Preface

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About *Intermediate Algebra 2e*

*Intermediate Algebra 2e* is designed to meet the scope and sequence requirements of a one-semester Intermediate Algebra course. The book’s organization makes it easy to adapt to a variety of course syllabi. The text expands on the fundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles. Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics.

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About the Authors

Foundations

Introduction

Use the Language of Algebra

Abstract:
By the end of this section, you will be able to:

* Find factors, prime factorizations, and least common multiples
* Use variables and algebraic symbols
* Simplify expressions using the order of operations
* Evaluate an expression
* Identify and combine like terms
* Translate an English phrase to an algebraic expression

Find Factors, Prime Factorizations, and Least Common Multiples

The numbers 2, 4, 6, 8, 10, 12 are called multiples of 2. A multiple of 2 can be written as the product of 2 and a counting number.

Use Variables and Algebraic Symbols

In algebra, we use a letter of the alphabet to represent a number whose value may change. We call this a variable and letters commonly used for variables are $x,y,a,b,c.$

Simplify Expressions Using the Order of Operations

To simplify an expression means to do all the math possible. For example, to simplify $4·2+1$ we would first multiply $4·2$ to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

Evaluate an Expression

In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

Identify and Combine Like Terms

Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.

Translate an English Phrase to an Algebraic Expression

We listed many operation symbols that are used in algebra. Now, we will use them to translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. [link] summarizes them.

Integers

Abstract:
By the end of this section, you will be able to:

* Simplify expressions with absolute value
* Add and subtract integers
* Multiply and divide integers
* Simplify expressions with integers
* Evaluate variable expressions with integers
* Translate phrases to expressions with integers
* Use integers in applications

Simplify Expressions with Absolute Value

A negative number is a number less than 0. The negative numbers are to the left of zero on the number line. See [link].

Add and Subtract Integers

So far, we have only used the counting numbers and the whole numbers.

Multiply and Divide Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we are using the model just to help us discover the pattern.

Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember Please Excuse My Dear Aunt Sally?

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

Fractions

Abstract:
By the end of this section, you will be able to:

* Simplify fractions
* Multiply and divide fractions
* Add and subtract fractions
* Use the order of operations to simplify fractions
* Evaluate variable expressions with fractions

Simplify Fractions

A fraction is a way to represent parts of a whole. The fraction $\frac{2}{3}$ represents two of three equal parts. See [link]. In the fraction $\frac{2}{3},$ the 2 is called the numerator and the 3 is called the denominator. The line is called the fraction bar.

Multiply and Divide Fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions.

Add and Subtract Fractions

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

Use the Order of Operations to Simplify Fractions

The fraction bar in a fraction acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Decimals

Abstract:
By the end of this section, you will be able to:

* Round decimals
* Add and subtract decimals
* Multiply and divide decimals
* Convert decimals, fractions, and percents
* Simplify expressions with square roots
* Identify integers, rational numbers, irrational numbers, and real numbers
* Locate fractions and decimals on the number line

Round Decimals

Decimals are another way of writing fractions whose denominators are powers of ten.

Add and Subtract Decimals

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.

Multiply and Divide Decimals

When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. We multiply the numbers temporarily ignoring the decimal point and then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product. Finally, we write the product with the appropriate sign.

Convert Decimals, Fractions, and Percents

In our work, it is often necessary to change the form of a number. We may have to change fractions to decimals or decimals to percent.

Simplify Expressions with Square Roots

Remember that when a number $n$ is multiplied by itself, we write ${n}^{2}$ and read it “$n$ squared.” The result is called the square of a number *n*. For example, ${8}^{2}$ is read “8 squared” and 64 is called the *square* of 8. Similarly, 121 is the square of 11 because ${11}^{2}$ is 121. It will be helpful to learn to recognize the perfect square numbers.

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

We have already described numbers as *counting number**s*, *whole number**s*, and *integers*. What is the difference between these types of numbers? Difference could be confused with subtraction. How about asking how we distinguish between these types of numbers?

Locate Fractions and Decimals on the Number Line

We now want to include fractions and decimals on the number line. Let’s start with fractions and locate $\frac{1}{5},-\frac{4}{5},3,\frac{7}{4},-\frac{9}{2},-5$ and $\frac{8}{3}$ on the number line.

Properties of Real Numbers

Abstract:
By the end of this section, you will be able to:

* Use the commutative and associative properties
* Use the properties of identity, inverse, and zero
* Simplify expressions using the Distributive Property

Use the Commutative and Associative Properties

The order we add two numbers doesn’t affect the result. If we add $8+9$ or $9+8,$ the results are the same—they both equal 17. So, $8+9=9+8.$ The order in which we add does not matter!

Use the Properties of Identity, Inverse, and Zero

What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity. The Identity Property of Addition that states that for any real number $a,a+0=a$ and $0+a=a.$

Simplify Expressions Using the Distributive Property

Suppose that three friends are going to the movies. They each need \$9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?

Solving Linear Equations

Introduction

Use a General Strategy to Solve Linear Equations

Abstract:
By the end of this section, you will be able to:

* Solve linear equations using a general strategy
* Classify equations
* Solve equations with fraction or decimal coefficients

Solve Linear Equations Using a General Strategy

Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that makes it a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle!

Classify Equations

Whether or not an equation is true depends on the value of the variable. The equation $7x+8=-13$ is true when we replace the variable, *x*, with the value $-3,$ but not true when we replace *x* with any other value. An equation like this is called a conditional equation. All the equations we have solved so far are conditional equations.

Solve Equations with Fraction or Decimal Coefficients

We could use the General Strategy to solve the next example. This method would work fine, but many students do not feel very confident when they see all those fractions. So, we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.

Use a Problem Solving Strategy

Abstract:
By the end of this section, you will be able to:

* Use a problem solving strategy for word problems
* Solve number word problems
* Solve percent applications
* Solve simple interest applications

Use a Problem Solving Strategy for Word Problems

Now that we can solve equations, we are ready to apply our new skills to word problems. We will develop a strategy we can use to solve any word problem successfully.

Solve Number Word Problems

We will now apply the problem solving strategy to “number word problems.” Number word problems give some clues about one or more numbers and we use these clues to write an equation. Number word problems provide good practice for using the Problem Solving Strategy.

Solve Percent Applications

There are several methods to solve percent equations. In algebra, it is easiest if we just translate English sentences into algebraic equations and then solve the equations. Be sure to change the given percent to a decimal before you use it in the equation.

Solve Simple Interest Applications

Interest is a part of our daily lives. From the interest earned on our savings to the interest we pay on a car loan or credit card debt, we all have some experience with interest in our lives.

