# Factoring

## Factor Trinomials

### Factor Trinomials of the Form

You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication. To factor the trinomial means to start with the product, and end with the factors.
To figure out how we would factor a trinomial of the form  such as  and factor it to  let’s start with two general binomials of the form  and

This tells us that to factor a trinomial of the form  we need two factors  and  where the two numbers m and n multiply to c and add to b.
Let’s summarize the steps we used to find the factors.
In the first example, 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.
How do you get a positive product and a negative sum? We use two negative numbers.
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.
How do you get a negative product and a positive sum? We use one positive and one negative number.
When we factor trinomials, we must have the terms written in descending order—in order from highest degree to lowest degree.
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 How To Factor trinomials.
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 ax2 + bx + c using Trial and Error

Our next step is 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 we’ve used so far. Let’s do an example to see how this works.
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, 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  the factors in the first case will work. Let’s use FOIL to check.
Our result of the factoring is:
Remember, 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 and remember if the leading coefficient is negative, so is the GCF.

### Factor Trinomials of the Form  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!
The “ac” method is summarized here.
Don’t forget to look for a common factor!

### Factor Using Substitution

Sometimes a trinomial does not appear to be in the  form. However, we can often make a thoughtful substitution that will allow us to make it fit the  form. This is called factoring by substitution. It is standard to use u for the substitution.
In the  the middle term has a variable, x, and its square,  is the variable part of the first term. Look for this relationship as you try to find a substitution.
Sometimes the expression to be substituted is not a monomial.

### Key Concepts

1. How to factor trinomials of the form

2. Strategy for Factoring Trinomials of the Form  When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.

Notice that, in the case when m and n have opposite signs, the sign of the one with the larger absolute value matches the sign of b.
3. How to factor trinomials of the form

4. How to factor trinomials of the form

### Practice Makes Perfect

Factor Trinomials of the Form
In the following exercises, factor each trinomial of the form
In the following exercises, factor each trinomial of the form
Factor Trinomials of the Form
In the following exercises, factor completely using trial and error.
Factor Trinomials of the Form
In the following exercises, factor using the ‘ac’ method.
Factor Using Substitution
In the following exercises, factor using substitution.
Mixed Practice
In the following exercises, factor each expression using any method.

### 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?


# Factoring

## Factor Special Products

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. We squared a binomial using the Binomial Squares pattern in a previous chapter.
The trinomial  is called a perfect square trinomial. It is the square of the binomial
In this chapter, 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  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 the product,  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 first step in factoring is to look for a greatest common factor. 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:
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!
The next example shows variables in both terms.
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.
Also, to completely factor the binomial in the next example, we’ll factor a difference of squares twice!
The next example has a polynomial with 4 terms. So far, when this occurred we grouped the terms in twos and factored from there. Here we will notice that the first three terms form a perfect square trinomial.

### 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 will 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 .
In the next example, we first factor out the GCF. Then we can recognize the sum of cubes.
The first term in the next example is a binomial cubed.

### Key Concepts

1. Perfect Square Trinomials Pattern: If a and b are real numbers,

2. How to factor perfect square trinomials.

3. Difference of Squares Pattern: If  are real numbers,

4. How to factor differences of squares.

5. Sum and Difference of Cubes Pattern

6. How to factor the sum or difference of cubes.

### Practice Makes Perfect

Factor Perfect Square Trinomials
In the following exercises, factor completely using the perfect square trinomials pattern.
Factor Differences of Squares
In the following exercises, factor completely using the difference of squares pattern, if possible.
Factor Sums and Differences of Cubes
In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
Mixed Practice
In the following exercises, factor completely.

### 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

## 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. The following chart summarizes all the factoring methods we have covered, and outlines a strategy you should use when factoring polynomials.
Remember, a polynomial is completely factored if, other than monomials, its factors are prime!
Be careful when you are asked to factor a binomial as there are several options!
The next example can be factored using several methods. Recognizing the trinomial squares pattern will make your work easier.
Remember, sums of squares do not factor, but sums of cubes do!
When using the sum or difference of cubes pattern, being careful with the signs.
Taking out the complete GCF in the first step will always make your work easier.
When we have factored a polynomial with four terms, most often we separated it into two groups of two terms. Remember that we can also separate it into a trinomial and then one term.

### Key Concepts

1. How to use a general strategy for factoring polynomials.

### Practice Makes Perfect

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

### 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

## Polynomial Equations

We have spent considerable time learning how to factor polynomials. We will now look at polynomial equations and solve them using factoring, if possible.
A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.
We have already solved polynomial equations of degree one. Polynomial equations of degree one are linear equations are of the form
We are now going to solve polynomial equations of degree two. A polynomial equation of degree two is called a quadratic equation. 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  with  (If  then  and we are left with no quadratic term.)
To solve quadratic equations we need methods different from the ones we used in solving linear equations. We will look at one method here and then several others in a later chapter.

### Use 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, then 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.

### Solve Quadratic Equations by Factoring

The Zero Product Property works very nicely to solve quadratic equations. The quadratic equation must be factored, with zero isolated on one side. So we must be sure to start with the quadratic equation in standard form,  Then we must 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?
In the next example, the left side of the equation 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.
In the next example, when we factor the quadratic equation we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.
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 greater than two by using the Zero Product Property, just like we solved quadratic equations.

### Solve Equations with Polynomial Functions

As our study of polynomial functions continues, it will often be important to know when the function will have a certain value or what points lie on the graph of the function. Our work with the Zero Product Property will be help us find these answers.
The Zero Product Property also helps us determine where the function is zero. A value of x where the function is 0, is called a zero of the function.
When  the point  is a point on the graph. This point is an  of the graph. It is often important to know where the graph of a function crosses the axes. We will see some examples later.

