses tell us to raise the  to the 4th power. In part ⓑ we raise just the 5 to the 4th power and then take the opposite.

### Simplify Expressions Using the Product Property for Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.
We’ll derive the properties of exponents by looking for patterns in several examples.
First, we will look at an example that leads to the Product Property.

We write:
The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.
An example with numbers helps to verify this property.
We can extend the Product Property for Exponents to more than two factors.

### Simplify Expressions Using the Power Property for Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

We write:
We multiplied the exponents. This leads to the Power Property for Exponents.
An example with numbers helps to verify this property.

### Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?
Notice that each factor was raised to the power and  is .
The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.
An example with numbers helps to verify this property:

### Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.
All exponent properties hold true for any real numbers . Right now, we only use whole number exponents.

### Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials.

### Key Concepts

1. Exponential Notation

2. Properties of Exponents

### Practice Makes Perfect

Simplify Expressions with Exponents
In the following exercises, simplify each expression with exponents.
Simplify Expressions Using the Product Property for Exponents
In the following exercises, simplify each expression using the Product Property for Exponents.
Simplify Expressions Using the Power Property for Exponents
In the following exercises, simplify each expression using the Power Property for Exponents.
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression using the Product to a Power Property.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
Multiply Monomials
In the following exercises, multiply the monomials.
Mixed Practice
In the following exercises, simplify each expression.

### Everyday Math

### Writing Exercises

### Self Check

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


# Polynomials

## Multiply Polynomials

### Multiply a Polynomial by a Monomial

We have used the Distributive Property to simplify expressions like . You multiplied both terms in the parentheses, , by 2, to get . With this chapter’s new vocabulary, you can say you were multiplying a binomial, , by a monomial, 2.
Multiplying a binomial by a monomial is nothing new for you! Here’s an example:

### Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.

### Multiply a Binomial by a Binomial Using the Distributive Property

Look at , where we multiplied a binomial by a monomial.

Notice that before combining like terms, you had four terms. You multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.

### Multiply a Binomial by a Binomial Using the FOIL Method

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes, like in , there are no like terms to combine.
Let’s look at the last example again and pay particular attention to how we got the four terms.
Where did the first term, , come from?
We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products.
Let’s look at .

Notice how the terms in third line fit the FOIL pattern.
Now we will do an example where we use the FOIL pattern to multiply two binomials.
We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!
When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.
The final products in the last four examples were trinomials because we could combine the two middle terms. This is not always the case.
Be careful of the exponents in the next example.

### Multiply a Binomial by a Binomial Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it only works for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.
Now we’ll apply this same method to multiply two binomials.
We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The methods are listed here all together, to help you remember them.

### Multiply a Trinomial by a Binomial

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

### Key Concepts

1. FOIL Method for Multiplying Two Binomials—To multiply two binomials:

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

3. Multiplying a Trinomial by a Binomial—To multiply a trinomial by a binomial, use the:

### Practice Makes Perfect

Multiply a Polynomial by a Monomial
In the following exercises, multiply.
Multiply a Binomial by a Binomial
In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.
In the following exercises, multiply the binomials. Use any method.
Multiply a Trinomial by a Binomial
In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.
In the following exercises, multiply. Use either method.
Mixed Practice

### Everyday Math

### Writing Exercises

### Self Check

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


# Polynomials

## Special Products

### Square a Binomial Using the Binomial Squares Pattern

Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and using the methods of the last section, there is less work to do if you learn to use a pattern.
Look at these results. Do you see any patterns?
What about the number of terms? In each example we squared a binomial and the result was a trinomial.
Now look at the  in each result. Where did it come from?
The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!
To get the  of the product, .
Where did the  come from? Look at the examples and find the pattern.
The last term is the product of the last terms, which is the square of the last term.
To get the .
Finally, look at the . Notice it came from adding the “outer” and the “inner” terms—which are both the same! So the middle term is double the product of the two terms of the binomial.
To get the .
Putting it all together:
To square a binomial:
1. square the first term
2. square the last term
3. double their product
A number example helps verify the pattern.
To multiply  usually you’d follow the Order of Operations.
The pattern works!

### Multiply Conjugates Using the Product of Conjugates Pattern

We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.
What do you notice about these pairs of binomials?
Look at the first term of each binomial in each pair.
Notice the first terms are the same in each pair.
Look at the last terms of each binomial in each pair.
Notice the last terms are the same in each pair.
Notice how each pair has one sum and one difference.
A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference has a special name. It is called a conjugate pair and is of the form .
There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier.
Let’s look for the pattern by using FOIL to multiply some conjugate pairs.
Each first term is the product of the first terms of the binomials, and since they are identical it is the square of the first term.
The last term came from multiplying the last terms, the square of the last term.
What do you observe about the products?
The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.
Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.
The product of conjugates is always of the form . This is called a difference of squares.
This leads to the pattern:
Let’s test this pattern with a numerical example.
Notice, the result is the same!
The binomials in the next example may look backwards – the variable is in the second term. But the two binomials are still conjugates, so we use the same pattern to multiply them.
Now we’ll multiply conjugates that have two variables.

### Recognize and Use the Appropriate Special Product Pattern

We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.

### Key Concepts

1. Binomial Squares Pattern

2. Product of Conjugates Pattern

3. To multiply conjugates:

### Practice Makes Perfect

Square a Binomial Using the Binomial Squares Pattern
In the following exercises, square each binomial using the Binomial Squares Pattern.
Multiply Conjugates Using the Product of Conjugates Pattern
In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.
Recognize and Use the Appropriate Special Product Pattern
In the following exercises, find each product.

### Everyday Math

### Writing Exercises

### Self Check

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


# Polynomials

## Divide Monomials

### Simplify Expressions Using the Quotient Property for Exponents

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties below.
Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions—which are also quotients.
As before, we’ll try to discover a property by looking at some examples.
Notice, in each case the bases were the same and we subtracted exponents.
When the larger exponent was in the numerator, we were left with factors in the numerator.
When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1.
We write:
This leads to the Quotient Property for Exponents.
A couple of examples with numbers may help to verify this property.
Notice the difference in the two previous examples:
1. If we start with more factors in the numerator, we will end up with factors in the numerator.
2. If we start with more factors in the denominator, we will end up with factors in the denominator.

The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator.

