Basic Algebra I Classnotes:A companion to Beginning Algebra by Lial, Hornsby, and McGinnis

Prepared by Richard G. Ligo

Dear reader,

Thank you for using this resource! These classnotes strive to be a source of assistancefor both instructors and students, bringing focus and emphasis to key ideas necessary forfurthering one’s education in mathematics, algebra in particular.

To instructors: You will hopefully find that these notes have the ability to streamline yourteaching. Effectively covering the entirety of Lial, Hornsby, and McGinnis’s text in a semestercan initially be an intimidating goal, but these notes attempt to boil their work down toits essentials. Ultimately, the notes help provide structure to one’s lecture while providingadditional examples to the students. If you have an easy method by which to disseminatecompleted copies of these notes to students, I highly recommend that one do so. I havefound that such tactics have the ability to significantly minimize problems caused by studentabsences.

To students: I’m happy that you have decided to step into the world of algebra! While itmay not seem like the most thrilling topic in the world, I encourage you to find the beautyburied in the structure and organization of the subject. With regard to these notes, I wouldhope that you would find them both straightforward and enlightening. These notes arenot intended to replace your text, but rather, provide an additional (sometimes simpler)perspective. Good luck!

Best,

Rich

Disclaimer: The problems contained within these notes are both of my creation, as well asthat of Lial, Hornsby, and McGinnis. I also do not claim that these notes (or my accompa-nying solutions) are entirely free of typographical errors.

2

1.1 Fractions Name:

What is the simplest number?

Our ancestors’ first numbers were , , and .

From this idea, we get the , which are the set:

The next important number after 1 is .

By adding zero to the set of natural numbers, we get the ,which are the set:

If we divide whole numbers, we sometimes get other whole numbers.In we have the case where integers do not divide evenly, we get a .

Any decimal that can be represented as a or eventuallydecimal is a rational number.

Several techniques useful for dealing with fractions, but is one of the mostimportant. Build some factor trees:

Factoring helps us to write a fraction in . Consider thefollowing fractions:

12

18=

36

54=

3

This idea can be connected with the multiplication of fractions:

3

8· 4

9=

When we fractions, we simply add an extra step before multiplication:

5

9÷ 2

3=

Adding and subtracting fractions is somewhat more complicated. To add or subtract tofractions, the fractions MUST

Ex 1:7

15+

4

15=

Ex 2: 42

3+2

1

6=

Ex 3:7

12−1

9=

Ex 4: 51

3−4

1

2=

Ex 5:5

6+

2

9=

4

1.2 Order of Operations Name:

Build the factor tree for 16:

We can then also represent 16 as . Then we have 33 = .

The tells us how to evaluate a mathematicalexpression like

2[7+3(4+5)] =

The order of operations as four steps:

1)

2)

3)

4)

Use the order of operations to evaluate

42[(13+4)−8] =

We have six symbols to compare numbers, which are:

Give an example using each symbol.

5

The following are several examples intended to build skills.

Ex 1: Reduce 144120

Ex 2: Evaluate2

3· 15

16

Ex 3: Evaluate3

4÷ 12

Ex 4: Evaluate7

12+

1

3

Ex 5: Evaluate5

8− 1

2

Ex 6: Evaluate6(5 + 1)− 9(1 + 1)

5(8− 6)− 23

Ex 7: Evaluate both sides and determine if the inequality is true:3 + 5(4− 1)

2 · 4 + 1≥ 3

6

1.3 Variables Name:

It is important to distinguish between equations and expressions. Give an example of each:

A represents an unknown or unspecified quantity.

Given that x = 4 and y = 1, evaluate the following expressions:

Ex 1: 2y2 +4x

2

Ex 2: 2(x+ 3y)

Determine if x = 2 is a solution to the following equations:

Ex 3:3

8x+

1

4= 1

Ex 4: 6x+ 2(x+ 3) = 14

7

1.4 Real Numbers & The Line Name:

We can classify all the numbers we know into several types:

The is one of the most important tools for under-standing the real numbers. Construct it and illustrate the points 0, −1, π, and 1

2:

As the real numbers are , we may write a corresponding an in-equality for any pair of real numbers. Specify a correct inequality for the following pairs ofnumbers:

1 0 −13 −10 −3 4 π4

3

We describe a number’s distance from 0 using

Evaluate the following expressions:

| − 5| |1− 8| −2|4− 7| 6

∣∣∣∣−4

3

∣∣∣∣

8

1.5 & 1.6 More on Real Numbers Name:

Real numbers become significantly more useful once we understand how to add, subtract,multiply and divide them. There are two main keys to victory here:

1) When adding and subtracting, think of the .

2) When multiplying and dividing, we have an important rule:

Evaluate the following expressions using our new knowledge!

Ex 1: −9 + [(3− 2)− (−4 + 2)]

Ex 2: (8− 1)− 12

Ex 3:32 − 42

7(−8 + 9)

Ex 4: (5− 12)(19− 4)

Ex 5:−21(3)

−3− 6

9

1.7 Real Number Properties Name:

There are properties of the real numbers that have the ability to make our work both fasterand easier. Some of them you may already use without realizing it, in fact. Rewrite anequivalent formulation of each of the following expressions:

1 + 2 = The property

2 · 3 =

1 + (2 + 3) = The property

5 · (2 · 3) =

1 + 0 = The property

1

2· 1 =

Fill in the blank:

5 + = 0 The property

3 · = 1

We can write these properties for arbitrary numbers a, b, and c as

The commutative property:

The associative property:

The identity property:

The inverse property:

10

The fifth property we will learn today is a bit more interesting than the previous four.Evaluate the following expressions:

2(3 + 4) =

2 · 3 + 2 · 4 =

What do you notice here? Rewrite the following expression using what you just observed:

6(2+5)=

This is called the property. Written in terms of arbitrary num-bers a, b, and c, this looks like:

The distributive property is most useful when we are dealing with variables, as it allows us tosimplify some messy expressions into nicer forms. Use the distributive property to simplifythe following examples:

Ex 1: 8(x+ 5)

Ex 2: −2(2− x)

Ex 3: −(3x+ 2− y)

Ex 4:−1

3(21− 15x)

11

1.8 Simplifying Expressions Name:

Unecessary mathematical vocabulary can sometimes complicate what might otherwise besimple questions. There are certain occasions, however, when vocabulary is incredibly use-ful, as it helps us verbalize mathematics in a precise and unambiguous way.