Solve a Formula for a Specific Variable

Abstract:
By the end of this section, you will be able to:

* Solve a formula for a specific variable
* Use formulas to solve geometry applications

Solve a Formula for a Specific Variable

We have all probably worked with some geometric formulas in our study of mathematics. Formulas are used in so many fields, it is important to recognize formulas and be able to manipulate them easily.

Use Formulas to Solve Geometry Applications

In this objective we will use some common geometry formulas. We will adapt our problem solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve.

Solve Mixture and Uniform Motion Applications

Abstract:
By the end of this section, you will be able to:

* Solve coin word problems
* Solve ticket and stamp word problems
* Solve mixture word problems
* Solve uniform motion applications

Solve Coin Word Problems

Using algebra to find the number of nickels and pennies in a piggy bank may seem silly. You may wonder why we just don’t open the bank and count them. But this type of problem introduces us to some techniques that will be useful as we move forward in our study of mathematics.

Solve Ticket and Stamp Word Problems

Problems involving tickets or stamps are very much like coin problems. Each type of ticket and stamp has a value, just like each type of coin does. So to solve these problems, we will follow the same steps we used to solve coin problems.

Solve Mixture Word Problems

Now we’ll solve some more general applications of the mixture model. In mixture problems, we are often mixing two quantities, such as raisins and nuts, to create a mixture, such as trail mix. In our tables we will have a row for each item to be mixed as well as one for the final mixture.

Solve Uniform Motion Applications

When you are driving down the interstate using your cruise control, the speed of your car stays the same—it is uniform. We call a problem in which the speed of an object is constant a uniform motion application. We will use the distance, rate, and time formula, $D=rt,$ to compare two scenarios, such as two vehicles travelling at different rates or in opposite directions.

Solve Linear Inequalities

Abstract:
By the end of this section, you will be able to:

* Graph inequalities on the number line
* Solve linear inequalities
* Translate words to an inequality and solve
* Solve applications with linear inequalities

Graph Inequalities on the Number Line

What number would make the inequality $x>3$ true? Are you thinking, “*x* could be four”? That’s correct, but *x* could be 6, too, or 37, or even 3.001. Any number greater than three is a solution to the inequality $x>3.$

Solve Linear Inequalities

A linear inequality is much like a linear equation—but the equal sign is replaced with an inequality sign. A **linear inequality** is an inequality in one variable that can be written in one of the forms, $ax+b$ax+b\le c,$$ax+b>c,$ or $ax+b\ge c.$

Translate to an Inequality and Solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like “more than” and “less than.” But others are not as obvious. [link] shows some common phrases that indicate inequalities.

Solve Applications with Linear Inequalities

Many real-life situations require us to solve inequalities. The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations.

Solve Compound Inequalities

Abstract:
By the end of this section, you will be able to:

* Solve compound inequalities with “and”
* Solve compound inequalities with “or”
* Solve applications with compound inequalities

Solve Compound Inequalities with “and”

Now that we know how to solve linear inequalities, the next step is to look at compound inequalities. A compound inequality is made up of two inequalities connected by the word “and” or the word “or.” For example, the following are compound inequalities.

Solve Compound Inequalities with “or”

To solve a compound inequality with “or”, we start out just as we did with the compound inequalities with “and”—we solve the two inequalities. Then we find all the numbers that make *either* inequality true.

Solve Applications with Compound Inequalities

Situations in the real world also involve compound inequalities. We will use the same problem solving strategy that we used to solve linear equation and inequality applications.

Solve Absolute Value Inequalities

Abstract:
By the end of this section, you will be able to:

* Solve absolute value equations
* Solve absolute value inequalities with “less than”
* Solve absolute value inequalities with “greater than”
* Solve applications with absolute value

Solve Absolute Value Equations

As we prepare to solve absolute value equations, we review our definition of absolute value.

Solve Absolute Value Inequalities with “Less Than”

Let’s look now at what happens when we have an absolute value inequality. Everything we’ve learned about solving inequalities still holds, but we must consider how the absolute value impacts our work.

Solve Absolute Value Inequalities with “Greater Than”

What happens for absolute value inequalities that have “greater than”? Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line.

Solve Applications with Absolute Value

Absolute value inequalities are often used in the manufacturing process. An item must be made with near perfect specifications. Usually there is a certain *tolerance* of the difference from the specifications that is allowed. If the difference from the specifications exceeds the tolerance, the item is rejected.

Graphs and Functions

Introduction

Graph  Linear Equations in Two Variables

Abstract:
By the end of this section, you will be able to:

* Plot points in a rectangular coordinate system
* Graph a linear equation by plotting points
* Graph vertical and horizontal lines
* Find the x- and y-intercepts
* Graph a line using the intercepts

Plot Points on a Rectangular Coordinate System

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the *xy*-plane or the “coordinate plane.”

Graph a Linear Equation by Plotting Points

There are several methods that can be used to graph a linear equation. The first method we will use is called plotting points, or the Point-Plotting Method. We find three points whose coordinates are solutions to the equation and then plot them in a rectangular coordinate system. By connecting these points in a line, we have the graph of the linear equation.

Graph Vertical and Horizontal Lines

Some linear equations have only one variable. They may have just *x* and no *y*, or just *y* without an *x*. This changes how we make a table of values to get the points to plot.

Find *x*- and *y*-intercepts

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the *x-* and *y-* intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.

Slope of a Line

Abstract:
By the end of this section, you will be able to:

* Find the slope of a line
* Graph a line given a point and the slope
* Graph a line using its slope and intercept
* Choose the most convenient method to graph a line
* Graph and interpret applications of slope–intercept
* Use slopes to identify parallel and perpendicular lines

Find the Slope of a Line

When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter.

Graph a Line Given a Point and the Slope

Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

Graph a Line Using its Slope and Intercept

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using one point and the slope of the line. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.

Choose the Most Convenient Method to Graph a Line

Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?

Graph and Interpret Applications of Slope–Intercept

Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope–intercept form relate to real world situations.

Use Slopes to Identify Parallel and Perpendicular Lines

Two lines that have the same slope are called parallel lines. Parallel lines have the same steepness and never intersect.

Find the Equation of a Line

Abstract:
By the end of this section, you will be able to:

* Find an equation of the line given the slope and $y\text{-intercept}$
* Find an equation of the line given the slope and a point
* Find an equation of the line given two points
* Find an equation of a line parallel to a given line
* Find an equation of a line perpendicular to a given line

Find an Equation of the Line Given the Slope and *y*-Intercept

We can easily determine the slope and intercept of a line if the equation is written in slope-intercept form, $y=mx+b.$ Now we will do the reverse—we will start with the slope and *y*-intercept and use them to find the equation of the line.

Find an Equation of the Line Given the Slope and a Point

Finding an equation of a line using the slope-intercept form of the equation works well when you are given the slope and *y*-intercept or when you read them off a graph. But what happens when you have another point instead of the *y*-intercept?