### Simplify Expressions with an Exponent of Zero

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like . From your earlier work with fractions, you know that:
In words, a number divided by itself is 1. So, , for any , since any number divided by itself is 1.
The Quotient Property for Exponents shows us how to simplify  when  and when  by subtracting exponents. What if ?
Consider , which we know is 1.
Now we will simplify  in two ways to lead us to the definition of the zero exponent. In general, for :
We see  simplifies to  and to 1. So .
In this text, we assume any variable that we raise to the zero power is not zero.
Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.
What about raising an expression to the zero power? Let’s look at . We can use the product to a power rule to rewrite this expression.
This tells us that any nonzero expression raised to the zero power is one.

### Simplify Expressions Using the Quotient to a Power Property

Now we will look at an example that will lead us to the Quotient to a Power Property.
Notice that the exponent applies to both the numerator and the denominator.
This leads to the Quotient to a Power Property for Exponents.
An example with numbers may help you understand this property:

### Simplify Expressions by Applying Several Properties

We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

### Divide Monomials

You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you’ll see how to use these properties to divide monomials. Later, you’ll use them to divide polynomials.
Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.
In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we’ll first find the product of two monomials in the numerator before we simplify the fraction. This follows the order of operations. Remember, a fraction bar is a grouping symbol.

### Key Concepts

1. Quotient Property for Exponents:

2. Zero Exponent

3. Quotient to a Power Property for Exponents:

4. Summary of Exponent Properties

### Practice Makes Perfect

Simplify Expressions Using the Quotient Property for Exponents
In the following exercises, simplify.
Simplify Expressions with Zero Exponents
In the following exercises, simplify.
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
Divide Monomials
In the following exercises, divide the monomials.
Mixed Practice

### Everyday Math

### Writing Exercises

### Self Check

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


# Polynomials

## Divide Polynomials

### Divide a Polynomial by a Monomial

In the last section, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials the next procedure is to divide a polynomial of two or more terms by a monomial.
The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition.
Now we will do this in reverse to split a single fraction into separate fractions.
We’ll state the fraction addition property here just as you learned it and in reverse.
We use the form on the left to add fractions and we use the form on the right to divide a polynomial by a monomial.
We use this form of fraction addition to divide polynomials by monomials.
Remember that division can be represented as a fraction. When you are asked to divide a polynomial by a monomial and it is not already in fraction form, write a fraction with the polynomial in the numerator and the monomial in the denominator.
When we divide by a negative, we must be extra careful with the signs.

### Divide a Polynomial by a Binomial

To divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.

We check division by multiplying the quotient by the divisor.
If we did the division correctly, the product should equal the dividend.
Now we will divide a trinomial by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.
When the divisor has subtraction sign, we must be extra careful when we multiply the partial quotient and then subtract. It may be safer to show that we change the signs and then add.
When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In , we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.
Look back at the dividends in , , and . The terms were written in descending order of degrees, and there were no missing degrees. The dividend in  will be . It is missing an  term. We will add in  as a placeholder.
In , we will divide by . As we divide we will have to consider the constants as well as the variables.

### Key Concepts

1. Fraction Addition

2. Division of a Polynomial by a Monomial

### Practice Makes Perfect

In the following exercises, divide each polynomial by the monomial.
Divide a Polynomial by a Binomial
In the following exercises, divide each polynomial by the binomial.

### Everyday Math

### Writing Exercises

### Self Check

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


# Polynomials

## Integer Exponents and Scientific Notation

### Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents introduced earlier in this chapter, has two forms depending on whether the exponent is larger in the numerator or the denominator.
What if we just subtract exponents regardless of which is larger?
Let’s consider .
We subtract the exponent in the denominator from the exponent in the numerator.
We can also simplify  by dividing out common factors:
This implies that  and it leads us to the definition of a negative exponent.
The negative exponent tells us we can re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent.
Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.
For example, if after simplifying an expression we end up with the expression , we will take one more step and write . The answer is considered to be in simplest form when it has only positive exponents.
In  we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.
This leads to the Property of Negative Exponents.
Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.
To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.
This leads us to the Quotient to a Negative Power Property.
When simplifying an expression with exponents, we must be careful to correctly identify the base.
We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different.
When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. We will assume all variables are non-zero.
When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the Order of Operations, we simplify expressions in parentheses before applying exponents. We’ll see how this works in the next example.
With negative exponents, the Quotient Rule needs only one form , for . When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.

### Simplify Expressions with Integer Exponents

All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.
In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property.
In the next two examples, we’ll use the Power Property and the Product to a Power Property.
To simplify a fraction, we use the Quotient Property and subtract the exponents.

### Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and . We know that 4,000 means  and 0.004 means .
If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:
When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.
It is customary in scientific notation to use as the  multiplication sign, even though we avoid using this sign elsewhere in algebra.
If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.
In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.

### Convert Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.
If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.
In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.
The steps are summarized below.

### Multiply and Divide Using Scientific Notation

Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

### Key Concepts

1. Property of Negative Exponents

2. Quotient to a Negative Exponent

3. To convert a decimal to scientific notation:

4. To convert scientific notation to decimal form:

### Section Exercises

### Practice Makes Perfect

Use the Definition of a Negative Exponent
In the following exercises, simplify.
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
Multiply and Divide Using Scientific Notation
In the following exercises, multiply. Write your answer in decimal form.
In the following exercises, divide. Write your answer in decimal form.

### Everyday Math

### Writing Exercises

### Self Check

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

### Chapter 6 Review Exercises

### Add and Subtract Polynomials

Identify Polynomials, Monomials, Binomials and Trinomials
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.
Determine the Degree of Polynomials
In the following exercises, determine the degree of each polynomial.
Add and Subtract Monomials
In the following exercises, add or subtract the monomials.
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
Evaluate a Polynomial for a Given Value of the Variable
In the following exercises, evaluate each polynomial for the given value.

### Use Multiplication Properties of Exponents

Simplify Expressions with Exponents
In the following exercises, simplify.
Simplify Expressions Using the Product Property for Exponents
In the following exercises, simplify each expression.
Simplify Expressions Using the Power Property for Exponents
In the following exercises, simplify each expression.
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
Multiply Monomials
In the following exercises 8, multiply the monomials.