Circle the coefficients and box the terms in the following expression:

2x+1

2y2 − z + 10

How do we deal with −z?

We can then operationally define the following terms:

coefficient:

term:

We say that two terms are if they have exactly the same vari-ables. Group the following terms into like terms:

7x2, 7y, 7x, 7xy, 4x, −2xy, x2, and 50000x.

If we know that two terms are like terms, we are allowed to add their coefficients. Simplifythe following by combining like terms:

15x2 + 3x2 =

y − 2y + 10y =

20123529x5y9z2 − x5y9z2 =

x+ y =

2x+ 3y − 5x+ 7z =

12

All of the knowledge acquired in Chapter 1 is combined in the task of simplifying expressions,and these skills will become invaluable later when we begin solving equations.

Ex 1: −10 + x+ 4x− 7− 4x =

Ex 2:−5

6+ 8x+

1

6x− 7− 7

6=

Ex 3: −5(5y − 9) + 3(3y + 6) =

Ex 4: −4− 5(t− 13) =

Ex 5: 15000(x+ y + z) =

Ex 6:1

6x− 1

8x+

1

2x2

13

2.1-2.3 Solving Linear Equations Name:

Solving linear equations is a universal skill that is crucial to success in algebra.

An equation is said to be if the variable is to the power.Determine the value of x satisfying the following equation:

x+ 2 = 6 x− 2 = 6 2x = 6x

2= 6

There is a method for approaching these problems in a more mechanical way. Consider thefollowing equation:

x− 15 = 3

It is crucial to remember that anything done to one side of the equation must be done to theother side of the equation. (This is because we must maintain equality!) Solve the followingequations for x:

5 = x+ 6 9− x = 3 3 = −x+ 3

Sometimes x is multiplied by something. Fortunately, we also have a mathematical way fordealing with this situation. Consider the following equation:

7x = −28

As before, it is crucial to perform the same operation on both sides of the equation. Solvethe following equations for x:

80 = 4x 57x = 10 0 = 4323x

14

We can now combine what we know to solve more complicated equations. Solve for x ineach of the following examples:

4x+ 3x = 21 8x+ x = −3 + 4x 6(2x+ 8) = 4(x− 6)

There are two strange cases that can occur when attempting to solve for x. Attempt to solvethe following equations for x:

4(x+ 2)− 2 = 4x+ 6 6x+ 1 = 3(2x+ 2)

Do either of these statements make sense? Think about what values of x satisfy the firstequation, then think about what values of x satisfy the second equation.

The first equation has solutions.

The second equation has solutions.

If an equation contains annoying fractions,can transform the equation into something much nicer. Solve the following equation for x:

2

3x− 1

2x = −1

6x− 2

15

Linear Equation Practice Name:

Solve the following equations for x.

Ex 1: 11x− 5(x+ 2) = 6x+ 5x Ex 2: 7x− 5x+ 2 = 5x+ 2− x

Ex 3: 2x+ 9 = 4x+ 11 Ex 4: 6(3x+ 5) = 2(10x+ 10)

Ex 5: 11x− 5(x+ 2) = 6x+ 5 Ex 6: 3(2x− 4) = 6(x− 2)

16

Ex 7: −(4x+ 2)− (−3x− 5) = 3 Ex 8: 12x− 5 = 11x+ 5− x

Ex 9:−1

4(x− 12) +

1

2(x+ 2) = x+ 4 Ex 10: 5x+ 8 = 7 + 3x

Ex 11: −16x− 3 = 13− 8x Ex 12: 19(x+ 18) + 1

3(2x+ 3) = x+ 3

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2.4 Linear Equation Applications Name:

A tired refrain found in many math classrooms is “when are we ever going to use this?” Thefollowing questions represent an attempt to answer that question.

Ex 1: If 4 is multiplied by a number decreased by 7, the product is 100. Find the number.

There are six steps to solving word problems.

1)

2)

3)

4)

5)

6)

Step 3 is typically the most challenging part, so let’s look at some more examples.

Ex 2: Rich plays way too much Halo 4. At the end of a match, Rich’s team had 60 kills,and Rich’s teammates had twice as many kills as him. Determine how many kills Rich had.

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Ex 3: Rich needs to put antifreeze in his old Subaru so it won’t overheat. He knows that themixture he should use will contain 3 oz of water for every 2 oz of antifreeze. Additionally, heknows it will take 65 oz to fill his Subaru’s coolant reservoir. Determine how much antifreezeRich needs to buy.

Ex 4: Rich is both very hungry and somewhat obsessive-compulsive. He wants to eat aSnickers bar, but he decides that he needs two extra pieces for later. In particular, he wantsthe largest piece to be three times larger than the middle piece, which is two inches longerthan the shortest piece. Given that a Snickers bar in nine inches long, determine the size ofeach piece.

Ex 5: The first test for this class will take place near the end of the September, and Rich mighthold a review session the night before the test. The sum of the dates of these consecutivedays is 51; determine when the review session will occur.

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2.5 Linear Equations and Geometry Name:

Mathematics is often at its most powerful at the intersection point of seemingly disparatesubjects, and this is certainly true of algebra and geometry. Geometry allows mathematiciansto precisely describe physical phenomenon with formulas, whereas algebra allows mathemati-cians to precisely solve these formulas when only limited information is available.

Note that it very helpful to apply the same problem-solving strategies from the previoussection.

Ex 1: The Pentacrest is approximately 800 feet long from west to east. Given that the totalarea of the Pentacrest is 640000 square feet, determine its length from north to south. Thenuse this new information to determine its perimeter.

Sometimes problems are written in a way that all the geometric components are related viaa common variable. For example:

Ex 2: The playing field in Kinnick Stadium, being a regulation NCAA football field, has awidth that is 4

9ths of is length (including endzones). Given that its perimeter is 1040 feet,

determine the dimensions of the field.

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Ex 3: Here is a classic geometry problem: what Papa John’s pizza is the best value? A tableof sizes and prices is found below:

Size Diameter PriceSmall 10” $8.99Medium 12” $10.99Large 14” $12.99

Hint: First determine the area of each pizza, then determine the cost per square inch.

Ex 4: Rich really likes nachos, and decides to buy an enormous block of cheddar cheese tosupplement his addiction. He knows he wants 240 cubic inches of cheese, and the block itwill be cut from is 3 inches tall and 8 inches wide. How long of a piece should Rich buy?