### Solve Applications Modeled by Polynomial 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 polynomial 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 polynomial 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 positive results.
In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.
In the next example, we will use the Pythagorean Theorem  This formula gives the relation between the legs and the hypotenuse of a right triangle.
We will use this formula to in the next example.
The next example uses the function that gives the height of an object as a function of time when it is thrown from 80 feet above the ground.

### Key Concepts

1. Polynomial Equation: A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.
2. Quadratic Equation: An equation of the form  is called a quadratic equation.

3. Zero Product Property: If  then either  or  or both.
4. How to use the Zero Product Property

5. How to solve a quadratic equation by factoring.

6. Zero of a Function: For any function f, if  then x is a zero of the function.
7. How to use a problem solving strategy to solve word problems.

### 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 Equations with Polynomial Functions
In the following exercises, solve.
In the following exercises, for each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.

### 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 Review Exercises

### 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.

### Factor Trinomials

Factor Trinomials of the Form
In the following exercises, factor each trinomial of the form
In the following examples, factor each trinomial of the form
Factor Trinomials of the Form
In the following exercises, factor completely using trial and error.
Factor Trinomials of the Form
In the following exercises, factor.
Factor using substitution
In the following exercises, factor using substitution.

### Factor Special Products

Factor Perfect Square Trinomials
In the following exercises, factor completely using the perfect square trinomials pattern.
Factor Differences of Squares
In the following exercises, factor completely using the difference of squares pattern, if possible.
Factor Sums and Differences of Cubes
In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

### General Strategy for Factoring Polynomials

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

### Polynomial Equations

Use the Zero Product Property
In the following exercises, solve.
Solve Quadratic Equations by Factoring
In the following exercises, solve.
Solve Equations with Polynomial Functions
In the following exercises, solve.
In each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.

### Chapter Practice Test

In the following exercises, factor completely.
In the following exercises, solve


# Rational Expressions and Functions

## Introduction

Twelve goals last season. Fifteen home runs. Nine touchdowns. Whatever the statistics, sports analysts know it. Their jobs depend on it. Compiling and analyzing sports data not only help fans appreciate their teams but also help owners and coaches decide which players to recruit, how to best use them in games, how much they should be paid, and which players to trade. Understanding this kind of data requires a knowledge of specific types of expressions and functions. In this chapter, you will work with rational expressions and perform operations on them. And you will use rational expressions and inequalities to solve real-world problems.


# Rational Expressions and Functions

## Multiply and Divide Rational Expressions

We previously reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression.
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 do the same operations with rational expressions that we did with fractions. We will simplify, add, subtract, multiply, divide and use them in applications.

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

If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.
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.
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.

### Simplify Rational Expressions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than 1, in its numerator and denominator.
For example,
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.
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.
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 have learned to factor the polynomials in the numerators and denominators in the following examples.
Every time we write a rational expression, we should make a 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.
Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. We previously introduced opposite notation: the opposite of a is  and
The numerical fraction, say  simplifies to . We also recognize that the numerator and denominator are opposites.
The fraction , whose numerator and denominator are opposites also simplifies to .
This tells us that  is the opposite of
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. Be careful not to treat  and  as opposites. Recall that in addition, order doesn’t matter so . So if , then

### 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.
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

Just like we did for numerical fractions, to divide rational expressions, we multiply the first fraction by the reciprocal of the second.
Once we rewrite the division as multiplication of the first expression by the reciprocal of the second, we then factor everything and look for common factors.
Recall from Use the Language of Algebra that a complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, 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.

### Multiply and Divide Rational Functions

We started this section stating that a rational expression is an expression of the form  where p and q are polynomials and  Similarly, we define a rational function as a function of the form  where  and  are polynomial functions and  is not zero.
The domain of a rational function is all real numbers except for those values that would cause division by zero. We must eliminate any values that make
To multiply rational functions, we multiply the resulting rational expressions on the right side of the equation using the same techniques we used to multiply rational expressions.
To divide rational functions, we divide the resulting rational expressions on the right side of the equation using the same techniques we used to divide rational expressions.

### Key Concepts

1. Determine the values for which a rational expression is undefined.

2. Equivalent Fractions Property
If a, b, and c are numbers where  then  and
3. How to simplify a rational expression.

4. Opposites in a Rational Expression
The opposite of  is         An expression and its opposite divide to
5. Multiplication of Rational Expressions
If p, q, r, and s are polynomials where  then

6. How to multiply rational expressions.

7. Division of Rational Expressions
If p, q, r, and s are polynomials where  then

8. How to divide rational expressions.

9. How to determine the domain of a rational function.

### Practice Makes Perfect

Determine the Values for Which a Rational Expression is Undefined
In the following exercises, determine the values for which the rational expression is undefined.
Simplify Rational Expressions
In the following exercises, simplify each rational expression.
Multiply Rational Expressions
In the following exercises, multiply the rational expressions.
Divide Rational Expressions
In the following exercises, divide the rational expressions.
For the following exercises, perform the indicated operations.
Multiply and Divide Rational Functions
In the following exercises, find the domain of each function.
For the following exercises, find  where  and  are given.
For the following exercises, find  where  and  are given.

### 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 Functions

## Add and Subtract Rational Expressions

### Add and Subtract 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.
To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result 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.
Remember, too, we do not allow values that would make the denominator zero. What value of x should be excluded in the next example?
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. Be careful of the signs when you subtract a binomial or trinomial.

### 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.

Be careful with the signs as you work with the opposites when the fractions are being subtracted.

### 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 this example:  Since the denominators are not the same, the first step was to find the least common denominator (LCD).
To find the LCD of the fractions, we factored 12 and 18 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.
When we add numerical fractions, once we found the LCD, we rewrote each fraction as an equivalent fraction with the LCD by multiplying the numerator and denominator by the same number. We are now ready to add.
We do the same thing for rational expressions. However, we leave the LCD in factored form.
Remember, we always exclude values that would make the denominator zero. What values of  should we exclude in this next example?