### Multiply Polynomials

Multiply a Polynomial by a Monomial
In the following exercises, multiply.
Multiply a Binomial by a Binomial
In the following exercises, multiply the binomials using: ⓐ the Distributive Property, ⓑ the FOIL method, ⓒ the Vertical Method.
In the following exercises, multiply the binomials. Use any method.
Multiply a Trinomial by a Binomial
In the following exercises, multiply using ⓐ the Distributive Property, ⓑ the Vertical Method.
In the following exercises, multiply. Use either method.

### Special Products

Square a Binomial Using the Binomial Squares Pattern
In the following exercises, square each binomial using the Binomial Squares Pattern.
Multiply Conjugates Using the Product of Conjugates Pattern
In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.
Recognize and Use the Appropriate Special Product Pattern
In the following exercises, find each product.

### Divide Monomials

Simplify Expressions Using the Quotient Property for Exponents
In the following exercises, simplify.
Simplify Expressions with Zero Exponents
In the following exercises, simplify.
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
Divide Monomials
In the following exercises, divide the monomials.

### Divide Polynomials

Divide a Polynomial by a Monomial
In the following exercises, divide each polynomial by the monomial.
Divide a Polynomial by a Binomial
In the following exercises, divide each polynomial by the binomial.

### Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent
In the following exercises, simplify.
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
Multiply and Divide Using Scientific Notation
In the following exercises, multiply and write your answer in decimal form.
In the following exercises, divide and write your answer in decimal form.

### Chapter Practice Test

In the following exercises, simplify each expression.
In the following exercises, simplify, and write your answer in decimal form.


# Factoring

## Introduction

Quadratic expressions may be used to model physical properties of a large bridge, the trajectory of a baseball or rocket, and revenue and profit of a business. By factoring these expressions, specific characteristics of the model can be identified. In this chapter, you will explore the process of factoring expressions and see how factoring is used to solve certain types of equations.


# Factoring

## Greatest Common Factor and Factor by Grouping

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

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.
We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.
First we’ll find the GCF of two numbers.
We summarize the steps we use to find the GCF below.
In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

### Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as  in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:
Now we will start with a product, like , and end with its factors, . To do this we apply the Distributive Property “in reverse.”
We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”
So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!
The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.
Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.
When the leading coefficient is negative, we factor the negative out as part of the GCF.

### Factor by Grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.
(Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.)

### Key Concepts

1. Finding the Greatest Common Factor (GCF): To find the GCF of two expressions:

2. Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial:

3. Factor by Grouping: To factor a polynomial with 4 four or more terms

### Practice Makes Perfect

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

### Everyday Math

### Writing Exercises

### Self Check

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


# Factoring

## Factor Trinomials of the Form x2+bx+c

### Factor Trinomials of the Form x2 + bx + c

You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication—to start with the product and end up with the factors. Let’s look at an example of multiplying binomials to refresh your memory.
To factor the trinomial means to start with the product, , and end with the factors, . You need to think about where each of the terms in the trinomial came from.
The first term came from multiplying the first term in each binomial. So to get  in the product, each binomial must start with an x.
The last term in the trinomial came from multiplying the last term in each binomial. So the last terms must multiply to 6.
What two numbers multiply to 6?
The factors of 6 could be 1 and 6, or 2 and 3. How do you know which pair to use?
Consider the middle term. It came from adding the outer and inner terms.
So the numbers that must have a product of 6 will need a sum of 5. We’ll test both possibilities and summarize the results in —the table will be very helpful when you work with numbers that can be factored in many different ways.
We see that 2 and 3 are the numbers that multiply to 6 and add to 5. So we have the factors of . They are .
You should check this by multiplying.
Looking back, we started with , which is of the form , where  and . We factored it into two binomials of the form .
To get the correct factors, we found two numbers m and n whose product is c and sum is b.
Let’s summarize the steps we used to find the factors.

### Factor Trinomials of the Form x2 + bx + c with b Negative, c Positive

In the examples so far, all terms in the trinomial were positive. What happens when there are negative terms? Well, it depends which term is negative. Let’s look first at trinomials with only the middle term negative.
Remember: To get a negative sum and a positive product, the numbers must both be negative.
Again, think about FOIL and where each term in the trinomial came from. Just as before,
1. the first term, , comes from the product of the two first terms in each binomial factor, x and y;
2. the positive last term is the product of the two last terms
3. the negative middle term is the sum of the outer and inner terms.
How do you get a positive product and a negative sum? With two negative numbers.

### Factor Trinomials of the Form  with c Negative

Now, what if the last term in the trinomial is negative? Think about FOIL. The last term is the product of the last terms in the two binomials. A negative product results from multiplying two numbers with opposite signs. You have to be very careful to choose factors to make sure you get the correct sign for the middle term, too.
Remember: To get a negative product, the numbers must have different signs.
Let’s make a minor change to the last trinomial and see what effect it has on the factors.
Some trinomials are prime. The only way to be certain a trinomial is prime is to list all the possibilities and show that none of them work.
Let’s summarize the method we just developed to factor trinomials of the form .

### Factor Trinomials of the Form x2 + bxy + cy2

Sometimes you’ll need to factor trinomials of the form  with two variables, such as  The first term, , is the product of the first terms of the binomial factors, . The  in the last term means that the second terms of the binomial factors must each contain y. To get the coefficients b and c, you use the same process summarized in the previous objective.

### Key Concepts

1. Factor trinomials of the form

### Practice Makes Perfect

Factor Trinomials of the Form
In the following exercises, factor each trinomial of the form .
Factor Trinomials of the Form
In the following exercises, factor each trinomial of the form .
Mixed Practice
In the following exercises, factor each expression.

### Everyday Math

### Writing Exercises

### Self Check

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


# Factoring

## Factor Trinomials of the Form ax2+bx+c

### Recognize a Preliminary Strategy for Factoring

Let’s summarize where we are so far with factoring polynomials. In the first two sections of this chapter, we used three methods of factoring: factoring the GCF, factoring by grouping, and factoring a trinomial by “undoing” FOIL. More methods will follow as you continue in this chapter, as well as later in your studies of algebra.
How will you know when to use each factoring method? As you learn more methods of factoring, how will you know when to apply each method and not get them confused? It will help to organize the factoring methods into a strategy that can guide you to use the correct method.
As you start to factor a polynomial, always ask first, “Is there a greatest common factor?” If there is, factor it first.
The next thing to consider is the type of polynomial. How many terms does it have? Is it a binomial? A trinomial? Or does it have more than three terms?
If it is a trinomial where the leading coefficient is one, , use the “undo FOIL” method.
If it has more than three terms, try the grouping method. This is the only method to use for polynomials of more than three terms.
Some polynomials cannot be factored. They are called “prime.”
Below we summarize the methods we have so far. These are detailed in Choose a strategy to factor polynomials completely.
Use the preliminary strategy to completely factor a polynomial. A polynomial is factored completely if, other than monomials, all of its factors are prime.