21

2.6-2.7 Additional applications Name:

Ratios, proportions, and percents are useful mathematical tools that can be accessed throughthe use of linear equations.

A is a comparison of two quantities.

A says that two ratios are equal.

Ex 1: Rich is planning on driving back to Pennsylvania, but first he will need to get gas.He asks his roommate what gas cost at the Kum & Go, but all he remembered was that hepaid $32.31 for 9 gallons. Rich knows it will take 8 gallons to fill his Subaru. Determine howmuch Rich will spend on gas using a proportion.

Ex 2: The gravity on the Moon is 16th of that on the Earth. Given that Rich weighs 25

pounds on the Moon, determine how much he weighs on the Earth.

A is a ratio where the second number is always 100. Convert thefollowing percents to decimals, and vice-versa:

50% .12 2% 3.14 .1% .8888

Ex 3: The retail price for Halo 4 is $60, but Rich bought it on Amazon for 33% off. Determinehow much Rich paid.

22

Before we continue, we must recall the formulas for simple interest and distance travelled:

Ex 4: Rich is secretly a multi-millionaire who just does math for fun. He decides to invest$275,000 at 5% per year for a single year. Determine the interest this money makes.

Ex 5: Rich keeps all of his money in a walk-in vault in his basement. He grabs a suitcase ofmoney and knows that he has 80 more $100 bills than $50 bills and $98,000 total. Determinehow many Benjamins Rich is carrying.

Ex 6: Rich takes the 98k to the closest Audi dealer and purchases a 2014 Audi R8. Richthen drives home on I-80 at a constant speed of 110 mph. Given that home is 660 milesaway, determine how long it takes Rich to get home.

Ex 7: Rich’s brother Joe leaves from Pennsylvania at same time and drives at a constantspeed of 65 mph. How long will it be until Rich and Joe meet each other?

23

2.8 Solving Linear Inequalities Name:

We begin our brief study of linear inequalities by talking about interval notation. Rewriteeach interval using inequalities, then illustrate each of the following intervals on a numberline:

(−1, 2)

(−2,∞)

(−∞, 3]

[−2, 4)

A means that the endpoint is NOT included.

A means that the endpoint IS included.

Solving linear inequalities is very similar to solving linear equations, with one key difference:

Ex 1: Solve 3x+ 2− 5 > −x+ 7 + 2x, and graph the solution set.

24

Ex 2: Solve −4 ≤ 6− 2x ≤ 10, and graph the solution set.

Ex 3: Rich is dominating his topology class. He got a 98 on his first test, and a 96 on hissecond. Determine what he scores he could get on his last test and averge at least a 90overall.

25

3.1-3.2 Graphing Linear Equations Name:

We’ve previously talked about linear equations with only one variable, but we can also con-sider linear equations with two variables. For example:

A few facts about linear equations with more than one variable:

1) Such equations will have solutions.

2) Solutions for such equations are written as .

These pairs are called “ordered” because . Thefirst part will represent the value of x, whereas the second part will represent the value of y.Decide if the following ordered pairs represent a solution for the equation above:

(0,2) (0,1) (-3,4) (6,-2) (1,1)

If we are given only half of an ordered pair, we can use our equation to determine whatsecond value will make the ordered pair into a solution. Finish the following ordered pairs:

(0,?) (?,1)

This skill allows us to complete a table of values representing solutions to our original equa-tion. Complete the following table of values:

x y

-6

4

0

0

6

26

Ordered pairs have both an algebraic and geometric meaning. Plot the ordered pairs fromthe previous table on the following coordinate plane:

The solutions of the equation form a !

We can use a linear equation to describe real-world phenomenon. Consider the followingexample:

Rich’s stupid landlord requires him to steam-clean his carpet before moving out (its actuallywritten into the lease). Dollar General rents steam cleaners for $30 per day, and it also costs$12 for the cleaning solution. Write an equation relating the number of days Rich has thesteam cleaner and the total amount Rich must spend. Next, construct a table of values andgraph the function.

We now have three main techniques for graphing linear equations.

1)

2)

3)

27

We’ve already seen the first technique, so let’s talk about the second.

First, an is the point at which a line intersects an axis. If weknow both the x-intercept and y-intercept of a line, then we almost always know what therest of the line looks like.

Find the intercepts in the following examples and graph the corresponding line.

Ex 1: 2x− y = 4.

Ex 2: 6x+ 2y = 0.

As can be seen from Example 2, the “find the intercept” method fails when...

28

There happens to be two other exceptions that cause the “find the intercept” method to fail.Graph an example of each and write the corresponding equation:

In general, we may then conclude the following for any number n:

The equation will be a vertical line.

The equation will be a horizontal line.

Complete the following examples using whatever method you like:

Ex 3: x+ 3y = 9 Ex 4: x = 3

Ex 5: 8x− 2y = 0 Ex 6: 6x− 5y = 5

29

3.3-3.4 More About Graphing Name:

Thankfully, the slope of a line is a very intuitive idea. Illustrate and label an example ofpositive, negative, and zero slope:

We can talk precisely about the slope of a line through the use of the following formula:

This makes it very easy to determine the slope of a line through two points. Determine theslope through the following pairs of points:

(0, 0),(1, 1) (4,−2),(2, 4) (51, 27),(522, 27) (−1, 3),(5, 4) (1, 3),(1, 6)

Graph the line through the last pair of points.

A is said to have an slope.

30

Sometimes we are given an equation instead of ordered pairs or a graph. In this case, wemay find the slope by

The resulting equation is said to be in form. Inthis form, the coefficient of x will be the slope.

Graph 2x − y = 1 using a table of values, then determine the slope of the resulting line.Next, solve the equation for y and determine the coefficient of x.

We can also use the slope-intercept form to graph a line! This is significantly less tediousthan constructing a table of values. Slope-intercept form is given by the general equation:

Graph the following equations using the slope-intercept form method:

2x+ 3y = 3 2x− y = 4

31

Now graph the equations x− y = 0, x− y = 2, and x+ y = 2 on the same coordinate planeusing slope-intercept form:

Make some observations about the resulting graph.

Two lines will be when their slopes are equal.

Two lines will be when their slops are negative reciprocals.