Find an Equation of the Line Given Two Points

When real-world data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when just two points are given.

Find an Equation of a Line Parallel to a Given Line

Suppose we need to find an equation of a line that passes through a specific point and is parallel to a given line. We can use the fact that parallel lines have the same slope. So we will have a point and the slope—just what we need to use the point-slope equation.

Find an Equation of a Line Perpendicular to a Given Line

Now, let’s consider perpendicular lines. Suppose we need to find a line passing through a specific point and which is perpendicular to a given line. We can use the fact that perpendicular lines have slopes that are negative reciprocals. We will again use the point-slope equation, like we did with parallel lines.

Graph Linear Inequalities in Two Variables

Abstract:
By the end of this section, you will be able to:

* Verify solutions to an inequality in two variables.
* Recognize the relation between the solutions of an inequality and its graph.
* Graph linear inequalities in two variables
* Solve applications using linear inequalities in two variables

Verify Solutions to an Inequality in Two Variables

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at **linear inequalities** in two variables which are very similar to linear equations in two variables.

Recognize the Relation Between the Solutions of an Inequality and its Graph

Now, we will look at how the solutions of an inequality relate to its graph.

Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.

Solve Applications using Linear Inequalities in Two Variables

Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how they might be used.

Relations and Functions

Abstract:
By the end of this section, you will be able to:

* Find the domain and range of a relation
* Determine if a relation is a function
* Find the value of a function

Find the Domain and Range of a Relation

As we go about our daily lives, we have many data items or quantities that are paired to our names. Our social security number, student ID number, email address, phone number and our birthday are matched to our name. There is a relationship between our name and each of those items.

Determine if a Relation is a Function

A special type of relation, called a function, occurs extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each *x*-value is matched with only one *y*-value.

Find the Value of a Function

It is very convenient to name a function and most often we name it *f*, *g*, *h*, *F*, *G*, or *H*. In any function, for each *x*-value from the domain we get a corresponding *y*-value in the range. For the function *f*, we write this range value *y* as $f\left(x\right).$ This is called function notation and is read *f* of *x* or the value of *f* at *x*. In this case the parentheses does not indicate multiplication.

Graphs of Functions

Abstract:
By the end of this section, you will be able to:

* Use the vertical line test
* Identify graphs of basic functions
* Read information from a graph of a function

Use the Vertical Line Test

In the last section we learned how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping or an equation. We will now look at how to tell if a graph is that of a function.

Identify Graphs of Basic Functions

We used the equation $y=2x-3$ and its graph as we developed the vertical line test. We said that the relation defined by the equation $y=2x-3$ is a function.

Read Information from a Graph of a Function

In the sciences and business, data is often collected and then graphed. The graph is analyzed, information is obtained from the graph and then often predictions are made from the data.

Systems of Linear Equations

Introduction

Solve Systems of Linear Equations with Two Variables

Abstract:
By the end of this section, you will be able to:

* Determine whether an ordered pair is a solution of a system of equations
* Solve a system of linear equations by graphing
* Solve a system of equations by substitution
* Solve a system of equations by elimination
* Choose the most convenient method to solve a system of linear equations

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

In Solving Linear Equations, we learned how to solve linear equations with one variable. Now we will work with two or more linear equations grouped together, which is known as a system of linear equations.

Solve a System of Linear Equations by Graphing

In this section, we will use three methods to solve a system of linear equations. The first method we’ll use is graphing.

Solve a System of Equations by Substitution

We will now solve systems of linear equations by the substitution method.

Solve a System of Equations by Elimination

We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression.

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

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

Solve Applications with Systems of Equations

Abstract:
By the end of this section, you will be able to:

* Solve direct translation applications
* Solve geometry applications
* Solve uniform motion applications

Solve Direct Translation Applications

Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two variables easier than setting them up with just one variable. To solve an application, we’ll first translate the words into a system of linear equations. Then we will decide the most convenient method to use, and then solve the system.

Solve Geometry Applications

We will now solve geometry applications using systems of linear equations. We will need to add complementary angles and supplementary angles to our list some properties of angles.

Solve uniform motion applications

We used a table to organize the information in uniform motion problems when we introduced them earlier. We’ll continue using the table here. The basic equation was $D=rt$ where *D* is the distance traveled, *r* is the rate, and *t* is the time.

Solve Mixture Applications with Systems of Equations

Abstract:
By the end of this section, you will be able to:

* Solve mixture applications
* Solve interest applications
* Solve applications of cost and revenue functions

Solve Mixture Applications

Mixture application involve combining two or more quantities. When we solved mixture applications with coins and tickets earlier, we started by creating a table so we could organize the information. For a coin example with nickels and dimes, the table looked like this:

Solve Interest Applications

The formula to model simple interest applications is $I=Prt.$ Interest, *I*, is the product of the principal, *P*, the rate, *r*, and the time, *t*. In our work here, we will calculate the interest earned in one year, so *t* will be 1.

Solve applications of cost and revenue functions

Suppose a company makes and sells *x* units of a product. The cost to the company is the total costs to produce *x* units. This is the cost to manufacture for each unit times *x*, the number of units manufactured, plus the fixed costs.

Solve Systems of Equations with Three Variables

Abstract:
By the end of this section, you will be able to:

* Determine whether an ordered triple is a solution of a system of three linear equations with three variables
* Solve a system of linear equations with three variables
* Solve applications using systems of linear equations with three variables

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

In this section, we will extend our work of solving a system of linear equations. So far we have worked with systems of equations with two equations and two variables. Now we will work with systems of three equations with three variables. But first let's review what we already know about solving equations and systems involving up to two variables.

Solve a System of Linear Equations with Three Variables

To solve a system of linear equations with three variables, we basically use the same techniques we used with systems that had two variables. We start with two pairs of equations and in each pair we eliminate the same variable. This will then give us a system of equations with only two variables and then we know how to solve that system!

Solve Applications using Systems of Linear Equations with Three Variables

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

Solve Systems of Equations Using Matrices

Abstract:
By the end of this section, you will be able to:

* Write the augmented matrix for a system of equations
* Use row operations on a matrix
* Solve systems of equations using matrices

Write the Augmented Matrix for a System of Equations

Solving a system of equations can be a tedious operation where a simple mistake can wreak havoc on finding the solution. An alternative method which uses the basic procedures of elimination but with notation that is simpler is available. The method involves using a matrix. A matrix is a rectangular array of numbers arranged in rows and columns.

Use Row Operations on a Matrix

Once a system of equations is in its augmented matrix form, we will perform operations on the rows that will lead us to the solution.