### Factor Trinomials of the form ax2 + bx + c with a GCF

Now that we have organized what we’ve covered so far, we are ready to factor trinomials whose leading coefficient is not 1, trinomials of the form .
Remember to always check for a GCF first! Sometimes, after you factor the GCF, the leading coefficient of the trinomial becomes 1 and you can factor it by the methods in the last section. Let’s do a few examples to see how this works.
Watch out for the signs in the next two examples.
In the next example the GCF will include a variable.

### Factor Trinomials using Trial and Error

What happens when the leading coefficient is not 1 and there is no GCF? There are several methods that can be used to factor these trinomials. First we will use the Trial and Error method.
Let’s factor the trinomial .
From our earlier work we expect this will factor into two binomials.
We know the first terms of the binomial factors will multiply to give us . The only factors of  are . We can place them in the binomials.
Check. Does ?
We know the last terms of the binomials will multiply to 2. Since this trinomial has all positive terms, we only need to consider positive factors. The only factors of 2 are 1 and 2. But we now have two cases to consider as it will make a difference if we write 1, 2, or 2, 1.
Which factors are correct? To decide that, we multiply the inner and outer terms.
Since the middle term of the trinomial is 5x, the factors in the first case will work. Let’s FOIL to check.
Our result of the factoring is:
When the middle term is negative and the last term is positive, the signs in the binomials must both be negative.
When we factor an expression, we always look for a greatest common factor first. If the expression does not have a greatest common factor, there cannot be one in its factors either. This may help us eliminate some of the possible factor combinations.
Don’t forget to look for a GCF first.

### Factor Trinomials using the “ac” Method

Another way to factor trinomials of the form  is the “ac” method. (The “ac” method is sometimes called the grouping method.) The “ac” method is actually an extension of the methods you used in the last section to factor trinomials with leading coefficient one. This method is very structured (that is step-by-step), and it always works!
When the third term of the trinomial is negative, the factors of the third term will have opposite signs.
Don’t forget to look for a common factor!
We can now update the Preliminary Factoring Strategy, as shown in  and detailed in Choose a strategy to factor polynomials completely (updated), to include trinomials of the form . Remember, some polynomials are prime and so they cannot be factored.

### Key Concepts

1. Factor Trinomials of the Form  See .

2. Factor Trinomials of the Form  See .

3. Choose a strategy to factor polynomials completely (updated):

### Practice Makes Perfect

Recognize a Preliminary Strategy to Factor Polynomials Completely
In the following exercises, identify the best method to use to factor each polynomial.
Factor Trinomials of the form
In the following exercises, factor completely.
Factor Trinomials Using Trial and Error
In the following exercises, factor.
Factor Trinomials using the ‘ac’ Method
In the following exercises, factor.
Mixed Practice
In the following exercises, factor.

### Everyday Math

### Writing Exercises

### Self Check

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


# Factoring

## Factor Special Products

The strategy for factoring we developed in the last section will guide you as you factor most binomials, trinomials, and polynomials with more than three terms. We have seen that some binomials and trinomials result from special products—squaring binomials and multiplying conjugates. If you learn to recognize these kinds of polynomials, you can use the special products patterns to factor them much more quickly.

### Factor Perfect Square Trinomials

Some trinomials are perfect squares. They result from multiplying a binomial times itself. You can square a binomial by using FOIL, but using the Binomial Squares pattern you saw in a previous chapter saves you a step. Let’s review the Binomial Squares pattern by squaring a binomial using FOIL.
The first term is the square of the first term of the binomial and the last term is the square of the last. The middle term is twice the product of the two terms of the binomial.
The trinomial 9x2 + 24 +16 is called a perfect square trinomial. It is the square of the binomial 3x+4.
We’ll repeat the Binomial Squares Pattern here to use as a reference in factoring.
When you square a binomial, the product is a perfect square trinomial. In this chapter, you are learning to factor—now, you will start with a perfect square trinomial and factor it into its prime factors.
You could factor this trinomial using the methods described in the last section, since it is of the form ax2 + bx + c. But if you recognize that the first and last terms are squares and the trinomial fits the perfect square trinomials pattern, you will save yourself a lot of work.
Here is the pattern—the reverse of the binomial squares pattern.
To make use of this pattern, you have to recognize that a given trinomial fits it. Check first to see if the leading coefficient is a perfect square, . Next check that the last term is a perfect square, . Then check the middle term—is it twice the product, 2ab? If everything checks, you can easily write the factors.
The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern , which factors to .
The steps are summarized here.
We’ll work one now where the middle term is negative.
The next example will be a perfect square trinomial with two variables.
Remember the very first step in our Strategy for Factoring Polynomials? It was to ask “is there a greatest common factor?” and, if there was, you factor the GCF before going any further. Perfect square trinomials may have a GCF in all three terms and it should be factored out first. And, sometimes, once the GCF has been factored, you will recognize a perfect square trinomial.

### Factor Differences of Squares

The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here’s an example:
Remember, when you multiply conjugate binomials, the middle terms of the product add to 0. All you have left is a binomial, the difference of squares.
Multiplying conjugates is the only way to get a binomial from the product of two binomials.
To factor, we will use the product pattern “in reverse” to factor the difference of squares. A difference of squares factors to a product of conjugates.
Remember, “difference” refers to subtraction. So, to use this pattern you must make sure you have a binomial in which two squares are being subtracted.
It is important to remember that sums of squares do not factor into a product of binomials. There are no binomial factors that multiply together to get a sum of squares. After removing any GCF, the expression  is prime!

Don’t forget that 1 is a perfect square. We’ll need to use that fact in the next example.
The binomial in the next example may look “backwards,” but it’s still the difference of squares.
To completely factor the binomial in the next example, we’ll factor a difference of squares twice!
As always, you should look for a common factor first whenever you have an expression to factor. Sometimes a common factor may “disguise” the difference of squares and you won’t recognize the perfect squares until you factor the GCF.