Determine the slopes of a perpendicular line for each of the given lines:

y = −14x+ 2 y = 2x− 3 y = x+ 50000 y = 29

71x+ π

All of the ideas covered thus far in Chapter 3 now into combine into a powerful toolsetfor manipulating and understanding linear equations and their graphs. This allows us toconstruct both a graph and an equation for limited sets of information; for example

1)

2)

3)

32

We now examine an example of each case. Graph the line and write the correspondingequation for each set of information provided:

Ex 1: m = 2, b = −3

Ex 2: m = −13

, (6,−1)

Ex 3: (−1,−5), (2, 3)

33

3.6 Introduction to Functions Name:

Before we talk about functions, we must first visit about a more general idea.

A is a set of ordered pairs. Determine the domain and range of thefollowing relations:

Ex 1: {(0, 5), (1, 6), (2, 5), (3, 7), (4, 8)}

Ex 2: {(1,−2), (1,−5), (2, 2), (9, 7)}

The domain is the .

The range is the .

A function is a special type of relation where

Which of the above examples are functions?

Functions (and relations, for that matter) can even be represented by infinitely many or-dered pairs. In this case we often represent functions as an equation; write an example of afunction given by an equation:

We can naturally think of functions as a graph. Determine whether or not the followinggraphs are examples of functions:

The is a straightforward method used for de-termining whether or not a graph represents a function.

34

We can also talk about the domain and range of a graph. Determine the domains and rangesof the following graphs:

We can think of the domain as .

We can think of the range as .

Finally, we must cover a small piece of notation used to represent functions. For example,we will sometimes see a function given as

Evaluate this function at x = 0, x = 1, and x = −3:

Write three more examples of functions:

35

4.1 Solving Systems by Graphing Name:

Graph the equations y = x+2 and y = 2x−1 on the first two coordinate planes, then graphthem together.

What does the intersection of these two graphs represent?

Determine if the following ordered pairs are solutions of both y = x+2 and y = 2x−1 usingonly the equations.

(3, 5) (1, 3) (−1,−3) (0, 0) (−2, 0)

Graph the following systems of equations and determine the solution.

Ex 1: −x+ 2y = 4 and x− 2y = 2

36

Ex 2: 2x+ y = 1 and 6x+ 3y = 3

We can conclude that there are three possible cases:

1) There is when the lines intersect once.

2) There are when the lines are parallel and distinct.

3) There are when the lines are the same.

Solve the following system of equations by graphing; check your answer using the equations.Ex 3: x+ 3y = 9 and 3x− y = 2

What else can we notice about these lines?

37

4.2 Solving Systems by Substitution Name:

Substitution is a more algebraic method for determining the solution to a system of equa-tions. This can be especially useful if the solutions aren’t integer values.

Find the solution of the system y = x+ 2 and y = 2x− 1 without using a graph.

Now use substitution to solve the following systems of equations.Ex 1: −x+ 2y = 4 and x− 2y = 2

Ex 2: 2x+ y = 1 and 6x+ 3y = 3

These results allow us to draw a connection between what we saw with the graphs! We havethe same three cases as before:

1)

2)

3)

Ex 3: x+ 3y = 9 and 3x− y = 2

38

Solve the following systems of equations using substitution.Ex 4: 2x+ 8y = 3 and x = 8− 4y Ex 5: 2y = 4x+ 24 and 2x− y = −12

Ex 6: 3x− y = 5 and y = 3x− 5 Ex 7:x+ y = 0 and 4x+ 2y = 3

Ex 8: x+ 3y = −28 and y = −5x Ex 9:x

2+y

3=

7

6and

x

4− 3y

2=

9

4

39

4.3 Solving Systems by Elimination Name:

Elimination is our third method for determining the solution to a system of equations. Whileusually faster than substitution, it can sometimes be a bit tricky.

Find the solution of the system −x + y = 2 and 2x − y = 1 without using a graph orsubstitution.

Now use elimination to solve the following systems of equations.Ex 1: −x+ 2y = 4 and x− 2y = 2

Ex 2: 2x+ y = 1 and 6x+ 3y = 3

Again, we still have the same three cases as before:

1)

2)

3)

Ex 3: x+ 3y = 9 and 3x− y = 2

40

Solve the following systems of equations using elimination.Ex 4: 2y = −3x and −3x− y = 3 Ex 5: 3x+ 3y = 33 and 5x− 2y = 27

Ex 6: 3x = 3 + 2y and −4

3x+ y =

1

3Ex 7: x+ 3y = 6 and −2x+ 12 = 6y

Ex 8: 24x+12y = −7 and 16x−18y = 17 Ex 9:1

5x+y =

6

5and

1

10x+

1

3y =

5

6

41

4.4 Applications of Linear Systems Name:

We now re-enter the world of applications of linear equations, this time with the knowledgeof how to solve a system of equations! We will be using the same general attack outline asbefore, which we recall as the following:

1)

2)

3)

4)

5)

6)

Ex 1: Rich’s diet consists heavily of ramen noodles. During a recent shopping trip, Richbought 8 more cases of beef ramen than chicken ramen. He also bought a total of 20 cases.How much of each kind did Rich buy?

Ex 2: The rest of Rich’s diet consists of frozen pizza. Cheese pizzas cost $1.50, whereassupremes cost $2.00. During the trip described above, Rich bought a total of 10 frozenpizzas, which altogether cost $19.00. How many of each kind did Rich buy?

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Ex 3: Rich is once again (presumably) speeding in his Audi R8. He leaves from Iowa Cityat the same time his brother Joe leaves from Pennsylvania (660 miles away), and they drivetowards each other on I-80 at a constant speed. If Rich was driving 45 mph faster than Joeand they drive for 4 hours before they meet, what were there individual speeds?

43

5.1-5.2 Integer Exponents Name:

We begin with a rather extensive list of facts. For any integers m and n and any real numbera we have the following properties:

Property Example

1. am · an = an+m

2. a0 = 1

3. a−m = 1am

4. am

an= am−n

5. (am)n = amn

6. (ab)m = ambm

7. (ab)m = am

bm

These properties can also be used on expressions containing variables. Simplify the followingexpressions:

Ex 1: (102y4)2(10xy2)3 Ex 2: (4x−2y−3)−2

Ex 3: −40 + (−4)0 Ex 4:(3−1x−3y)−1(2x2y−3)2

(5x−2y2)−2

44

Ex 5: 5−1 + 6−1 Ex 6:(x−4y2)3(x2y)−1

(xy2)−3

Ex 7: 5−2 + 6−2 Ex 8: −(−19)0

Ex 9:(2xy−3)−2

(3x−2y4)−3Ex 10: (2p−2qr−3)(2p)−4

Ex 11:(7−1x−3)−2(x4)−6

7−1x−3Ex 12: 8−1 + 6−1

Ex 13:(8pq−2)4

(8p−2q−3)3Ex 14:

013

130

45

5.3 Scientific Notation Name:

Anyone who ever hopes to take a science class cannot escape from scientific notation. Ulti-mately, scientific notation is a tool that helps us

Evaluate the following expressions:

3.19× 103 =

3.19× 102 =

3.19× 101 =

3.19× 100 =

3.19× 10−1 =

3.19× 10−2 =

3.19× 10−3 =

There is a helpful pattern that we can recognize here!