To solve a system of equations using matrices, we transform the augmented matrix into a matrix in row-echelon form using row operations. For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

Solve Systems of Equations Using Determinants

Abstract:
By the end of this section, you will be able to:

* Evaluate the determinant of a $2{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}2$ matrix
* Evaluate the determinant of a $3{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}3$ matrix
* Use Cramer’s Rule to solve systems of equations
* Solve applications using determinants

Evaluate the Determinant of a $2{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}2$ Matrix

If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant. To find the determinant of the square matrix $\left[\begin{array}{c}a{\rule{1.5em}{0ex}}b\hfill \\ c{\rule{1.5em}{0ex}}d\hfill \end{array}\right],$ we first write it as $|\begin{array}{c}a{\rule{1.5em}{0ex}}b\hfill \\ c{\rule{1.5em}{0ex}}d\hfill \end{array}|.$ To get the real number value of the determinant we subtract the products of the diagonals, as shown.

Evaluate the Determinant of a $3{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}3$ Matrix

To evaluate the determinant of a $3{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}3$ matrix, we have to be able to evaluate the minor of an entry in the determinant. The minor of an entry is the $2{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}2$ determinant found by eliminating the row and column in the $3{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}3$ determinant that contains the entry.

Use Cramer’s Rule to Solve Systems of Equations

Cramer’s Rule is a method of solving systems of equations using determinants. It can be derived by solving the general form of the systems of equations by elimination. Here we will demonstrate the rule for both systems of two equations with two variables and for systems of three equations with three variables.

Solve Applications using Determinants

An interesting application of determinants allows us to test if points are collinear. Three points $\left({x}_{1},{y}_{1}\right),$ $\left({x}_{2},{y}_{2}\right)$ and $\left({x}_{3},{y}_{3}\right)$ are collinear if and only if the determinant below is zero.

Graphing Systems of Linear Inequalities

Abstract:
By the end of this section, you will be able to:

* Determine whether an ordered pair is a solution of a system of linear inequalities
* Solve a system of linear inequalities by graphing
* Solve applications of systems of inequalities

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

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.

Solve a System of Linear Inequalities by Graphing

The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.

Solve Applications of Systems of Inequalities

The first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so we add inequalities to the system as additional requirements.

Polynomials and Polynomial Functions

Introduction

Add and Subtract Polynomials

Abstract:
By the end of this section, you will be able to:

* Determine the degree of polynomials
* Add and subtract polynomials
* Evaluate a polynomial function for a given value
* Add and subtract polynomial functions

Determine the Degree of Polynomials

We have learned that a *term* is a constant or the product of a constant and one or more variables. A monomial is an algebraic expression with one term. When it is of the form $a{x}^{m},$ where *a* is a constant and *m* is a whole number, it is called a monomial in one variable. Some examples of monomials in one variable are
 $2x,5y,17z,$ and $4{y}^{2}$. Monomials can also have more than one variable such as
 $5abc$ and $-4{a}^{2}{b}^{3}{c}^{2}.$

Add and Subtract Polynomials

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

Evaluate a Polynomial Function for a Given Value

A polynomial function is a function defined by a polynomial. For example, $f\left(x\right)={x}^{2}+5x+6$ and $g\left(x\right)=3x-4$ are polynomial functions, because ${x}^{2}+5x+6$ and $3x-4$ are polynomials.

Add and Subtract Polynomial Functions

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

Properties of Exponents and Scientific Notation

Abstract:
By the end of this section, you will be able to:

* Simplify expressions using the properties for exponents
* Use the definition of a negative exponent
* Use scientific notation

Simplify Expressions Using the Properties for Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression ${a}^{m},$ the *exponent m* tells us how many times we use the *base a* as a factor.

Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents has two forms depending on whether the exponent is larger in the numerator or the denominator. What if we just subtract exponents regardless of which is larger?

Use Scientific Notation

Working with very large or very small numbers can be awkward. Since our number system is base ten we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers 4,000 and 0.004.

Multiply Polynomials

Abstract:
By the end of this section, you will be able to:

* Multiply monomials
* Multiply a polynomial by a monomial
* Multiply a binomial by a binomial
* Multiply a polynomial by a polynomial
* Multiply special products
* Multiply polynomial functions

Multiply Monomials

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

Multiply a Polynomial by a Monomial

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

Multiply a Binomial by a Binomial

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

Multiply a Polynomial by a Polynomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.

Multiply Special Products

Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and multiplying them, there is less work to do if you learn to use a pattern. Let’s start by looking at three examples and look for a pattern.

Multiply Polynomial Functions

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

Dividing Polynomials

Abstract:
By the end of this section, you will be able to:

* Dividing monomials
* Dividing a polynomial by a monomial
* Dividing polynomials using long division
* Dividing polynomials using synthetic division
* Dividing polynomial functions
* Use the remainder and factor theorems

Dividing Monomials

We are now familiar with all the properties of exponents and used them to multiply polynomials. Next, we’ll use these properties to divide monomials and polynomials.

Divide a Polynomial by a Monomial

Now that we know how to divide a monomial by a monomial, the next procedure is to divide a polynomial of two or more terms by a monomial.

Divide Polynomials Using Long Division

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

Divide Polynomials using Synthetic Division

As we have mentioned before, mathematicians like to find patterns to make their work easier. Since long division can be tedious, let’s look back at the long division we did in [link] and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.

Divide Polynomial Functions

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

Use the Remainder and Factor Theorem

Let’s look at the division problems we have just worked that ended up with a remainder. They are summarized in the chart below. If we take the dividend from each division problem and use it to define a function, we get the functions shown in the chart. When the divisor is written as $x-c,$ the value of the function at $c,f\left(c\right),$ is the same as the remainder from the division problem.

Factoring

Introduction to Factoring

Greatest Common Factor and Factor by Grouping

Abstract:
By the end of this section, you will be able to:

* Find the greatest common factor of two or more expressions
* Factor the greatest common factor from a polynomial
* Factor by grouping

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will reverse this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

Factor the Greatest Common Factor from a Polynomial

It is sometimes useful to represent a number as a product of factors, for example, 12 as $2·6$ or $3·4.$ In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as $3{x}^{2}+15x,$ and end with its factors, $3x\left(x+5\right).$ To do this we apply the Distributive Property “in reverse.”

Factor by Grouping

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

Factor Trinomials

Abstract:
By the end of this section, you will be able to:

* Factor trinomials of the form ${x}^{2}+bx+c$
* Factor trinomials of the form $a{x}^{2}+bx+c$ using trial and error
* Factor trinomials of the form $a{x}^{2}+bx+c$ using the ‘ac’ method
* Factor using substitution

Factor Trinomials of the Form ${x}^{2}+bx+c$

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.