### Factor Sums and Differences of Cubes

There is another special pattern for factoring, one that we did not use when we multiplied polynomials. This is the pattern for the sum and difference of cubes. We will write these formulas first and then check them by multiplication.
We’ll check the first pattern and leave the second to you.

The two patterns look very similar, don’t they? But notice the signs in the factors. The sign of the binomial factor matches the sign in the original binomial. And the sign of the middle term of the trinomial factor is the opposite of the sign in the original binomial. If you recognize the pattern of the signs, it may help you memorize the patterns.
The trinomial factor in the sum and difference of cubes pattern cannot be factored.
It can be very helpful if you learn to recognize the cubes of the integers from 1 to 10, just like you have learned to recognize squares. We have listed the cubes of the integers from 1 to 10 in .
Be careful to use the correct signs in the factors of the sum and difference of cubes.
In the next example, we first factor out the GCF. Then we can recognize the sum of cubes.

### Key Concepts

1. Factor perfect square trinomials See .

2. Factor differences of squares See .

3. Factor sum and difference of cubes To factor the sum or difference of cubes: See .

### Practice Makes Perfect

Factor Perfect Square Trinomials
In the following exercises, factor.
Factor Differences of Squares
In the following exercises, factor.
Factor Sums and Differences of Cubes
In the following exercises, factor.
Mixed Practice
In the following exercises, factor.

### Everyday Math

### Writing Exercises

### Self Check

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


# Factoring

## General Strategy for Factoring Polynomials

### Recognize and Use the Appropriate Method to Factor a Polynomial Completely

You have now become acquainted with all the methods of factoring that you will need in this course. (In your next algebra course, more methods will be added to your repertoire.) The figure below summarizes all the factoring methods we have covered.  outlines a strategy you should use when factoring polynomials.
Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

### Key Concepts

1. General Strategy for Factoring Polynomials See
2. How to Factor Polynomials

### Practice Makes Perfect

Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.

### Everyday Math

### Writing Exercises

### Self Check

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


# Factoring

## Quadratic Equations

We have already solved linear equations, equations of the form . In linear equations, the variables have no exponents. Quadratic equations are equations in which the variable is squared. Listed below are some examples of quadratic equations:
The last equation doesn’t appear to have the variable squared, but when we simplify the expression on the left we will get .
The general form of a quadratic equation is .
To solve quadratic equations we need methods different than the ones we used in solving linear equations. We will look at one method here and then several others in a later chapter.

### Solve Quadratic Equations Using the Zero Product Property

We will first solve some quadratic equations by using the Zero Product Property. The Zero Product Property says that if the product of two quantities is zero, it must be that at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.
We will now use the Zero Product Property, to solve a quadratic equation.
We usually will do a little more work than we did in this last example to solve the linear equations that result from using the Zero Product Property.
Notice when we checked the solutions that each of them made just one factor equal to zero. But the product was zero for both solutions.
It may appear that there is only one factor in the next example. Remember, however, that  means .

### Solve Quadratic Equations by Factoring

Each of the equations we have solved in this section so far had one side in factored form. In order to use the Zero Product Property, the quadratic equation must be factored, with zero on one side. So we must be sure to start with the quadratic equation in standard form, . Then we can factor the expression on the left.
Before we factor, we must make sure the quadratic equation is in standard form.
Solving quadratic equations by factoring will make use of all the factoring techniques you have learned in this chapter! Do you recognize the special product pattern in the next example?
The left side in the next example is factored, but the right side is not zero. In order to use the Zero Product Property, one side of the equation must be zero. We’ll multiply the factors and then write the equation in standard form.
The Zero Product Property also applies to the product of three or more factors. If the product is zero, at least one of the factors must be zero. We can solve some equations of degree more than two by using the Zero Product Property, just like we solved quadratic equations.
When we factor the quadratic equation in the next example we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.

### Solve Applications Modeled by Quadratic Equations

The problem solving strategy we used earlier for applications that translate to linear equations will work just as well for applications that translate to quadratic equations. We will copy the problem solving strategy here so we can use it for reference.
We will start with a number problem to get practice translating words into a quadratic equation.
Were you surprised by the pair of negative integers that is one of the solutions to the previous example? The product of the two positive integers and the product of the two negative integers both give 132.
In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.
In an earlier chapter, we used the Pythagorean Theorem . It gave the relation between the legs and the hypotenuse of a right triangle.
We will use this formula to in the next example.

### Key Concepts

1. Zero Product Property If , then either  or  or both. See .
2. Solve a quadratic equation by factoring To solve a quadratic equation by factoring: See .

3. Use a problem solving strategy to solve word problems See .

### Section Exercises

### Practice Makes Perfect

Use the Zero Product Property
In the following exercises, solve.
Solve Quadratic Equations by Factoring
In the following exercises, solve.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.
Mixed Practice
In the following exercises, solve.

### Everyday Math

### Writing Exercises

### Self Check

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

### Chapter 7 Review Exercises

### 7.1 Greatest Common Factor and Factor by Grouping

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

### 7.2 Factor Trinomials of the form

Factor Trinomials of the Form
In the following exercises, factor each trinomial of the form .
Factor Trinomials of the Form
In the following examples, factor each trinomial of the form .

### 7.3 Factoring Trinomials of the form

Recognize a Preliminary Strategy to Factor Polynomials Completely
In the following exercises, identify the best method to use to factor each polynomial.
Factor Trinomials of the Form
In the following exercises, factor completely.
Factor Trinomials Using the “ac” Method
In the following exercises, factor.
Factor Trinomials with a GCF Using the “ac” Method
In the following exercises, factor.

### 7.4 Factoring Special Products

Factor Perfect Square Trinomials
In the following exercises, factor.
Factor Differences of Squares
In the following exercises, factor.
Factor Sums and Differences of Cubes
In the following exercises, factor.

### 7.5 General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.

### 7.6 Quadratic Equations

Use the Zero Product Property
In the following exercises, solve.
Solve Quadratic Equations by Factoring
In the following exercises, solve.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.

### Practice Test

In the following exercises, find the Greatest Common Factor in each expression.
In the following exercises, factor completely.
In the following exercises, solve.