Write the following numbers in standard notation:

5.62× 104 −4.2× 107 5.86× 10−4 −9.99× 10−1 3× 10−3

This special form is called . A number is in this formwhen it is written as

46

Determine if the following numbers are in scientific notation; if they are not, write them sothat they are.

56.2× 104 −.42× 107 1× 100 −999× 10−1 10× 10−3

Change the following numbers from standard notation to scientific notation.

42, 000, 000 78, 900, 000, 000 0.00123 −0.0000888 −22

Here we see an interesting connection with the sign of the exponent:

One advantage of scientific notation is that it greatly simplifies the multiplication and divi-sion of extremely small or large numbers. Evaluate the following expressions and write theresult in both scientific notation and standard notation.

Ex 1: (5× 108)(8× 109)

Ex 2: (6× 10−7)÷ (2× 103)

Ex 3: (9× 10−5)(3× 105)

47

5.4 Polynomials and Graphing Name:

Simplify each of the following polynomials, write in descending form, then determine its de-gree and state whether it is a monomial, binomial, trinomial, or none of these.

5x2y + 2x

2x+ 5 + 6x2 + 6x

9000x9000 + x9000 + 9000

x+ x2 + x4 + x8 + x16 + x32

5xy2 − 2xy2 + 3xy2

10x2y3z4

This gives us the following definitions:

The is the sum of its exponents.

The is the greatest degree of any term in the polynomial.

A has one term, a has two terms, and ahas three terms.

Evaluating a polynomial is similar to evaluating a function. Evaluate the following polyno-mials at x = 2 and x = −1:

4x3 − 3x2 + 2x− 1

−2x3 + 10x+ 1

48

We can also add and subtract polynomials. This task is simply combining like terms.

Add and subtract the polynomials 4x3 − 3x2 + 2x− 1 and −2x3 + 10x+ 1.

We now make a brief detour to talk about the graph of degree 2 polynomials.

Graph each of the following functions on the axes provided (use a table of values).

Ex 1: y = x2.

Ex 2: y = −x2 + 5.

49

5.5-5.6 Multiplying Polynomials Name:

The textbook describes a variety of techniques for multiplying polynomials, but it can besimpler to approach all polynomial multiplication from the perspective of the distributiveproperty.

Ex 1: (4x3)(2x2)(−x5) Ex 2: 7x3y2(3x2 + 2xy − y3)

Ex 3: (x+ 6)(x+ 5) Ex 4: (5− 2x)(2 + 7x)

Ex 5: (5x2 + 2x+ 1)(x2 − 3x+ 5) Ex 6: (8x+ 3y)2

50

Ex 7: (8x6 + 4x4 − 12x2)(34x2 + 2) Ex 8: (x− 2)3

Ex 9: (x+ 3)(x− 3) Ex 10: (10x+ 3y)(10x− 3y)

We can see from these examples that there are a couple special rules we can describe:

51

5.7 Dividing Polynomials Name:

Dividing polynomials, while appearing highly technical, is actually a seriously valuable skillthat you will likely employ in later algebra classes. This skill is most easily acquired viapractice.

Ex 1:18x5 + 12x3 − 6x2

−6x3

Ex 2:5x4 − 6x3 + 8x

3x2Ex 3:

200a5b6 − 160a4b7 − 120a3b9 + 40a2b2

40a2b

Ex 4:x2 − 4x+ 4

x− 2Ex 5:

2x2 − 5x− 12

2x+ 3

52

Ex 6:x2 + 2x+ 20

x+ 6Ex 7:

x3 + 1

x+ 1

Ex 8:2x5 + 9x4 + 8x3 + 10x2 + 14x+ 5

2x2 + 3x+ 1

53

6.1 Introduction to Factoring Name:

We talked about the prime factorization of a number on the first day of class, and today weintroduce a related concept. Determine the factorization of 24 and 30.

What’s the largest number that divides both?

This number has a special name: .

Find the greatest common factor for the terms 21x7, 18x6, 45x8, and 24x5.

This technique allows us to factor polynomials. Factor the following polynomials:

5x2 + 10x 20x5 + 10x4 + 15x3

Sometimes the greatest common factor can be a polynomial itself. Factor the following ex-ample:

x(x+ 5) + 3(x+ 5)

This leads to a more advanced technique we call “factoring by grouping.” Use this newtechnique to factor the following polynomials:

2x+ 6 + xy + 3y x3 + 2x2 − 3x− 6 10x2 − 12y + 15x− 8xy

54

Factor the following polynomials:

Ex 1: 8x2 + 6x Ex 2: 19x2y + 38x2y3

Ex 3: x2(y − 9) + 1(y − 9) Ex 4: 36x6y + 45x5y4 + 81x3y2

Ex 5: 16x3 − 4x2y2 − 4xy + y3 Ex 6: x2 + 2x+ xy + 2y

Ex 7: x5 − 3 + 2x5y − 6y Ex 8: x(x+ 4y) + y(x+ 4y)

55

6.2-6.3 More on Factoring Name:

We now continue our work on factoring polynomials.

We have already seen how to multiply two binomials and get a trinomial. Multiply thefollowing:

(x+ 4)(x+ 2)

In some sense, factoring is the “reverse” of multiplication. Factor the following:

Ex 1: x2 + 5x+ 6 Ex 2: x2 + 13x+ 12

Ex 3: x2 − 2x− 8 Ex 4: x2 − 6x+ 9

Ex 5: 3x2 + 3x− 36 Ex 6: 2x2 + x− 3

56

Ex 7: 24x4 + 17x2 − 20 Ex 8: 36x3y2 − 104x2y2 − 12xy2

Ex 9: 20x2 + 11x− 3 Ex 10: x2 + 10x− 30

Ex 11: 48x2 − 74x− 10 Ex 12: x2 + 4xy + 3y2

Ex 13: 4t2(k + 9)7 + 20ts(k + 9)7 + 25s2(k + 9)7

57

6.4 Special Factoring Name:

Multiply the following polynomials:

(x+ y)(x− y)

(x+ y)(x+ y)

(x− y)(x2 + xy + y2)

(x+ y)(x2 − xy + y2)

We notice that three of these multiplications give us .