Factor Trinomials of the form *ax*2 + *bx* + *c* using Trial and Error

Our next step is to factor trinomials whose leading coefficient is not 1, trinomials of the form $a{x}^{2}+bx+c.$

Factor Trinomials of the Form $a{x}^{2}+bx+c$ using the “ac” Method

Another way to factor trinomials of the form $a{x}^{2}+bx+c$ 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!

Factor Using Substitution

Sometimes a trinomial does not appear to be in the $a{x}^{2}+bx+c$ form. However, we can often make a thoughtful substitution that will allow us to make it fit the $a{x}^{2}+bx+c$ form. This is called factoring by substitution. It is standard to use *u* for the substitution.

Factor Special Products

Abstract:
By the end of this section, you will be able to:

* Factor perfect square trinomials
* Factor differences of squares
* Factor sums and differences of cubes

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.

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:

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.

General Strategy for Factoring Polynomials

Abstract:
By the end of this section, you will be able to:

* Recognize and use the appropriate method to factor a polynomial completely

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.

Polynomial Equations

Abstract:
By the end of this section, you will be able to:

* Use the Zero Product Property
* Solve quadratic equations by factoring
* Solve equations with polynomial functions
* Solve applications modeled by polynomial equations

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.

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, $a{x}^{2}+bx+c=0.$ Then we must factor the expression on the left.

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.

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.

Rational Expressions and Functions

Introduction

Multiply and Divide Rational Expressions

Abstract:
By the end of this section, you will be able to:

* Determine the values for which a rational expression is undefined
* Simplify rational expressions
* Multiply rational expressions
* Divide rational expressions
* Multiply and divide rational functions

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.

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.

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.

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.

Multiply and Divide Rational Functions

We started this section stating that a rational expression is an expression of the form $\frac{p}{q},$ where *p* and *q* are polynomials and $q\ne 0.$ Similarly, we define a rational function as a function of the form $R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ where $p\left(x\right)$ and $q\left(x\right)$ are polynomial functions and $q\left(x\right)$ is not zero.

Add and Subtract Rational Expressions

Abstract:
By the end of this section, you will be able to:

* Add and subtract rational expressions with a common denominator
* Add and subtract rational expressions whose denominators are opposites
* Find the least common denominator of rational expressions
* Add and subtract rational expressions with unlike denominators
* Add and subtract rational functions

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.

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 $\frac{-1}{-1}.$

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.

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.

Add and subtract rational functions

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

Simplify Complex Rational Expressions

Abstract:
By the end of this section, you will be able to:

* Simplify a complex rational expression by writing it as division
* Simplify a complex rational expression by using the LCD

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:

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.

Solve Rational Equations

Abstract:
By the end of this section, you will be able to:

* Solve rational equations
* Use rational functions
* Solve a rational equation for a specific variable

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.

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.

Solve Applications with Rational Equations

Abstract:
By the end of this section, you will be able to:

* Solve proportions
* Solve similar figure applications
* Solve uniform motion applications
* Solve work applications
* Solve direct variation problems
* Solve inverse variation problems

Solve Proportions

When two rational expressions are equal, the equation relating them is called a proportion.

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.

Solve Uniform Motion Applications

We have solved uniform motion problems using the formula $D=rt$ in previous chapters. We used a table like the one below to organize the information and lead us to the equation.

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?

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.

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 $y=\frac{k}{x}.$

Solve Rational Inequalities

Abstract:
By the end of this section, you will be able to:

* Solve rational inequalities
* Solve an inequality with rational functions

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.

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.

Roots and Radicals

Introduction

Simplify Expressions with Roots

Abstract:
By the end of this section, you will be able to:

* Simplify expressions with roots
* Estimate and approximate roots
* Simplify variable 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 ${n}^{2}$ and read it ‘*n* squared’. This number is called the square of *n*, and *n* is called the square root. For example,

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 $\sqrt{4}=2,$ what is the value of $\sqrt[]{21}$ or $\sqrt[3]{50}?$ In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.

Simplify Variable Expressions with Roots

The odd root of a number can be either positive or negative. For example,

Simplify Radical Expressions

Abstract:
By the end of this section, you will be able to:

* Use the Product Property to simplify radical expressions
* Use the Quotient Property to 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.

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.

Simplify Rational Exponents

Abstract:
By the end of this section, you will be able to:

* Simplify expressions with ${a}^{\frac{1}{n}}$
* Simplify expressions with ${a}^{\frac{m}{n}}$
* Use the properties of exponents to simplify expressions with rational exponents

Simplify Expressions with ${a}^{\frac{1}{n}}$

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.

Simplify Expressions with ${a}^{\frac{m}{n}}$

We can look at ${a}^{\frac{m}{n}}$ in two ways. Remember the Power Property tells us to multiply the exponents and so ${\left({a}^{\frac{1}{n}}\right)}^{m}$ and ${\left({a}^{m}\right)}^{{}^{\frac{1}{n}}}$ both equal ${a}^{\frac{m}{n}}.$ If we write these expressions in radical form, we get

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.

Add, Subtract, and Multiply Radical Expressions

Abstract:
By the end of this section, you will be able to:

* Add and subtract radical expressions
* Multiply radical expressions
* Use polynomial multiplication to 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.

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.

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.

Divide Radical Expressions

Abstract:
By the end of this section, you will be able to:

* Divide radical expressions
* Rationalize a one term denominator
* Rationalize a two term denominator

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.

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!

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.

Solve Radical Equations

Abstract:
By the end of this section, you will be able to:

* Solve radical equations
* Solve radical equations with two radicals
* Use radicals in applications

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.

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.

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.

Use Radicals in Functions

Abstract:
By the end of this section, you will be able to:

* Evaluate a radical function
* Find the domain of a radical function
* Graph radical 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.

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.

Graph Radical Functions

Before we graph any radical function, we first find the domain of the function. For the function, $f\left(x\right)=\sqrt{x},$ the index is even, and so the radicand must be greater than or equal to 0.

Use the Complex Number System

Abstract:
By the end of this section, you will be able to:

* Evaluate the square root of a negative number
* Add and subtract complex numbers
* Multiply complex numbers
* Divide complex numbers
* Simplify powers of $i$

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 $\sqrt{-1},$ we are looking for a real number *x* so that *x*2 = –1. Since all real numbers squared are positive numbers, there is no real number that equals –1 when squared.

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.

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.

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.

Simplify Powers of *i*

The powers of $i$ make an interesting pattern that will help us simplify higher powers of *i*. Let’s evaluate the powers of $i$ to see the pattern.

Quadratic Equations and Functions

Introduction

Solve Quadratic Equations Using the Square Root Property

Abstract:
By the end of this section, you will be able to:

* Solve quadratic equations of the form ${ax}^{2}=k$ using the Square Root Property
* Solve quadratic equations of the form $a{\left(x–h\right)}^{2}=k$ using the Square Root Property

Solve Quadratic Equations of the form $a{x}^{2}=k$ using the Square Root Property

We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation *x*2 = 9.