# Rational Expressions and Equations

## Introduction

Like rowing a boat, riding a bicycle is a situation in which going in one direction, downhill, is easy, but going in the opposite direction, uphill, can be more work. The trip to reach a destination may be quick, but the return trip whether upstream or uphill will take longer.
Rational equations are used to model situations like these. In this chapter, we will work with rational expressions, solve rational equations, and use them to solve problems in a variety of applications.


# Rational Expressions and Equations

## Simplify Rational Expressions

In Chapter 1, we reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers, and the denominator is not zero.
In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call these rational expressions.
Remember, division by 0 is undefined.
Here are some examples of rational expressions:
Notice that the first rational expression listed above,  is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.
We will perform the same operations with rational expressions that we do with fractions. We will simplify, add, subtract, multiply, divide, and use them in applications.

### Determine the Values for Which a Rational Expression is Undefined

When we work with a numerical fraction, it is easy to avoid dividing by zero, because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.
If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.
So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.

### Evaluate Rational Expressions

To evaluate a rational expression, we substitute values of the variables into the expression and simplify, just as we have for many other expressions in this book.
Remember that a fraction is simplified when it has no common factors, other than 1, in its numerator and denominator. When we evaluate a rational expression, we make sure to simplify the resulting fraction.

### Simplify Rational Expressions

Just like a fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator, a rational expression is simplified if it has no common factors, other than 1, in its numerator and denominator.
For example:
1.  is simplified because there are no common factors of 2 and 3.
2.  is not simplified because x is a common factor of 2x and 3x.
We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.
Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see  clearly stated. Every time we write a rational expression, we should make a similar statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.
Let’s start by reviewing how we simplify numerical fractions.
Throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example,  and .
To simplify rational expressions we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.
Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.
Note that removing the x’s from  would be like cancelling the 2’s in the fraction !
We now summarize the steps you should follow to simplify rational expressions.
Usually, we leave the simplified rational expression in factored form. This way it is easy to check that we have removed all the common factors!
We’ll use the methods we covered in Factoring to factor the polynomials in the numerators and denominators in the following examples.

### Simplify Rational Expressions with Opposite Factors

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. Let’s start with a numerical fraction, say . We know this fraction simplifies to . We also recognize that the numerator and denominator are opposites.
In Foundations, we introduced opposite notation: the opposite of  is . We remember, too, that .
We simplify the fraction , whose numerator and denominator are opposites, in this way:

So, in the same way, we can simplify the fraction :

But the opposite of  could be written differently:

This means the fraction  simplifies to .
In general, we could write the opposite of  as . So the rational expression  simplifies to .
We will use this property to simplify rational expressions that contain opposites in their numerators and denominators.
Remember, the first step in simplifying a rational expression is to factor the numerator and denominator completely.

### Key Concepts

1. Determine the Values for Which a Rational Expression is Undefined

2. Simplified Rational Expression

3. Simplify a Rational Expression

4. Opposites in a Rational Expression

### Practice Makes Perfect

In the following exercises, determine the values for which the rational expression is undefined.
Evaluate Rational Expressions
In the following exercises, evaluate the rational expression for the given values.
Simplify Rational Expressions
In the following exercises, simplify.
Simplify Rational Expressions with Opposite Factors
In the following exercises, simplify each rational expression.

### Everyday Math

### Writing Exercises

### Self Check

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


# Rational Expressions and Equations

## Multiply and Divide Rational Expressions

### Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.
We’ll do the first example with numerical fractions to remind us of how we multiplied fractions without variables.
Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example,  and .

### Divide Rational Expressions

To divide rational expressions we multiply the first fraction by the reciprocal of the second, just like we did for numerical fractions.
Remember, the reciprocal of  is . To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.
Remember, first rewrite the division as multiplication of the first expression by the reciprocal of the second. Then factor everything and look for common factors.
Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide , we first write 4 as a fraction so that we can find its reciprocal.
We do the same thing when we divide rational expressions.
Remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.
If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then we factor and multiply.

### Key Concepts

1. Multiplication of Rational Expressions

2. Multiply a Rational Expression

3. Division of Rational Expressions

4. Divide Rational Expressions

### Practice Makes Perfect

Multiply Rational Expressions
In the following exercises, multiply.
Divide Rational Expressions
In the following exercises, divide.

### Everyday Math

### Writing Exercises

### Self Check

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


# Rational Expressions and Equations

## Add and Subtract Rational Expressions with a Common Denominator

### Add Rational Expressions with a Common Denominator

What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.
It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.
We will add two numerical fractions first, to remind us of how this is done.
Remember, we do not allow values that would make the denominator zero. What value of  should be excluded in the next example?

### Subtract Rational Expressions with a Common Denominator

To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator.
We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.
Be careful of the signs when you subtract a binomial!

### Add and Subtract Rational Expressions whose Denominators are Opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by .
Let’s see how this works.

### Key Concepts

1. Rational Expression Addition

2. Rational Expression Subtraction

### Practice Makes Perfect

Add Rational Expressions with a Common Denominator
In the following exercises, add.
Subtract Rational Expressions with a Common Denominator
In the following exercises, subtract.
Add and Subtract Rational Expressions whose Denominators are Opposites
In the following exercises, add.
In the following exercises, subtract.

### Everyday Math

### Writing Exercises

### Self Check

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


# Rational Expressions and Equations

## Add and Subtract Rational Expressions with Unlike Denominators

### Find the Least Common Denominator of Rational Expressions

When we add or subtract rational expressions with unlike denominators we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.
Let’s look at the example  from Foundations. Since the denominators are not the same, the first step was to find the least common denominator (LCD). Remember, the LCD is the least common multiple of the denominators. It is the smallest number we can use as a common denominator.
To find the LCD of 12 and 18, we factored each number into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD.
We do the same thing for rational expressions. However, we leave the LCD in factored form.

### Find Equivalent Rational Expressions

When we add numerical fractions, once we find the LCD, we rewrite each fraction as an equivalent fraction with the LCD.
We will do the same thing for rational expressions.

### Add Rational Expressions with Different Denominators

Now we have all the steps we need to add rational expressions with different denominators. As we have done previously, we will do one example of adding numerical fractions first.
Now we will add rational expressions whose denominators are monomials.
Now we are ready to tackle polynomial denominators.
The steps to use to add rational expressions are summarized in the following procedure box.
Avoid the temptation to simplify too soon! In the example above, we must leave the first rational expression as  to be able to add it to . Simplify only after you have combined the numerators.