### Add and Subtract Rational Expressions with Unlike Denominators

Now we have all the steps we need to add or subtract rational expressions with unlike denominators.
The steps used to add rational expressions are summarized here.
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.
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.
There are lots of negative signs in the next example. Be extra careful.
Things can get very messy when both fractions must be multiplied by a binomial to get the common denominator.
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.

### Add and subtract rational functions

To add or subtract rational functions, we use the same techniques we used to add or subtract polynomial functions.

### Key Concepts

1. Rational Expression Addition and Subtraction
If p, q, and r are polynomials where  then
 and
2. How to find the least common denominator of rational expressions.

3. How to add or subtract rational expressions.

### Practice Makes Perfect

Add and Subtract Rational Expressions with a Common Denominator
In the following exercises, add.
In the following exercises, subtract.
Add and Subtract Rational Expressions whose Denominators are Opposites
In the following exercises, add or subtract.
Find the Least Common Denominator of Rational Expressions
In the following exercises, ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.
Add and Subtract Rational Expressions with Unlike Denominators
In the following exercises, perform the indicated operations.
Add and Subtract Rational Functions
In the following exercises, find ⓐ  ⓑ

### 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 Functions

## Simplify Complex Rational Expressions

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

Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously 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.
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 complex rational expressions. We make sure the complex rational expression is of the form where one fraction is over one fraction. We then 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.
We follow the same procedure when the complex rational expression contains variables.
We summarize the steps here.

### 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 the 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.
We will use the same example as in . Decide which method works better for you.
Be sure to start by factoring all the denominators so you can find the LCD.
Be sure to factor the denominators first. Proceed carefully as the math can get messy!

### Key Concepts

1. How to simplify a complex rational expression by writing it as division.

2. How 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 each complex rational expression by writing it as division.
Simplify a Complex Rational Expression by Using the LCD
In the following exercises, simplify each complex rational expression by using the LCD.
In the following exercises, simplify each complex rational expression using either method.

### 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 Functions

## Solve Rational Equations

After defining the terms ‘expression’ and ‘equation’ earlier, 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 a rational equation.
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.
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 to a rational equation.
We note any possible extraneous solutions, c, by writing  next to the equation.
The steps of this method are shown.
We always start by noting the values that would cause any denominators to be zero.
In the next example, the last denominators is a difference of squares. Remember to factor it first to find the LCD.
In the next example, the first denominator is a trinomial. Remember to factor it first to find the LCD.
The equation we solved in the previous example had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. In the next example we get two algebraic solutions. Here one or both could be extraneous solutions.
In some cases, all the algebraic solutions are extraneous.

### Use Rational Functions

Working with functions that are defined by rational expressions often lead to rational equations. Again, we use the same techniques to solve them.

### 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.
When we developed the point-slope formula from our slope formula, we cleared the fractions by multiplying by the LCD.

In the next example, we will use the same technique with the formula for slope that we used to get the point-slope form of an equation of a line through the point  We will add one more step to solve for y.
Remember to multiply both sides by the LCD in the next example.

### Key Concepts

1. How to solve equations with rational expressions.

### Practice Makes Perfect

Solve Rational Equations
In the following exercises, solve each rational equation.
Solve Rational Equations that Involve Functions
Solve a Rational Equation for a Specific Variable
In the following exercises, solve.

### 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  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 Functions

## Solve Applications with Rational Equations

### 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.”
Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved rational equations. We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.
Notice in the last example that when we cleared the fractions by multiplying by the LCD, the result is the same as if we had cross-multiplied.
For any proportion,  we get the same result when we clear the fractions by multiplying by the LCD as when we cross-multiply.
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 each other and the units in the denominators must also match each other.

### 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 have the same ratio.
For example, the two triangles in  are similar. Each side of  is four 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.
On the map, Seattle, Portland, and Boise form a triangle. The distance between the cities is measured in inches. The figure on the left below represents the triangle formed by the cities on the map. The actual distance from Seattle to Boise is 400 miles.
We can use similar figures to find heights that we cannot directly measure.

### 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 the wind affects the speed of a plane. 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

### Solve Work Applications

The weekly gossip magazine has a big story about the Princess’ baby and the editor wants the magazine to be printed as soon as possible. She has asked the printer to run an extra printing press to get the printing done more quickly. Press #1 takes 6 hours to do the job and Press #2 takes 12 hours to do the job. How long will it take the printer to get the magazine printed with both presses running together?
This is a typical ‘work’ application. There are three quantities involved here—the time it would take each of the two presses to do the job alone and the time it would take for them to do the job together.
If Press #1 can complete the job in 6 hours, in one hour it would complete  of the job.
If Press #2 can complete the job in 12 hours, in one hour it would complete  of the job.
We will let t be the number of hours it would take the presses to print the magazines with both presses running together. So in 1 hour working together they have completed  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 the presses to complete the job if they worked together, we solve for
A chart will help us organize the information. We are looking for how many hours it would take to complete the job with both presses running together.
Keep in mind, it should take less time for two presses to complete a job working together than for either press to do it alone.

### Solve Direct Variation Problems

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.
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 here.
Now we’ll solve an application of direct variation.

### 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’.

### Key Concepts

1. A proportion is an equation of the form  where  The proportion is read “a is to b as c is to d.”
2. Property of Similar Triangles
If  is similar to  then their corresponding angle measure are equal and their corresponding sides have the same ratio.

3. Direct Variation

4. Inverse Variation

### Practice Makes Perfect

Solve Proportions
In the following exercises, solve each proportion.
In the following exercises, solve.
Solve Similar Figure Applications
In the following exercises, the triangles are similar. Find the length of the indicated side.
In the following exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle. The actual distance from New York to Chicago is 800 miles.
In the following exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle. The actual distance from Atlanta to New Orleans is 420 miles.
In the following exercises, answer each question.
Solve Uniform Motion Applications
In the following exercises, solve the application problem provided.
Solve Work Applications
Solve Direct Variation Problems
In the following exercises, solve.
Solve Inverse Variation Problems
In the following exercises, solve.
In the following exercises, write an inverse variation equation to solve the following problems.