We can then run this idea in reverse to factor special examples of binomials! Factor thefollowing binomials using what we learned above:

Ex 1: x2 − 9 Ex 2: x2 − 25y2

Ex 3: 400x10 − 144y20 Ex 4: x2 + 4x+ 4

58

Ex 5: 18x2 − 48xy + 32y2 Ex 6: x3 − 27

Ex 7: x3 + 8 Ex 8: 1000x6 + 27y3

Ex 9: 27x6 + 8y3 Ex 10: x9 − y9

One case you cannot factor is .

Write an example of this type:

59

6.5 Solving Quadratic Equations Name:

We can now employ our factoring techniques to solve quadratic equations! Write an exampleof a quadratic equation:

A quadratic equation has the form: .

Now solve the quadratic equation you wrote above using factoring:

The reason we can solve quadratic equations like this is because

Solve the following quadratic equations:

Ex 1: (2x− 7)(x− 3) = 0

Ex 3: x2 = 7x

Ex 2: x2 = 3 + 2x

Ex 4: x2 − 9 = 0

60

Ex 5: (2x+ 7)(x2 + 2x− 3) = 0

Ex 7: x4 = 2x3 + 15x2

Ex 9: x2 + 3x = −2

Ex 6: 9x3 − 49x = 0

Ex 8: 2x(3x− 4) = 0

Ex 10: x2 + (x+ 1)2 = (x+ 2)2

61

6.6 Applications of Quadratics Name:

We again return to application problems, this time with the power of knowing how to solvequadratic equations. As we will see, combining old techniques with new is often the key tosuccess in mathematics. Solve the following application problems:

Ex 1: Rich wants to buy an Ellie Goulding poster that covers his entire bedroom door. Forsome strange reason, he knows that the area of the door is 21 square feet, and that it is 4feet taller than it is wide. What are the dimensions of the door?

Ex 2: Rich and his older roommate Gabriel have ages that are given by two consecutiveintegers and whose product is 552. What are their ages?

Ex 3: After Westminster’s homecoming, Rich and his old roommate Will decided to raceback to their respective graduate schools. If Rich travels west to UI 70 mph faster than Willtravels south to WVU and they are 260 miles apart after 2 hours, how fast is each going?

62

Ex 4: Chargers punter Mike Scifres is a man-beast. We can model the height of his bestpunt using the equation

h = −14t2 + 84t,

where h is the height (in feet) of the ball and t is the time (in seconds) since it was punted.How long was the punt in the air? What was its maximum height?

63

7.1 Rational Expressions Name:

Write an example of a rational expression using the variable x:

Now evaluate the expression using x = −2 and then x = 3:

We know that we’re not allowed to . Determine what value of xwill make our expression undefined.

When we deal with typical fractions, we often want them in . Reduceour rational expression using factoring.

For each of the following examples, evaluate for x = −2, determine where the expression isundefined, and reduce the expression.

Ex 1:4x− 8

10x− 20

64

Ex 2:5x+ 20

3x+ 12

Ex 3:2x2 − 3x− 5

2x2 − 7x+ 5

Ex 4:1− 8x3

8x2 + 4x+ 2

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7.2 More Rational Expressions Name:

Multiply the following fractions and write them in lowest terms:

3

10· 5

94

x· x

2

6

Now divide the following fractions and write them in lowest terms:

14

15÷ 2

5

12

5x2÷ 3

10x

Simplify the following expressions, writing all fractions in lowest terms.

Ex 1:2(x+ y)

y· 18

6(x+ y)2

Ex 3:2− x

8÷ x− 2

6

Ex 2:4x2

2x5÷ (2x)3

−6

Ex 4:27− 3x

4· 12

2x− 18

66

Ex 5:2x2 + 3x− 2

6x2 − 7x+ 2· 4x2 − 5x+ 1

x2 + x− 2

Ex 6:2x2 − x− 1

2x2 + 5x+ 3÷ 4x2 − 1

2x2 + x− 3

Ex 7:−x3 − y3

x2 − 2xy + y2÷ 3y2 − 3xy

x2 − y2

Ex 8:x− 8

x− 4÷(x2 − 12x+ 32

8x· x2 − 8x

x2 − 8x+ 16

)

67

7.3 Even More Rational Expressions Name:

Before we add or subtract fractions, we must find a

While we learned this idea implicitly in our first encounter with fractions, it’s now time tocompletely expose the concept. A common denominator is good, but as our fractions havebecome significantly more complicated, we’d really like a

Find the LCD of the following pairs of fractions:

1

20,

7

16

1

9x,

5

12x2

We can get the LCD via the following process:

1)

2)

3)

Find the LCD of the folllowing sets of rational expressions:

Ex 1:9

10,13

25Ex 2:

17

250,

21

300,

1

360

68

Ex 3:4

25x2,

7

10x4

Ex 5:3

7x2 + 21x,

2

5x2 + 15x

Ex 4:7

3x4y5,

23

9x6y8

Ex 6:13

x2 + 7x,−3

5x+ 35,

−4

x2 + 14x+ 49

Ex 7:10

x2 − 10x+ 21,

2

x2 − 2x− 3,

15

x2 − 6x− 7

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7.4 Still More Rational Expressions Name:

Perform the following operations, writing the resulting fraction in lowest terms:

1

20+

7

16

1

9x+

5

12x2

All of our old ideas about adding and subtracting fractions still hold here. Most importantly,

Perform the indicated operation, writing your answer in lowest terms:

Ex 1:3x− 2

x2 − 4+−2

x+ 2

Ex 2:x− 2

x2 − 2x− 3− 2x

x2 − x− 6

70

Ex 3:2x

x− 3+

6

3− x

Ex 4:x+ 3

x2 − 9− −1

x2 + x− 12

Ex 5:x+ y

18x2 + 9xy − 2y2+

3x− y36x2 − y2

71

7.5 Complex Fractions Name:

Can we not escape fractions? While they never completely disappear, this is the beginningof their end for this semester. Simplify the following expressions by adding the fractions andthen dividing:

23

+ 59

14

+ 112

6 + 3x

x4

+ 18

There is an alternate method of performing this simplification involving the LCD. Simplifyagain, this time using the alternate method:

23

+ 59

14

+ 112

6 + 3x

x4

+ 18

Simplify the following examples using either method:

Ex 1:

xpq3

p2

qx2

Ex 2:35m− 2

m2

92m

+ 34m2

72

Ex 3:

1y

+ 2y+2

4y− 3

y+2

Ex 4:1− 2

x− 3

x2

1− 5x

+ 6x2

Ex 5:x+2x−3x2−4x2−9

73

7.6 Solving Rational Equations Name:

A long, long time ago (when it was over 100 ◦F) we learned how to solve very basic equa-tions with fractions in them, and today we travel a little deeper into that subject. Solve thefollowing equations:

Ex 1:3

4x− 2

3x =

1

2Ex 2:

p

2− p− 1

3= 1

We are allowed to clear when we are .You absolutely cannot use this technique when simplifying an expression.