### Subtract Rational Expressions with Different Denominators

The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.
The steps to take to subtract rational expressions are listed below.
There are lots of negative signs in the next example. Be extra careful!
When one expression is not in fraction form, we can write it as a fraction with denominator 1.
We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example we will start by factoring all three denominators to find their LCD.

### Key Concepts

1. Find the Least Common Denominator of Rational Expressions

2. Add or Subtract Rational Expressions

### Practice Makes Perfect

In the following exercises, find the LCD.
In the following exercises, write as equivalent rational expressions with the given LCD.
In the following exercises, add.
In the following exercises, subtract.
In the following exercises, add and subtract.
In the following exercises, simplify.

### Everyday Math

### Writing Exercises

### Self Check

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


# Rational Expressions and Equations

## Simplify Complex Rational Expressions

Complex fractions are fractions in which the numerator or denominator contains a fraction. In Chapter 1 we simplified complex fractions like these:
In this section we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.
Here are a few complex rational expressions:
Remember, we always exclude values that would make any denominator zero.
We will use two methods to simplify complex rational expressions.

### Simplify a Complex Rational Expression by Writing it as Division

We have already seen this complex rational expression earlier in this chapter.
We noted that fraction bars tell us to divide, so rewrote it as the division problem
Then we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.
This is one method to simplify rational expressions. We write it as if we were dividing two fractions.
Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.

### Simplify a Complex Rational Expression by Using the LCD

We “cleared” the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by LCD of all the rational expressions.
Let’s look at the complex rational expression we simplified one way in . We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by  we are multiplying by 1, so the value stays the same.
Be sure to start by factoring all the denominators so you can find the LCD.

### Key Concepts

1. To Simplify a Rational Expression by Writing it as Division

2. To Simplify a Complex Rational Expression by Using the LCD

### Practice Makes Perfect

Simplify a Complex Rational Expression by Writing It as Division
In the following exercises, simplify.
Simplify a Complex Rational Expression by Using the LCD
In the following exercises, simplify.
Simplify
In the following exercises, use either method.

### Everyday Math

### Writing Exercises

### Self Check

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


# Rational Expressions and Equations

## Solve Rational Equations

After defining the terms expression and equation early in Foundations, we have used them throughout this book. We have simplified many kinds of expressions and solved many kinds of equations. We have simplified many rational expressions so far in this chapter. Now we will solve rational equations.
The definition of a rational equation is similar to the definition of equation we used in Foundations.
You must make sure to know the difference between rational expressions and rational equations. The equation contains an equal sign.

### Solve Rational Equations

We have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to “clear” the fractions.
Here is an example we did when we worked with linear equations:

We will use the same strategy to solve rational equations. We will multiply both sides of the equation by the LCD. Then we will have an equation that does not contain rational expressions and thus is much easier for us to solve.
But because the original equation may have a variable in a denominator we must be careful that we don’t end up with a solution that would make a denominator equal to zero.
So before we begin solving a rational equation, we examine it first to find the values that would make any denominators zero. That way, when we solve a rational equation we will know if there are any algebraic solutions we must discard.
An algebraic solution to a rational equation that would cause any of the rational expressions to be undefined is called an extraneous solution.
We note any possible extraneous solutions, c, by writing  next to the equation.
The steps of this method are shown below.
We always start by noting the values that would cause any denominators to be zero.
When one of the denominators is a quadratic, remember to factor it first to find the LCD.
The equation we solved in  had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. Some equations have no solution.

### Solve a Rational Equation for a Specific Variable

When we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable.
We’ll start with a formula relating distance, rate, and time. We have used it many times before, but not usually in this form.
 uses the formula for slope that we used to get the point-slope form of an equation of a line.
Be sure to follow all the steps in . It may look like a very simple formula, but we cannot solve it instantly for either denominator.

### Key Concepts

1. Strategy to Solve Equations with Rational Expressions

### Practice Makes Perfect

Solve Rational Equations
In the following exercises, solve.
Solve a Rational Equation for a Specific Variable
In the following exercises, solve.

### Everyday Math

### Writing Exercises

### Self Check

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


# Rational Expressions and Equations

## Solve Proportion and Similar Figure Applications

### Solve Proportions

When two rational expressions are equal, the equation relating them is called a proportion.
The equation  is a proportion because the two fractions are equal. The proportion  is read “1 is to 2 as 4 is to 8.”
Proportions are used in many applications to ‘scale up’ quantities. We’ll start with a very simple example so you can see how proportions work. Even if you can figure out the answer to the example right away, make sure you also learn to solve it using proportions.
Suppose a school principal wants to have 1 teacher for 20 students. She could use proportions to find the number of teachers for 60 students. We let x be the number of teachers for 60 students and then set up the proportion:
We are careful to match the units of the numerators and the units of the denominators—teachers in the numerators, students in the denominators.
Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved equations in Solve Rational Equations. We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.
So let’s finish solving the principal’s problem now. We will omit writing the units until the last step.
Now we’ll do a few examples of solving numerical proportions without any units. Then we will solve applications using proportions.
When we work with proportions, we exclude values that would make either denominator zero, just like we do for all rational expressions. What value(s) should be excluded for the proportion in the next example?
To solve applications with proportions, we will follow our usual strategy for solving applications. But when we set up the proportion, we must make sure to have the units correct—the units in the numerators must match and the units in the denominators must match.
In the example above, we related the number of pesos to the number of dollars by using a proportion. We could say the number of pesos is proportional to the number of dollars. If two quantities are related by a proportion, we say that they are proportional.

### Solve Similar Figure Applications

When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with similar figures. If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides are in the same ratio.
For example, the two triangles in  are similar. Each side of  is 4 times the length of the corresponding side of .
This is summed up in the Property of Similar Triangles.
To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.
The next example shows how similar triangles are used with maps.
We can use similar figures to find heights that we cannot directly measure.

### Key Concepts

1. Property of Similar Triangles

2. Problem Solving Strategy for Geometry Applications

### Practice Makes Perfect

Solve Proportions
In the following exercises, solve.
Solve Similar Figure Applications
In the following exercises,  is similar to . Find the length of the indicated side.
In the following exercises,  is similar to .
In the following two exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle whose sides are shown in the figure below. The actual distance from New York to Chicago is 800 miles.
In the following two exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle whose sides are shown in the figure below. The actual distance from Atlanta to New Orleans is 420 miles.