### 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 Functions

## Solve Rational Inequalities

### Solve Rational Inequalities

We learned to solve linear inequalities after learning to solve linear equations. The techniques were very much the same with one major exception. When we multiplied or divided by a negative number, the inequality sign reversed.
Having just learned to solve rational equations we are now ready to solve rational inequalities. A rational inequality is an inequality that contains a rational expression.
Inequalities such as  and  are rational inequalities as they each contain a rational expression.
When we solve a rational inequality, we will use many of the techniques we used solving linear inequalities. We especially must remember that when we multiply or divide by a negative number, the inequality sign must reverse.
Another difference is that we must carefully consider what value might make the rational expression undefined and so must be excluded.
When we solve an equation and the result is  we know there is one solution, which is 3.
When we solve an inequality and the result is  we know there are many solutions. We graph the result to better help show all the solutions, and we start with 3. Three becomes a critical point and then we decide whether to shade to the left or right of it. The numbers to the right of 3 are larger than 3, so we shade to the right.
To solve a rational inequality, we first must write the inequality with only one quotient on the left and 0 on the right.
Next we determine the critical points to use to divide the number line into intervals. A critical point is a number which make the rational expression zero or undefined.
We then will evaluate the factors of the numerator and denominator, and find the quotient in each interval. This will identify the interval, or intervals, that contains all the solutions of the rational inequality.
We write the solution in interval notation being careful to determine whether the endpoints are included.
We summarize the steps for easy reference.
The next example requires that we first get the rational inequality into the correct form.
In the next example, the numerator is always positive, so the sign of the rational expression depends on the sign of the denominator.
The next example requires some work to get it into the needed form.

### Solve an Inequality with Rational Functions

When working with rational functions, it is sometimes useful to know when the function is greater than or less than a particular value. This leads to a rational inequality.
In economics, the function  is used to represent the cost of producing x units of a commodity. The average cost per unit can be found by dividing  by the number of items  Then, the average cost per unit is

### Key Concepts

1. Solve a rational inequality.

### Section Exercises

### Practice Makes Perfect

Solve Rational Inequalities
In the following exercises, solve each rational inequality and write the solution in interval notation.
Solve an Inequality with Rational Functions
In the following exercises, solve each rational function inequality and write the solution in interval notation.

### Writing Exercises

### Self Check

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

### Chapter Review Exercises

### Simplify, Multiply, and Divide 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.
Simplify Rational Expressions
In the following exercises, simplify.
Multiply Rational Expressions
In the following exercises, multiply.
Divide Rational Expressions
In the following exercises, divide.
Multiply and Divide Rational Functions

### Add and Subtract Rational Expressions

Add and Subtract Rational Expressions with a Common Denominator
In the following exercises, perform the indicated operations.
Add and Subtract Rational Expressions Whose Denominators Are Opposites
In the following exercises, add and subtract.
Find the Least Common Denominator of Rational Expressions
In the following exercises, find the LCD.
Add and Subtract Rational Expressions with Unlike Denominators
In the following exercises, perform the indicated operations.
Add and Subtract Rational Functions
In the following exercises, find  where  and  are given.
In the following exercises, find  where  and  are given.

### 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.

### 7.4 Solve Rational Equations

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

### Solve Applications with Rational Equations

Solve Proportions
In the following exercises, solve.
Solve Using Proportions
In the following exercises, solve.
Solve Similar Figure Applications
In the following exercises, solve.
Solve Uniform Motion Applications
In the following exercises, solve.
Solve Work Applications
In the following exercises, solve.
Solve Direct Variation Problems
In the following exercises, solve.
Solve Inverse Variation Problems
In the following exercises, solve.

### Solve Rational Inequalities

Solve Rational Inequalities
In the following exercises, solve each rational inequality and write the solution in interval notation.
Solve an Inequality with Rational Functions
In the following exercises, solve each rational function inequality and write the solution in interval notation

### 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 each rational inequality and write the solution in interval notation.
In the following exercises, find  given  and
In the following exercises, solve.


# Roots and Radicals

## Introduction

Imagine charging your cell phone is less than five seconds. Consider cleaning radioactive waste from contaminated water. Think about filtering salt from ocean water to make an endless supply of drinking water. Ponder the idea of bionic devices that can repair spinal injuries. These are just of few of the many possible uses of a material called graphene. Materials scientists are developing a material made up of a single layer of carbon atoms that is stronger than any other material, completely flexible, and conducts electricity better than most metals. Research into this type of material requires a solid background in mathematics, including understanding roots and radicals. In this chapter, you will learn to simplify expressions containing roots and radicals, perform operations on radical expressions and equations, and evaluate radical functions.


# Roots and Radicals

## Simplify Expressions with Roots

### Simplify Expressions with Roots

In Foundations, we briefly looked at square roots. Remember that when a real number n is multiplied by itself, we write  and read it ‘n squared’. This number is called the square of n, and n is called the square root. For example,
Notice (−13)2 = 169 also, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169.
So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? We use a radical sign, and  write,  which denotes the positive square root of m. The positive square root is also called the principal square root. This symbol, as well as other radicals to be introduced later, are grouping symbols.
We also use the radical sign for the square root of zero. Because   Notice that zero has only one square root.
We know that every positive number has two square roots and the radical sign indicates the positive one. We write  If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example,
Can we simplify  Is there a number whose square is
Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to  The square root of a negative number is not a real number.
So far we have only talked about squares and square roots. Let’s now extend our work to include higher powers and higher roots.
Let’s review some vocabulary first.