Sometimes our denominator contains variables. We can solve the equation as before, butwhen we finish we must check to make sure our solution doesn’t cause a division by zero.Solve the following equations and check the solutions:

Ex 3:x

x− 2=

2

x− 2+ 2 Ex 4:

2

x2 − x=

1

x2 − 1

74

Solve the following equations and check the solutions:

Ex 5:1

x− 1+

1

2=

2

x2 − 1

Ex 6:1

x2 + 4x+ 3+

1

2x+ 2=

3

4x+ 12

We can occasionally be asked to solve for a variable. Solve the following equation for c.

Ex 7:1

a=

1

b+

1

c

75

7.7 Rational Equation Applications Name:

Now it’s time to put our knowledge to work in some application problems. Solve the following:

Ex 1: Rich needs to put antifreeze in his old Subaru so it won’t overheat. He knows that themixture he should use will contain 3 oz of water for every 2 oz of antifreeze. Additionally, heknows it will take 65 oz to fill his Subaru’s coolant reservoir. Determine how much antifreezeRich needs to buy.

Ex 2: If the same number is added to both the numerator and denominator of the fraction57

is equivalent to 35, determine the unknown number.

Ex 3: An F-22 Raptor traveling at top speed during severe weather (100 mph winds) goes8000 miles with the wind in the same amount of time that it takes to go 7000 miles againstthe wind. Determine its top speed.

76

Two useful concepts are the ideas of the rate of work and the amount of work. (Note thatthis is not the same type of “work” that is referred to in physics.)

Ex 4: Rich’s brother Joe is known for being an exceedingly slow farm worker. In fact, Rich’sfather describes Joe as someone who moves at “glacial speed.” If Rich can stack a load ofhay in 20 minutes and Joe stacks the same load in 1 hour, determine how long will it takethem to stack it when working together.

Ex 5: Rich and Joe have 2048 bales of hay to unload. If Rich unloads at a rate of 115 balesper 20 minutes and Joe unloads at a rate of 90 bales per 40 minutes, determine how longwill it take them to finish.

Ex 6: Coty, Franklin, and Bobby have a 12-pack of Yuengling bottles. Coty drinks 1 bottleevery 10 minutes, Franklin drinks 1 bottle every 12 minutes, and Bobby drinks 1 bottleevery 15 minutes. If Rich is bringing them Yuengling at a constant rate of 3 bottles every20 minutes, determine how long it will be till they run out, and then determine how manytotal bottles they drank during that time.

77

8.1-8.3 Roots Name:

Just as a quick review, evaluate the following square roots:

√9

√64

√81

√144

We can think of the square root as asking the following question:

We can make things slightly more interesting by considering the following examples:

−√

100√

49

√−36

Square roots don’t always give us integers, or even rational numbers. For instance, one canprove that the square root of any prime number is an irrational number. However, we dohave the ability to estimate a square root of a number that is not square. Consider thefollowing examples:

√7

√35

√69

Sometimes we need to “undo” a square root. This can be done by squaring. Square thefollowing examples:

√9

√2 −

√3

√17

Square roots are also very important to a few of math’s most frequently used formulas:

Use these formulas to solve the following problems:

Ex 1: Determine the distance between the points (2,3) and (-6,1).

78

Ex 2: Rich (while playing CoD:MW2) is hiding behind a dumpster with a FAMAS. He is 40feet away from the base of a warehouse, which happens to be 30 feet tall. Some n00b climbedup on the roof and shot Rich with an RPG. Determine the distance the rocket traveled.

We can extend the idea of square roots to roots with higher index. Consider the followingexamples:

3√

8

4√

81

5√

32

3√−8

4√−81

5√−32

3√

27

− 4√

16

− 5√

32

There are two very important properties that change how we work with radicals:

Use those properties to simplify the following expressions:

√2 ·√

3√288√2

√35·√

15

These properties also give us the ability to simplify square roots by “breaking down” theexpression inside the root. Simplify the following examples:

√24

√63

√72

79

Similar ideas still hold if we have expressions containing variables or roots with an indexgreater than two. Simplify the following examples:

√x5

√8x10

√x8

400

3√x6

3√

27x12

3√

32x10

Adding and subtracting square roots works much differently than multiplying and dividing.When adding and subtracting square roots it can be helpful to think about

Most importantly, you absolutely cannot .

Simplify the following examples:

3√

6 + 5√

6

√10 + 9

√10

7√

3 +√

12

4x√

63y + 6√

28x2y

√18−

√27

3√

16 + 4 3√

2

3x√

50 +√

2x2

24√

6x7 − x 4√

96x3

80

Simplify the following examples:

−√

144121

5√

3 · 2√

15

3√

125a15

35

√75− 2

3

√45

5 4√

32 + 2 4√

32 · 4√

4

√(−3− 1)2 + (1− 4)2

50√20

2√10

√7

x10

6√

40

√6 +√

7

3√

2 · 3√

4

√(−6)2 − 4(1)(−3)

81

8.4-8.5 More Radicals Name:

We now visit a rather antiquated topic related to square roots, called “rationalizing the de-nominator.” Rationalizing the denominator is a process that allows us to rewrite a fractioninvolving roots so that it has no roots in the denominator.