Solve Quadratic Equations of the Form *a*(*x* − *h*)2 = *k* Using the Square Root Property

We can use the Square Root Property to solve an equation of the form *a*(*x* − *h*)2 = *k* as well. Notice that the quadratic term, *x*, in the original form *ax*2 = *k* is replaced with (*x* − *h*).

Solve Quadratic Equations by Completing the Square

Abstract:
By the end of this section, you will be able to:

* Complete the square of a binomial expression
* Solve quadratic equations of the form ${x}^{2}+bx+c=0$ by completing the square
* Solve quadratic equations of the form ${ax}^{2}+bx+c=0$ by completing the square

Complete the Square of a Binomial Expression

In the last section, we were able to use the Square Root Property to solve the equation (*y* − 7)2 = 12 because the left side was a perfect square.

Solve Quadratic Equations of the Form *x*2 + *bx* + *c* = 0 by Completing the Square

In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a quadratic equation by completing the square too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.

Solve Quadratic Equations of the Form *ax*2 + *bx* + *c* = 0 by Completing the Square

The process of completing the square works best when the coefficient of *x*2 is 1, so the left side of the equation is of the form *x*2 + *bx* + *c*. If the *x*2 term has a coefficient other than 1, we take some preliminary steps to make the coefficient equal to 1.

Solve Quadratic Equations Using the Quadratic Formula

Abstract:
By the end of this section, you will be able to:

* Solve quadratic equations using the Quadratic Formula
* Use the discriminant to predict the number and type of solutions of a quadratic equation
* Identify the most appropriate method to use to solve a quadratic equation

When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.

Use the Discriminant to Predict the Number and Type of Solutions of a Quadratic Equation

When we solved the quadratic equations in the previous examples, sometimes we got two real solutions, one real solution, and sometimes two complex solutions. Is there a way to predict the number and type of solutions to a quadratic equation without actually solving the equation?

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

We summarize the four methods that we have used to solve quadratic equations below.

Solve Equations in Quadratic Form

Abstract:
By the end of this section, you will be able to:

* Solve equations in quadratic form

Solve Equations in Quadratic Form

Sometimes when we factored trinomials, the trinomial did not appear to be in the *ax*2 + *bx* + *c* form. So we factored by substitution allowing us to make it fit the *ax*2 + *bx* + *c* form. We used the standard $u$ for the substitution.

Solve Applications of Quadratic Equations

Abstract:
By the end of this section, you will be able to:

* Solve applications modeled by quadratic equations

Solve Applications Modeled by Quadratic Equations

We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications.

Graph Quadratic Functions Using Properties

Abstract:
By the end of this section, you will be able to:

* Recognize the graph of a quadratic function
* Find the axis of symmetry and vertex of a parabola
* Find the intercepts of a parabola
* Graph quadratic functions using properties
* Solve maximum and minimum applications

Recognize the Graph of a Quadratic Function

Previously we very briefly looked at the function $f\left(x\right)={x}^{2}$, which we called the square function. It was one of the first non-linear functions we looked at. Now we will graph functions of the form $f\left(x\right)=a{x}^{2}+bx+c$ if $a\ne 0.$ We call this kind of function a quadratic function.

Find the Axis of Symmetry and Vertex of a Parabola

Look again at [link]. Do you see that we could fold each parabola in half and then one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry of the parabola.

Find the Intercepts of a Parabola

When we graphed linear equations, we often used the *x*- and *y*-intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.

Now we have all the pieces we need in order to graph a quadratic function. We just need to put them together. In the next example we will see how to do this.

Solve Maximum and Minimum Applications

Knowing that the vertex of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic function. The *y*-coordinate of the vertex is the minimum value of a parabola that opens upward. It is the maximum value of a parabola that opens downward. See [link].

Graph Quadratic Functions Using Transformations

Abstract:
By the end of this section, you will be able to:

* Graph quadratic equations of the form $f\left(x\right)={x}^{2}+k$
* Graph quadratic functions of the form $f\left(x\right)={\left(x-h\right)}^{2}$
* Graph quadratic functions of the form $f\left(x\right)=a{x}^{2}$
* Graph quadratic functions using transformations
* Find a quadratic function from its graph

Graph Quadratic Functions of the form $f\left(x\right)={x}^{2}+k$

In the last section, we learned how to graph quadratic functions using their properties. Another method involves starting with the basic graph of $f\left(x\right)={x}^{2}$ and ‘moving’ it according to information given in the function equation. We call this graphing quadratic functions using transformations.

Graph Quadratic Functions of the form $f\left(x\right)={\left(x-h\right)}^{2}$

In the first example, we graphed the quadratic function $f\left(x\right)={x}^{2}$ by plotting points and then saw the effect of adding a constant *k* to the function had on the resulting graph of the new function $f\left(x\right)={x}^{2}+k.$

Graph Quadratic Functions of the Form $f\left(x\right)=a{x}^{2}$

So far we graphed the quadratic function $f\left(x\right)={x}^{2}$ and then saw the effect of including a constant *h* or *k* in the equation had on the resulting graph of the new function. We will now explore the effect of the coefficient *a* on the resulting graph of the new function $f\left(x\right)=a{x}^{2}.$

We have learned how the constants *a*, *h*, and *k* in the functions, $f\left(x\right)={x}^{2}+k,{\rule{0.2em}{0ex}}f\left(x\right)={\left(x-h\right)}^{2},$ and $f\left(x\right)=a{x}^{2}$ affect their graphs. We can now put this together and graph quadratic functions $f\left(x\right)=a{x}^{2}+bx+c$ by first putting them into the form $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ by completing the square. This form is sometimes known as the vertex form or standard form.

Find a Quadratic Function from its Graph

So far we have started with a function and then found its graph.

Solve Quadratic Inequalities

Abstract:
By the end of this section, you will be able to:

* Solve quadratic inequalities graphically
* Solve quadratic inequalities algebraically

Solve Quadratic Inequalities Graphically

A quadratic equation is in standard form when written as *ax*2 + *bx* + *c* = 0. If we replace the equal sign with an inequality sign, we have a quadratic inequality in standard form.

Solve Quadratic Inequalities Algebraically

The algebraic method we will use is very similar to the method we used to solve rational inequalities. We will find the critical points for the inequality, which will be the solutions to the related quadratic equation. Remember a polynomial expression can change signs only where the expression is zero.