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.
It will be helpful to have a table of the powers of the integers from −5 to 5. See .
Notice the signs in the table. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of −2 to help you see this.
We will now extend the square root definition to higher roots.
Just like we use the word ‘cubed’ for b3, we use the term ‘cube root’ for
We can refer to  to help find higher roots.
Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.
We will apply these properties in the next two examples.
In this example be alert for the negative signs as well as even and odd powers.

### Estimate and Approximate Roots

When we see a number with a radical sign, we often don’t think about its numerical value. While we probably know that the  what is the value of  or  In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.
To get a numerical estimate of a square root, we look for perfect square numbers closest to the radicand. To find an estimate of  we see 11 is between perfect square numbers 9 and 16, closer to 9. Its square root then will be between 3 and 4, but closer to 3.
Similarly, to estimate  we see 91 is between perfect cube numbers 64 and 125. The cube root then will be between 4 and 5.
There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the  key on your calculator. To find a cube root, or any root with higher index, you will use the  key.
When you use these keys, you get an approximate value. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is  and it is read ‘approximately’.
Suppose your calculator has a 10 digit display. You would see that
How do we know these values are approximations and not the exact values? Look at what happens when we square them:
Their squares are close to 5, but are not exactly equal to 5. The fourth powers are close to 93, but not equal to 93.

### Simplify Variable Expressions with Roots

The odd root of a number can be either positive or negative. For example,
But what about an even root? We want the principal root, so
But notice,
How can we make sure the fourth root of −5 raised to the fourth power is 5? We can use the absolute value.  So we say that when n is even  This guarantees the principal root is positive.
What about square roots of higher powers of variables? The Power Property of Exponents says  So if we square a, the exponent will become 2m.
Looking now at the square root,

We apply this concept in the next example.
The next example uses the same idea for highter roots.
In the next example, we now have a coefficient in front of the variable. The concept  works in much the same way.
But notice  and no absolute value sign is needed as u4 is always positive.
This example just takes the idea farther as it has roots of higher index.
The next examples have two variables.

### Key Concepts

1. Square Root Notation

2.

3. Properties of

4. Simplifying Odd and Even Roots

### Practice Makes Perfect

Simplify Expressions with Roots
In the following exercises, simplify.
Estimate and Approximate Roots
In the following exercises, estimate each root between two consecutive whole numbers.
In the following exercises, approximate each root and round to two decimal places.
Simplify Variable Expressions with Roots
In the following exercises, simplify using absolute values as necessary.

### 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.


# Roots and Radicals

## Simplify Radical Expressions

### Use the Product Property to Simplify Radical Expressions

We will simplify radical expressions in a way similar to how we simplified fractions. A fraction is simplified if there are no common factors in the numerator and denominator. To simplify a fraction, we look for any common factors in the numerator and denominator.
A radical expression,  is considered simplified if it has no factors of  So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index.
For example,  is considered simplified because there are no perfect square factors in 5. But  is not simplified because 12 has a perfect square factor of 4.
Similarly,  is simplified because there are no perfect cube factors in 4. But  is not simplified because 24 has a perfect cube factor of 8.
To simplify radical expressions, we will also use some properties of roots. The properties we will use to simplify radical expressions are similar to the properties of exponents. We know that  The corresponding of Product Property of Roots says that
We use the Product Property of Roots to remove all perfect square factors from a square root.
Notice in the previous example that the simplified form of  is  which is the product of an integer and a square root. We always write the integer in front of the square root.
Be careful to write your integer so that it is not confused with the index. The expression  is very different from
We will apply this method in the next example. It may be helpful to have a table of perfect squares, cubes, and fourth powers.
The next example is much like the previous examples, but with variables. Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.
We follow the same procedure when there is a coefficient in the radicand. In the next example, both the constant and the variable have perfect square factors.
In the next example, we continue to use the same methods even though there are more than one variable under the radical.
We have seen how to use the order of operations to simplify some expressions with radicals. In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer since one term contains a radical and the other does not. The next example also includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

### Use the Quotient Property to Simplify Radical Expressions

Whenever you have to simplify a radical expression, the first step you should take is to determine whether the radicand is a perfect power of the index. If not, check the numerator and denominator for any common factors, and remove them. You may find a fraction in which both the numerator and the denominator are perfect powers of the index.
In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents,
Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.
We can use a similar property to simplify a root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect power of the index, we simplify the numerator and denominator separately.
Be sure to simplify the fraction in the radicand first, if possible.
In the next example, there is nothing to simplify in the denominators. Since the index on the radicals is the same, we can use the Quotient Property again, to combine them into one radical. We will then look to see if we can simplify the expression.

### Key Concepts

1. Simplified Radical Expression

2. Product Property of n

3. How to simplify a radical expression using the Product Property

4. Quotient Property of Radical Expressions

5. How to simplify a radical expression using the Quotient Property.

### Practice Makes Perfect

Use the Product Property to Simplify Radical Expressions
In the following exercises, use the Product Property to simplify radical expressions.
In the following exercises, simplify using absolute value signs as needed.
Use the Quotient Property to Simplify Radical Expressions
In the following exercises, use the Quotient Property to simplify square roots.

### 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?


# Roots and Radicals

## Simplify Rational Exponents

### Simplify Expressions with

Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
The Power Property for Exponents says that  when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.
Suppose we want to find a number p such that  We will use the Power Property of Exponents to find the value of p.

So  But we know also  Then it must be that
This same logic can be used for any positive integer exponent n to show that
The denominator of the rational exponent is the index of the radical.
There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.
In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.
In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.
Be careful of the placement of the negative signs in the next example. We will need to use the property  in one case.

### Simplify Expressions with

We can look at  in two ways. Remember the Power Property tells us to multiply the exponents and so  and  both equal  If we write these expressions in radical form, we get
This leads us to the following definition.
Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller, before raising it to the power indicated.
Remember that  The negative sign in the exponent does not change the sign of the expression.