### Everyday Math

### Writing Exercises

### Self Check

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


# Rational Expressions and Equations

## Solve Uniform Motion and Work Applications

### Solve Uniform Motion Applications

We have solved uniform motion problems using the formula  in previous chapters. We used a table like the one below to organize the information and lead us to the equation.
The formula  assumes we know r and t and use them to find D. If we know D and r and need to find t, we would solve the equation for t and get the formula .
We have also explained how flying with or against a current affects the speed of a vehicle. We will revisit that idea in the next example.
In the next example, we will know the total time resulting from travelling different distances at different speeds.
Once again, we will use the uniform motion formula solved for the variable t.

### Solve Work Applications

Suppose Pete can paint a room in 10 hours. If he works at a steady pace, in 1 hour he would paint  of the room. If Alicia would take 8 hours to paint the same room, then in 1 hour she would paint  of the room. How long would it take Pete and Alicia to paint the room if they worked together (and didn’t interfere with each other’s progress)?
This is a typical ‘work’ application. There are three quantities involved here – the time it would take each of the two people to do the job alone and the time it would take for them to do the job together.
Let’s get back to Pete and Alicia painting the room. We will let t be the number of hours it would take them to paint the room together. So in 1 hour working together they have completed  of the job.
In one hour Pete did  of the job. Alicia did  of the job. And together they did  of the job.
We can model this with the word equation and then translate to a rational equation. To find the time it would take them if they worked together, we solve for t.

Keep in mind, it should take less time for two people to complete a job working together than for either person to do it alone.

### Practice Makes Perfect

Solve Uniform Motion Applications
In the following exercises, solve uniform motion applications
Solve Work Applications
In the following exercises, solve work applications.

### Everyday Math

### Writing Exercises

### Self Check

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


# Rational Expressions and Equations

## Use Direct and Inverse Variation

When two quantities are related by a proportion, we say they are proportional to each other. Another way to express this relation is to talk about the variation of the two quantities. We will discuss direct variation and inverse variation in this section.

### Solve Direct Variation Problems

Lindsay gets paid $15 per hour at her job. If we let s be her salary and h be the number of hours she has worked, we could model this situation with the equation
Lindsay’s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay’s salary varies directly with the number of hours she works. Two variables vary directly if one is the product of a constant and the other.
In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates x and y. Then we can use that equation to find values of y for other values of x.
We’ll list the steps below.
Now we’ll solve a few applications of direct variation.
In the previous example, the variables c and m were named in the problem. Usually that is not the case. We will have to name the variables in the next example as part of the solution, just like we do in most applied problems.
In some situations, one variable varies directly with the square of the other variable. When that happens, the equation of direct variation is . We solve these applications just as we did the previous ones, by substituting the given values into the equation to solve for k.

### Solve Inverse Variation Problems

Many applications involve two variable that vary inversely. As one variable increases, the other decreases. The equation that relates them is .
The word ‘inverse’ in inverse variation refers to the multiplicative inverse. The multiplicative inverse of x is .
We solve inverse variation problems in the same way we solved direct variation problems. Only the general form of the equation has changed. We will copy the procedure box here and just change ‘direct’ to ‘inverse’.

### Section Exercises

### Practice Makes Perfect

Solve Direct Variation Problems
In the following exercises, solve.
Solve Inverse Variation Problems
In the following exercises, solve.
Write an inverse variation equation to solve the following problems.

### Mixed Practice

### Everyday Math

### Writing Exercises

### Self Check

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

### Chapter 8 Review Exercises

### Simplify Rational Expressions

Determine the Values for Which a Rational Expression is Undefined
In the following exercises, determine the values for which the rational expression is undefined.
Evaluate Rational Expressions
In the following exercises, evaluate the rational expressions for the given values.
Simplify Rational Expressions
In the following exercises, simplify.
Simplify Rational Expressions with Opposite Factors
In the following exercises, simplify.

### Multiply and Divide Rational Expressions

Multiply Rational Expressions
In the following exercises, multiply.
Divide Rational Expressions
In the following exercises, divide.

### Add and Subtract Rational Expressions with a Common Denominator

Add Rational Expressions with a Common Denominator
In the following exercises, add.
Subtract Rational Expressions with a Common Denominator
In the following exercises, subtract.
Add and Subtract Rational Expressions whose Denominators are Opposites
In the following exercises, add and subtract.

### Add and Subtract Rational Expressions With Unlike Denominators

Find the Least Common Denominator of Rational Expressions
In the following exercises, find the LCD.
Find Equivalent Rational Expressions
In the following exercises, rewrite as equivalent rational expressions with the given denominator.
Add Rational Expressions with Different Denominators
In the following exercises, add.
Subtract Rational Expressions with Different Denominators
In the following exercises, subtract and add.

### Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing it as Division
In the following exercises, simplify.
Simplify a Complex Rational Expression by Using the LCD
In the following exercises, simplify.

### Solve Rational Equations

Solve Rational Equations
In the following exercises, solve.
Solve a Rational Equation for a Specific Variable
In the following exercises, solve for the indicated variable.

### Solve Proportion and Similar Figure Applications Similarity

Solve Proportions
In the following exercises, solve.
In the following exercises, solve using proportions.
Solve Similar Figure Applications
In the following exercises, solve.

### Solve Uniform Motion and Work Applications Problems

Solve Uniform Motion Applications
In the following exercises, solve.
Solve Work Applications
In the following exercises, solve.

### Use Direct and Inverse Variation

Solve Direct Variation Problems
In the following exercises, solve.
Solve Inverse Variation Problems
In the following exercises, solve.

### Practice Test

In the following exercises, simplify.
In the following exercises, perform the indicated operation and simplify.
In the following exercises, solve each equation.
In the following exercises, solve.


# Roots and Radicals

## Introduction

Suppose a stone falls from the edge of a cliff. The number of feet the stone has dropped after  seconds can be found by multiplying 16 times the square of . But to calculate the number of seconds it would take the stone to hit the land below, we need to use a square root. In this chapter, we will introduce and apply the properties of square roots, and extend these concepts to higher order roots and rational exponents.