Exponential and Logarithmic Functions

Introduction

Finding Composite and Inverse Functions

Abstract:
By the end of this section, you will be able to:

* Find and evaluate composite functions
* Determine whether a function is one-to-one
* Find the inverse of a function

Find and Evaluate Composite Functions

Before we introduce the functions, we need to look at another operation on functions called composition. In composition, the output of one function is the input of a second function. For functions $f$ and $g,$ the composition is written $f\circ g$ and is defined by $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right).$

Determine Whether a Function is One-to-One

When we first introduced functions, we said a function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each *x*-value is matched with only one *y*-value.

Find the Inverse of a Function

Let’s look at a one-to one function, $f$, represented by the ordered pairs $\left\{\left(0,5\right),\left(1,6\right),\left(2,7\right),\left(3,8\right)\right\}.$ For each $x$-value, $f$ adds 5 to get the $y$-value. To ‘undo’ the addition of 5, we subtract 5 from each $y$-value and get back to the original $x$-value. We can call this “taking the inverse of $f$” and name the function ${f}^{-1}.$

Evaluate and Graph Exponential Functions

Abstract:
By the end of this section, you will be able to:

* Graph exponential functions
* Solve Exponential equations
* Use exponential models in applications

Graph Exponential Functions

The functions we have studied so far do not give us a model for many naturally occurring phenomena. From the growth of populations and the spread of viruses to radioactive decay and compounding interest, the models are very different from what we have studied so far. These models involve exponential functions.

Solve Exponential Equations

Equations that include an exponential expression ${a}^{x}$ are called exponential equations. To solve them we use a property that says as long as $a>0$ and $a\ne 1,$ if ${a}^{x}={a}^{y}$ then it is true that $x=y.$ In other words, in an exponential equation, if the bases are equal then the exponents are equal.

Use Exponential Models in Applications

Exponential functions model many situations. If you own a bank account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, *P*, is invested at an interest rate, *r*, for *t* years, the new balance, *A*, will depend on how often the interest is compounded. If the interest is compounded *n* times a year we use the formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}.$ If the interest is compounded continuously, we use the formula $A=P{e}^{rt}.$ These are the formulas for compound interest.

Evaluate and Graph Logarithmic Functions

Abstract:
By the end of this section, you will be able to:

* Convert between exponential and logarithmic form
* Evaluate logarithmic functions
* Graph Logarithmic functions
* Solve logarithmic equations
* Use logarithmic models in applications

Convert Between Exponential and Logarithmic Form

Since the equations $y={\text{log}}_{a}x$ and $x={a}^{y}$ are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth let’s take a close look at the equations. See [link]. Notice the positions of the exponent and base.

Evaluate Logarithmic Functions

We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential equation.

Graph Logarithmic Functions

To graph a logarithmic function $y={\text{log}}_{a}x,$ it is easiest to convert the equation to its exponential form, $x={a}^{y}.$ Generally, when we look for ordered pairs for the graph of a function, we usually choose an *x*-value and then determine its corresponding *y*-value. In this case you may find it easier to choose *y*-values and then determine its corresponding *x*-value.

Solve Logarithmic Equations

When we talked about exponential functions, we introduced the number *e*. Just as *e* was a base for an exponential function, it can be used a base for logarithmic functions too. The logarithmic function with base *e* is called the natural logarithmic function. The function $f\left(x\right)={\text{log}}_{e}x$ is generally written $f\left(x\right)=\text{ln}{\rule{0.2em}{0ex}}x$ and we read it as “el en of $x.”$

Use Logarithmic Models in Applications

There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from 0, which is barely audible to 160, which can rupture an eardrum. The ${10}^{-12}$ in the formula represents the intensity of sound that is barely audible.

Use the Properties of Logarithms

Abstract:
By the end of this section, you will be able to:

* Use the properties of logarithms
* Use the Change of Base Formula

Use the Properties of Logarithms

Now that we have learned about exponential and logarithmic functions, we can introduce some of the properties of logarithms. These will be very helpful as we continue to solve both exponential and logarithmic equations.

Use the Change-of-Base Formula

To evaluate a logarithm with any other base, we can use the Change-of-Base Formula. We will show how this is derived.

Solve Exponential and Logarithmic Equations

Abstract:
By the end of this section, you will be able to:

* Solve logarithmic equations using the properties of logarithms
* Solve exponential equations using logarithms
* Use exponential models in applications

Solve Logarithmic Equations Using the Properties of Logarithms

In the section on logarithmic functions, we solved some equations by rewriting the equation in exponential form. Now that we have the properties of logarithms, we have additional methods we can use to solve logarithmic equations.

Solve Exponential Equations Using Logarithms

In the section on exponential functions, we solved some equations by writing both sides of the equation with the same base. Next we wrote a new equation by setting the exponents equal.

Use Exponential Models in Applications

In previous sections we were able to solve some applications that were modeled with exponential equations. Now that we have so many more options to solve these equations, we are able to solve more applications.

Conics

Introduction

Distance and Midpoint Formulas; Circles

Abstract:
By the end of this section, you will be able to:

* Use the Distance Formula
* Use the Midpoint Formula
* Write the equation of a circle in standard form
* Graph a circle

Use the Distance Formula

We have used the Pythagorean Theorem to find the lengths of the sides of a right triangle. Here we will use this theorem again to find distances on the rectangular coordinate system. By finding distance on the rectangular coordinate system, we can make a connection between the geometry of a conic and algebra—which opens up a world of opportunities for application.

Use the Midpoint Formula

It is often useful to be able to find the midpoint of a segment. For example, if you have the endpoints of the diameter of a circle, you may want to find the center of the circle which is the midpoint of the diameter. To find the midpoint of a line segment, we find the average of the *x*-coordinates and the average of the *y*-coordinates of the endpoints.

Write the Equation of a Circle in Standard Form

As we mentioned, our goal is to connect the geometry of a conic with algebra. By using the coordinate plane, we are able to do this easily.

Graph a Circle

Any equation of the form ${\left(x-h\right)}^{2}+{\left(y-k\right)}^{2}={r}^{2}$ is the standard form of the equation of a circle with center, $\left(h,k\right),$ and radius, *r.* We can then graph the circle on a rectangular coordinate system.

Parabolas

Abstract:
By the end of this section, you will be able to:

* Graph vertical parabolas
* Graph horizontal parabolas
* Solve applications with parabolas

Graph Vertical Parabolas

The next conic section we will look at is a parabola. We define a parabola as all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the **focus,** and the fixed line is called the **directrix** of the parabola.

Graph Horizontal Parabolas

Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the *x* and *y* in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.

Solve Applications with Parabolas

Many architectural designs incorporate parabolas. It is not uncommon for bridges to be constructed using parabolas as we will see in the next example.

Ellipses

Abstract:
By the end of this section, you will be able to:

* Graph an ellipse with center at the origin
* Find the equation of an ellipse with center at the origin
* Graph an ellipse with center not at the origin
* Solve application with ellipses

Graph an Ellipse with Center at the Origin

The next conic section we will look at is an ellipse. We define an ellipse as all points in a plane where the sum of the distances from two fixed points is constant. Each of the given points is called a **focus** of the ellipse.