### Use the Properties of Exponents to Simplify Expressions with Rational Exponents

The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponenets here to have them for reference as we simplify expressions.
We will apply these properties in the next example.
Sometimes we need to use more than one property. In the next example, we will use both the Product to a Power Property and then the Power Property.
We will use both the Product Property and the Quotient Property in the next example.

### Key Concepts

1. Rational Exponent

2. Rational Exponent

3. Properties of Exponents

### Practice Makes Perfect

Simplify expressions with
In the following exercises, write as a radical expression.
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Simplify Expressions with
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify. Assume all variables are positive.

### 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?


# Roots and Radicals

## Add, Subtract, and Multiply Radical Expressions

### Add and Subtract Radical Expressions

Adding radical expressions with the same index and the same radicand is just like adding like terms. We call radicals with the same index and the same radicand like radicals to remind us they work the same as like terms.
We add and subtract like radicals in the same way we add and subtract like terms. We know that  is  Similarly we add  and the result is
Think about adding like terms with variables as you do the next few examples. When you have like radicals, you just add or subtract the coefficients. When the radicals are not like, you cannot combine the terms.
For radicals to be like, they must have the same index and radicand. When the radicands contain more than one variable, as long as all the variables and their exponents are identical, the radicands are the same.
Remember that we always simplify radicals by removing the largest factor from the radicand that is a power of the index. Once each radical is simplified, we can then decide if they are like radicals.
In the next example, we will remove both constant and variable factors from the radicals. Now that we have practiced taking both the even and odd roots of variables, it is common practice at this point for us to assume all variables are greater than or equal to zero so that absolute values are not needed. We will use this assumption throughout the rest of this chapter.

### Multiply Radical Expressions

We have used the Product Property of Roots to simplify square roots by removing the perfect square factors. We can use the Product Property of Roots ‘in reverse’ to multiply square roots. Remember, we assume all variables are greater than or equal to zero.
We will rewrite the Product Property of Roots so we see both ways together.
When we multiply two radicals they must have the same index. Once we multiply the radicals, we then look for factors that are a power of the index and simplify the radical whenever possible.
Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply  we multiply the coefficients together and then the variables. The result is 12xy. Keep this in mind as you do these examples.
We follow the same procedures when there are variables in the radicands.

### Use Polynomial Multiplication to Multiply Radical Expressions

In the next a few examples, we will use the Distributive Property to multiply expressions with radicals. First we will distribute and then simplify the radicals when possible.
When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.
Recognizing some special products made our work easier when we multiplied binomials earlier. This is true when we multiply radicals, too. The special product formulas we used are shown here.
We will use the special product formulas in the next few examples. We will start with the Product of Binomial Squares Pattern.
In the next example, we will use the Product of Conjugates Pattern. Notice that the final product has no radical.

### Key Concepts

1. Product Property of Roots

2. Special Products

### Practice Makes Perfect

Add and Subtract Radical Expressions
In the following exercises, simplify.
Multiply Radical Expressions
In the following exercises, simplify.
Use Polynomial Multiplication to Multiply Radical Expressions
In the following exercises, multiply.
Mixed Practice

### 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?


# Roots and Radicals

## Divide Radical Expressions

### Divide Radical Expressions

We have used the Quotient Property of Radical Expressions to simplify roots of fractions. We will need to use this property ‘in reverse’ to simplify a fraction with radicals.
We give the Quotient Property of Radical Expressions again for easy reference. Remember, we assume all variables are greater than or equal to zero so that no absolute value bars are needed.
We will use the Quotient Property of Radical Expressions when the fraction we start with is the quotient of two radicals, and neither radicand is a perfect power of the index. When we write the fraction in a single radical, we may find common factors in the numerator and denominator.

### Rationalize a One Term Denominator

Before the calculator became a tool of everyday life, approximating the value of a fraction with a radical in the denominator was a very cumbersome process!
For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator.
This process is still used today, and is useful in other areas of mathematics, too.
Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a radical.
Similarly, a radical expression is not considered simplified if the radicand contains a fraction.
To rationalize a denominator with a square root, we use the property that  If we square an irrational square root, we get a rational number.
We will use this property to rationalize the denominator in the next example.
When we rationalized a square root, we multiplied the numerator and denominator by a square root that would give us a perfect square under the radical in the denominator. When we took the square root, the denominator no longer had a radical.
We will follow a similar process to rationalize higher roots. To rationalize a denominator with a higher index radical, we multiply the numerator and denominator by a radical that would give us a radicand that is a perfect power of the index. When we simplify the new radical, the denominator will no longer have a radical.
For example,
We will use this technique in the next examples.

### Rationalize a Two Term Denominator

When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates Pattern to rationalize the denominator.
When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.
Notice we did not distribute the 5 in the answer of the last example. By leaving the result factored we can see if there are any factors that may be common to both the numerator and denominator.
Be careful of the signs when multiplying. The numerator and denominator look very similar when you multiply by the conjugate.

### Key Concepts

1. Quotient Property of Radical Expressions

2. Simplified Radical Expressions

### Practice Makes Perfect

Divide Square Roots
In the following exercises, simplify.
Rationalize a One Term Denominator
In the following exercises, rationalize the denominator.
Rationalize a Two Term Denominator
In the following exercises, simplify.

### 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?


# Roots and Radicals

## Solve Radical Equations

### Solve Radical Equations

In this section we will solve equations that have a variable in the radicand of a radical expression. An equation of this type is called a radical equation.
As usual, when solving these equations, what we do to one side of an equation we must do to the other side as well. Once we isolate the radical, our strategy will be to raise both sides of the equation to the power of the index. This will eliminate the radical.
Solving radical equations containing an even index by raising both sides to the power of the index may introduce an algebraic solution that would not be a solution to the original radical equation. Again, we call this an extraneous solution as we did when we solved rational equations.
In the next example, we will see how to solve a radical equation. Our strategy is based on raising a radical with index n to the nth power. This will eliminate the radical.
When we use a radical sign, it indicates the principal or positive root. If an equation has a radical with an even index equal to a negative number, that equation will have no solution.
If one side of an equation with a square root is a binomial, we use the Product of Binomial Squares Pattern when we square it.
Don’t forget the middle term!
When the index of the radical is 3, we cube both sides to remove the radical.
Sometimes an equation will contain rational exponents instead of a radical. We use the same techniques to solve the equation as when we have a radical. We raise each side of the equation to the power of the denominator of the rational exponent. Since  we have for example,
Remember,  and
Sometimes the solution of a radical equation results in two algebraic solutions, but one of them may be an extraneous solution!
When there is a coefficient in front of the radical, we must raise it to the power of the index, too.