Rationalize the denominator, leaving your result in simplest form:

Ex 1: 3√2

Ex 2:√

403

Ex 3:√

173·√

176

Ex 4:√

16x

Ex 5:√

2x2z4

3y

Ex 6:√

x2

4y

82

Things can become slightly more complicated when dealing with roots whose indices arehigher than two. For the purposes of this class, the only such roots will be cube or fourthroots. Rationalize the denominator in the following cases:

Ex 7: 23√2

Ex 8: 3

√3

25x2

Ex 9: 4

√5x

27x5

By combining some of our old ideas about multiplying polynomials with sections 8.1-8.4 wecan extend our ability to deal with radicals into a variety of messier problems. One espe-cially useful technique comes from using the difference of squares formula to rationalize adenominator. Rationalize the denominator in the following example using the difference ofsquares formula.

3

2 +√

5

This technique is called .

Now simplify the following example:

Ex 10:

√5√

3−√

7

83

“By your powers combined” simplify the following examples:

Ex 11: 7√

6 ·√

3− 2√

18

Ex 13: (√

12−√

11)(√

12 +√

11)

Ex 15: 6−√5√

2+2

Ex 17: 5√7−105

Ex 12: (2√

7 + 3)2

Ex 14: (12−√x)2

Ex 16: 1√x+√y

Ex 18: 25+√75

10

84

Simplify the following examples:

Ex 19:√6+√5√

3+√5

Ex 21: ( 3√

2− 1)( 3√

4 + 3)

Ex 20: 3√

5(4 3√

5− 3√

25)

Ex 22: ( 3√

5− 3√

4)( 3√

25 + 3√

20 + 3√

16)

85

8.6 Solving Radical Equations Name:

Now that we know how to simplify expressions with roots, we move on to solving equationscontaining roots. The following is a key strategy in this process:

During this process we can sometimes introduce an extraneous solution. As a result, we

Now that we’ve gotten these key ideas on the table, let’s put them into action by attackingthe following examples:

Ex 1:√x+ 1− 3 = 0

Ex 2: 3√x =√x+ 8

Ex 3:√x+ 1 = −4

86

Ex 4:√x2 + 5x+ 10− x = 0

Ex 5:√

2x− 3 = x− 3

Ex 6:√

21 + x−√x = 3

Ex 7: 3√

5x = 3√

3x+ 1

87

8.7 Rational Exponents Name:

Rewrite the following expressions without fractional exponents:

x12 x

13 x

32

In general, we can say that

Additionally, we can extend one of old rules about integer exponents from Chapter 6:

This allows us to simplify expressions with terrible-looking exponents into much more man-agable forms. Use the above rules to simplify the following expressions, rewriting themwithout fractional exponents:

8112

−16−54

1534

1554

(−64)13

x13 · x 5

3

(y

12

x−4

) 43

3225

(725 )

53

x56 · x−1

6

x13

88

9.1 Solving Quadratics Using Roots Name:

We saw how to solve quadratics using factoring is section 6.5. Use this technique to solvethe following example:

x2 − 9 = 0

We can find another route to this solution by using a square root instead of factoring! (I’veleft you space above to try it with the same equation.) However, we must keep one crucialfact in mind:

We can write this rule mathematically as the following:

Now let’s put it into action! Solve the following examples:

Ex 1: x2 = 169 Ex 2: x2 = 254

89

Ex 3: x2 − 8 = 0

Ex 5: (x− 7)2 = 16

Ex 7: (7x− 5)2 = 25

Ex 4: 4x2 − 3 = 7

Ex 6: (x+ 2)2 = −17

Ex 8: (2x− 5)2 − 180 = 0

90

9.2 Completing the Square Name:

Completion of the square might be considered the pinnacle of our work this semester. Whileit is not as practical as the quadratic formula (which we’ll see in the next section), completingthe square is a technique that will combine many of the ideas we’ve learned this fall.

We haven’t frequently used the term perfect square trinomial, but we have seen plenty ofexamples (especially in Chapter 6). Write two examples of perfect square trinomials:

In general, we can write perfect square trinomials in one of two possible forms:

Completion of the square allows us to rewrite an quadratic equations in a way that thenturns them into the types of problems we say in Section 9.2. We’ll take the following examplestep-by-step:

Solve x2 + 6x+ 7 = 0 using completion of the square.

91

Use completion of the square to solve the following examples.

Ex 1: x2 − 8x = 5

Ex 3: 4x2 + 16x− 9 = 0

Ex 2: 2x2 − 7x− 9 = 0

Ex 4: (x+ 3)(x− 1) = 2

92

9.3 The Quadratic Formula Name:

At long last, we may lay our hands upon the ever-useful, never-tiring, quadratic formula,which is stated mathematically as:

It is important to note that this gives .

Use the quadratic formula to solve the following examples:

Ex 1: x2 − 8x− 9 = 0

Ex 3: 2x2 = 30 + 7x

Ex 2: x2 − 10x+ 25 = 0

Ex 4: 2x2 + 12x = −5

93

Solve the following examples using the quadratic formula:

Ex 5: 9x2 = 11x

Ex 7: 3x2 − 2x+ 5 = 10x+ 1

Ex 9:2

5x2 − 3

5x− 1 = 0

Ex 6: x2 − 96 = 0

Ex 8: (x+ 3)(x+ 2) = 15

Ex 10:2

3x2 − 4

9x =

1

3

94

9.4 Complex Numbers Name:

We have finally come full circle. On our very first day we discussed the “types of numbers”throughout mathematics, which lead to one of my favorite illustrations:

We now have the ability to discuss those numbers in the box marked “imaginary.” Suchnumbers provide solutions to equations that are not satisfied by real numbers. Consider thefollowing equation:

x2 = −1

No matter how we attack this problem, we find that x = ±√−1. This leads us to the

definition of the imaginary unit :

Now we can finally handle those annoying square roots containing negative numbers. Rewritethe following using the imaginary unit:

√−5

√−16

√−27

This gives us the general rule:

We also have two new definitions. Let a, b be real numbers and i be the imaginary unit.

An is any number of the form ia.

A is any number of the form a+ ib.

95

As we do with real numbers, we would like to be able to add, subtract, multiply and dividecomplex numbers. We can accomplish this by extending our rules for working with radicals.Simplify the following examples:

Ex 1: (2− 6i) + (7 + 4i)

Ex 3: 3i(2− 5i)

Ex 5: (3 + 2i)(3− 2i)

Ex 2: (2 + 6i)− (−4 + i)

Ex 4: (4− 3i)(2 + 5i)

Ex 6: 8+i1+2i

Finally, complex numbers allow us to get solutions to equations that have no real solutions.Solve the following equations:

Ex 7: (x+ 2)2 = −9 Ex 8: 2x2 = 4x− 5

96