Find the Equation of an Ellipse with Center at the Origin

If we are given the graph of an ellipse, we can find the equation of the ellipse.

Graph an Ellipse with Center Not at the Origin

The ellipses we have looked at so far have all been centered at the origin. We will now look at ellipses whose center is $\left(h,k\right).$

Hyperbolas

Abstract:
By the end of this section, you will be able to:

* Graph a hyperbola with center at $\left(0,0\right)$
* Graph a hyperbola with center at $\left(h,k\right)$
* Identify conic sections by their equations

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

The last conic section we will look at is called a hyperbola. We will see that the equation of a hyperbola looks the same as the equation of an ellipse, except it is a difference rather than a sum. While the equations of an ellipse and a hyperbola are very similar, their graphs are very different.

Graph a Hyperbola with Center at $\left(h,k\right)$

Hyperbolas are not always centered at the origin. When a hyperbola is centered at $\left(h,k\right)$ the equations changes a bit as reflected in the table.

Identify Conic Sections by their Equations

Now that we have completed our study of the conic sections, we will take a look at the different equations and recognize some ways to identify a conic by its equation. When we are given an equation to graph, it is helpful to identify the conic so we know what next steps to take.

Solve Systems of Nonlinear Equations

Abstract:
By the end of this section, you will be able to:

* Solve a system of nonlinear equations using graphing
* Solve a system of nonlinear equations using substitution
* Solve a system of nonlinear equations using elimination
* Use a system of nonlinear equations to solve applications

Solve a System of Nonlinear Equations Using Graphing

We learned how to solve systems of linear equations with two variables by graphing, substitution and elimination. We will be using these same methods as we look at nonlinear systems of equations with two equations and two variables. A **system of nonlinear equations** is a system where at least one of the equations is not linear.

Solve a System of Nonlinear Equations Using Substitution

The graphing method works well when the points of intersection are integers and so easy to read off the graph. But more often it is difficult to read the coordinates of the points of intersection. The substitution method is an algebraic method that will work well in many situations. It works especially well when it is easy to solve one of the equations for one of the variables.

Solve a System of Nonlinear Equations Using Elimination

When we studied systems of linear equations, we used the method of elimination to solve the system. We can also use elimination to solve systems of nonlinear equations. It works well when the equations have both variables squared. When using elimination, we try to make the coefficients of one variable to be opposites, so when we add the equations together, that variable is eliminated.

Use a System of Nonlinear Equations to Solve Applications

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

Sequences, Series and Binomial Theorem

Introduction

Sequences

Abstract:
By the end of this section, you will be able to:

* Write the first few terms of a sequence
* Find a formula for the general term (nth term) of a sequence
* Use factorial notation
* Find the partial sum
* Use summation notation to write a sum

Write the First Few Terms of a Sequence

Let’s look at the function $f\left(x\right)=2x$ and evaluate it for just the counting numbers.

Find a Formula for the General Term (*n*th Term) of a Sequence

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

Use Factorial Notation

Sequences often have terms that are products of consecutive integers. We indicate these products with a special notation called *factorial notation*. For example,$5!$, read 5 factorial, means $5·4·3·2·1.$ The exclamation point is not punctuation here; it indicates the factorial notation.

Find the Partial Sum

Sometimes in applications, rather than just list the terms, it is important for us to add the terms of a sequence. Rather than just connect the terms with plus signs, we can use summation notation.

Use Summation Notation to Write a Sum

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

Arithmetic Sequences

Abstract:
By the end of this section, you will be able to:

* Determine if a sequence is arithmetic
* Find the general term ($n$th term) of an arithmetic sequence
* Find the sum of the first $n$ terms of an arithmetic sequence

Determine if a Sequence is Arithmetic

The last section introduced sequences and now we will look at two specific types of sequences that each have special properties. In this section we will look at arithmetic sequences and in the next section, geometric sequences.

Find the General Term (*n*th Term) of an Arithmetic Sequence

Just as we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.

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

As with the general sequences, it is often useful to find the sum of an arithmetic sequence. The sum, ${S}_{n},$ of the first $n$ terms of any arithmetic sequence is written as ${S}_{n}={a}_{1}+{a}_{2}+{a}_{3}+...+{a}_{n}.$ To find the sum by merely adding all the terms can be tedious. So we can also develop a formula to find the sum of a sequence using the first and last term of the sequence.

Geometric Sequences and Series

Abstract:
By the end of this section, you will be able to:

* Determine if a sequence is geometric
* Find the general term (nth term) of a geometric sequence
* Find the sum of the first $n$ terms of a geometric sequence
* Find the sum of an infinite geometric series
* Apply geometric sequences and series in the real world

Determine if a Sequence is Geometric

We are now ready to look at the second special type of sequence, the geometric sequence.

Find the General Term (*n*th Term) of a Geometric Sequence

Just as we found a formula for the general term of a sequence and an arithmetic sequence, we can also find a formula for the general term of a geometric sequence.

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

We found the sum of both general sequences and arithmetic sequence. We will now do the same for geometric sequences. The sum,${S}_{n},$ of the first *n* terms of a geometric sequence is written as ${S}_{n}={a}_{1}+{a}_{2}+{a}_{3}+...+{a}_{n}.$ We can write this sum by starting with the first term, ${a}_{1},$ and keep multiplying by *r* to get the next term as:

Find the Sum of an Infinite Geometric Series

If we take a geometric sequence and add the terms, we have a sum that is called a geometric series. An infinite geometric series is an infinite sum whose first term is ${a}_{1}$ and common ratio is *r* and is written

Apply Geometric Sequences and Series in the Real World

One application of geometric sequences has to do with consumer spending. If a tax rebate is given to each household, the effect on the economy is many times the amount of the individual rebate.

Binomial Theorem

Abstract:
By the end of this section, you will be able to:

* Use Pascal’s Triangle to expand a binomial
* Evaluate a binomial coefficient
* Use the Binomial Theorem to expand a binomial

Use Pascal’s Triangle to Expand a Binomial

In our previous work, we have squared binomials either by using FOIL or by using the Binomial Squares Pattern. We can also say that we expanded ${\left(a+b\right)}^{2}.$

Evaluate a Binomial Coefficient

While Pascal’s Triangle is one method to expand a binomial, we will also look at another method. Before we get to that, we need to introduce some more factorial notation. This notation is not only used to expand binomials, but also in the study and use of probability.

Use the Binomial Theorem to Expand a Binomial

We are now ready to use the alternate method of expanding binomials. The Binomial Theorem uses the same pattern for the variables, but uses the binomial coefficient for the coefficient of each term.