### Solve Radical Equations with Two Radicals

If the radical equation has two radicals, we start out by isolating one of them. It often works out easiest to isolate the more complicated radical first.
In the next example, when one radical is isolated, the second radical is also isolated.
Sometimes after raising both sides of an equation to a power, we still have a variable inside a radical. When that happens, we repeat Step 1 and Step 2 of our procedure. We isolate the radical and raise both sides of the equation to the power of the index again.
We summarize the steps here. We have adjusted our previous steps to include more than one radical in the equation This procedure will now work for any radical equations.
Be careful as you square binomials in the next example. Remember the pattern is  or

### Use Radicals in Applications

As you progress through your college courses, you’ll encounter formulas that include radicals in many disciplines. We will modify our Problem Solving Strategy for Geometry Applications slightly to give us a plan for solving applications with formulas from any discipline.
One application of radicals has to do with the effect of gravity on falling objects. The formula allows us to determine how long it will take a fallen object to hit the gound.
For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting  into the formula.

It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.
Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes.

### Key Concepts

1. Binomial Squares

2. Solve a Radical Equation

3. Problem Solving Strategy for Applications with Formulas

4. Falling Objects

5. Skid Marks and Speed of a Car

### Practice Makes Perfect

Solve Radical Equations
In the following exercises, solve.
Solve Radical Equations with Two Radicals
In the following exercises, solve.
Use Radicals in Applications
In the following exercises, solve. Round approximations to one decimal place.

### 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?


# Roots and Radicals

## Use Radicals in Functions

### Evaluate a Radical Function

In this section we will extend our previous work with functions to include radicals. If a function is defined by a radical expression, we call it a radical function.
The square root function is
The cube root function is
To evaluate a radical function, we find the value of f(x) for a given value of x just as we did in our previous work with functions.
We follow the same procedure to evaluate cube roots.
The next example has fourth roots.

### Find the Domain of a Radical Function

To find the domain and range of radical functions, we use our properties of radicals. For a radical with an even index, we said the radicand had to be greater than or equal to zero as even roots of negative numbers are not real numbers. For an odd index, the radicand can be any real number. We restate the properties here for reference.
So, to find the domain of a radical function with even index, we set the radicand to be greater than or equal to zero. For an odd index radical, the radicand can be any real number.
The next example involves a cube root and so will require different thinking.

### Graph Radical Functions

Before we graph any radical function, we first find the domain of the function. For the function,  the index is even, and so the radicand must be greater than or equal to 0.
This tells us the domain is  and we write this in interval notation as
Previously we used point plotting to graph the function,  We chose x-values, substituted them in and then created a chart. Notice we chose points that are perfect squares in order to make taking the square root easier.
Once we see the graph, we can find the range of the function. The y-values of the function are greater than or equal to zero. The range then is
In our previous work graphing functions, we graphed  but we did not graph the function  We will do this now in the next example.

### Key Concepts

1. Properties of

2. Domain of a Radical Function

### Practice Makes Perfect

Evaluate a Radical Function
In the following exercises, evaluate each function.
Find the Domain of a Radical Function
In the following exercises, find the domain of the function and write the domain in interval notation.
Graph Radical Functions
In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

### 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?


# Roots and Radicals

## Use the Complex Number System

### Evaluate the Square Root of a Negative Number

Whenever we have a situation where we have a square root of a negative number we say there is no real number that equals that square root. For example, to simplify  we are looking for a real number x so that x2 = –1. Since all real numbers squared are positive numbers, there is no real number that equals –1 when squared.
Mathematicians have often expanded their numbers systems as needed. They added 0 to the counting numbers to get the whole numbers. When they needed negative balances, they added negative numbers to get the integers. When they needed the idea of parts of a whole they added fractions and got the rational numbers. Adding the irrational numbers allowed numbers like  All of these together gave us the real numbers and so far in your study of mathematics, that has been sufficient.
But now we will expand the real numbers to include the square roots of negative numbers. We start by defining the imaginary unit  as the number whose square is –1.
We will use the imaginary unit to simplify the square roots of negative numbers.
We will use this definition in the next example. Be careful that it is clear that the i is not under the radical. Sometimes you will see this written as  to emphasize the i is not under the radical. But the  is considered standard form.
Now that we are familiar with the imaginary number i, we can expand our concept of the number system to include imaginary numbers. The complex number system includes the real numbers and the imaginary numbers. A complex number is of the form a + bi, where a, b are real numbers. We call a the real part and b the imaginary part.
A complex number is in standard form when written as  where a and b are real numbers.
If  then  becomes  and is a real number.
If  then  is an imaginary number.
If  then  becomes  and is called a pure imaginary number.
We summarize this here.
The standard form of a complex number is  so this explains why the preferred form is  when
The diagram helps us visualize the complex number system. It is made up of both the real numbers and the imaginary numbers.

### Add or Subtract Complex Numbers

We are now ready to perform the operations of addition, subtraction, multiplication and division on the complex numbers—just as we did with the real numbers.
Adding and subtracting complex numbers is much like adding or subtracting like terms. We add or subtract the real parts and then add or subtract the imaginary parts. Our final result should be in standard form.
Remember to add both the real parts and the imaginary parts in this next example.

### Multiply Complex Numbers

Multiplying complex numbers is also much like multiplying expressions with coefficients and variables. There is only one special case we need to consider. We will look at that after we practice in the next two examples.
In the next example, we multiply the binomials using the Distributive Property or FOIL.
In the next example, we could use FOIL or the Product of Binomial Squares Pattern.
Since the square root of a negative number is not a real number, when we have the square roots of two negative numbers, we cannot use the Product Property for Radicals. In order to multiply square roots of negative numbers we should first write them as complex numbers, using  This is one place students tend to make errors, so be careful when you see multiplying with a negative square root.
In the next example, each binomial has a square root of a negative number. Before multiplying, each square root of a negative number must be written as a complex number.
We first looked at conjugate pairs when we studied polynomials. We said that a pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a conjugate pair and is of the form
A complex conjugate pair is very similar. For a complex number of the form  its conjugate is  Notice they have the same first term and the same last term, but one is a sum and one is a difference.
We will multiply a complex conjugate pair in the next example.
From our study of polynomials, we know the product of conjugates is always of the form  The result is called a difference of squares. We can multiply a complex conjugate pair using this pattern.
The last example we used FOIL. Now we will use the Product of Conjugates Pattern.
Notice this is the same result we found in .
When we multiply complex conjugates, the product of the last terms will always have an  which simplifies to
This leads us to the Product of Complex Conjugates Pattern:

### Divide Complex Numbers

Dividing complex numbers is much like rationalizing a denominator. We want our result to be in standard form with no imaginary numbers in the denominator.
We summarize the steps here.
Be careful as you find the conjugate of the denominator.

### Simplify Powers of i

The powers of  make an interesting pattern that will help us simplify higher powers of i. Let’s evaluate the powers of  to see the pattern.
We summarize this now.
If we continued, the pattern would keep repeating in blocks of four. We can use this pattern to help us simplify powers of i. Since i4 = 1, we rewrite each power, i, as a product using i4 to a power and another power of i.
We rewrite it in the form  where the exponent, q, is the quotient of n divided by 4 and the exponent, r, is the remainder from this division. For example, to simplify i57, we divide 57 by 4 and we get 14 with a remainder of 1. In other words,  So we write  and then simplify from there.

### Key Concepts

1. Square Root of a Negative Number

2. Product of Complex Conjugates

3. How to Divide Complex Numbers

### Section Exercises

### Practice Makes Perfect

Evaluate the Square Root of a Negative Number
In the following exercises, write each expression in terms of i and simplify if possible.
Add or Subtract Complex Numbers In the following exercises, add or subtract.
Multiply Complex Numbers
In the following exercises, multiply.
In the following exercises, multiply using the Product of Binomial Squares Pattern.
In the following exercises, multiply.
In the following exercises, multiply using the Product of Complex Conjugates Pattern.
Divide Complex Numbers
In the following exercises, divide.
Simplify Powers of
In the following exercises, simplify.

### 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  how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

### Chapter Review Exercises

### Simplify Expressions with Roots

Simplify Expressions with Roots
In the following exercises, simplify.
Estimate and Approximate Roots
In the following exercises, estimate each root between two consecutive whole numbers.
In the following exercises, approximate each root and round to two decimal places.
Simplify Variable Expressions with Roots
In the following exercises, simplify using absolute values as necessary.

### Simplify Radical Expressions

Use the Product Property to Simplify Radical Expressions
In the following exercises, use the Product Property to simplify radical expressions.
In the following exercises, simplify using absolute value signs as needed.
Use the Quotient Property to Simplify Radical Expressions
In the following exercises, use the Quotient Property to simplify square roots.

### Simplify Rational Exponents

Simplify expressions with
In the following exercises, write as a radical expression.
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Simplify Expressions with
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify.

### Add, Subtract and Multiply Radical Expressions

Add and Subtract Radical Expressions
In the following exercises, simplify.
Multiply Radical Expressions
In the following exercises, simplify.
Use Polynomial Multiplication to Multiply Radical Expressions
In the following exercises, multiply.

### Divide Radical Expressions

Divide Square Roots
In the following exercises, simplify.
Rationalize a One Term Denominator
In the following exercises, rationalize the denominator.
Rationalize a Two Term Denominator
In the following exercises, simplify.

### Solve Radical Equations

Solve Radical Equations
In the following exercises, solve.
Solve Radical Equations with Two Radicals
In the following exercises, solve.
Use Radicals in Applications
In the following exercises, solve. Round approximations to one decimal place.

### Use Radicals in Functions

Evaluate a Radical Function
In the following exercises, evaluate each function.
Find the Domain of a Radical Function
In the following exercises, find the domain of the function and write the domain in interval notation.
Graph Radical Functions
In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

### Use the Complex Number System

Evaluate the Square Root of a Negative Number
In the following exercises, write each expression in terms of i and simplify if possible.
Add or Subtract Complex Numbers
In the following exercises, add or subtract.
Multiply Complex Numbers
In the following exercises, multiply.
In the following exercises, multiply using the Product of Binomial Squares Pattern.
In the following exercises, multiply using the Product of Complex Conjugates Pattern.
Divide Complex Numbers
In the following exercises, divide.
Simplify Powers of
In the following exercises, simplify.

### Practice Test

In the following exercises, simplify using absolute values as necessary.
In the following exercises, simplify. Assume all variables are positive.
In the following exercises, solve.
In the following exercise, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.


# Quadratic Equations and Functions

## Introduction

Blink your eyes. You’ve taken a photo. That’s what will happen if you are wearing a contact lens with a built-in camera. Some of the same technology used to help doctors see inside the eye may someday be used to make cameras and other devices. These technologies are being developed by biomedical engineers using many mathematical principles, including an understanding of quadratic equations and functions. In this chapter, you will explore these kinds of equations and learn to solve them in different ways. Then you will solve applications modeled by quadratics, graph them, and extend your understanding to quadratic inequalities.