Algebra Beginn

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Title |

VML COLLEGE ALGEBRA INTERMEDIATE ALGEBRA BEGINNING

ALGEBRA GRE MATH THEA/ACCUPLACER

Beginning Algebra

Tutorial 1: How to Succeed in a Math Class

WTAMU > Virtual Math Lab > Beginning Algebra

Learning Objectives

After completing this tutorial, you should be able to:

Formulate a plan on how to approach your math class.

Disclaimer:


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WTAMU and Kim Seward are not responsible for how a student does on any test

or any class for any reason including not being able to access the website due to

any technology problems. We cannot guarantee that you will pass your math class

after you go through this website. However, it will definitely help you to better

understand the topics covered.

Introduction

This tutorial will give you some helpful suggestions on how you can be successful

in your math class. Hopefully this tutorial can convert some of you who are math

atheists, or at least try to help you get rid of some of your math phobia nightmares.

Those of you who are lucky enough not to have a math phobia can also benefit

from this tutorial. Now it's time to check out the "Tips on How to Succeed in aMath Class" listed below.

Tutorial


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Tips on How to Succeed in a Math Class

Yes, You Can Learn Math!!!

Note that these tips were written by Kim Seward and revised by A.P. 'Sissy'

Campbell, tutor coordinator and counselor for Student Support Services at

WTAMU, and Kim Seward.

Get a can do attitude:

If you can do it in sports, music, dance, etc., you can do it in math! Try not to let

fear or negative experiences turn you off to math.

Practice a little math every day:

It helps you build up your confidence and move your brain away from the panic

button at test time.

Take advantage of your math class:


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If you are a college or high school student, realize that most colleges and

universities require at least college algebra for any bachelor's degree. Some

classes, like chemistry, nursing, statistics, etc. will require some algebra skills to

succeed in them. If you are getting a bachelor's degree, then chances are you are

going for a professional job. Most professional jobs require at least some math.Granted, some more than others, but nonetheless math (problem solving, numbers,

etc...) is everywhere. So make sure that you embrace your math experience and

make the most of it.

Get help outside the classroom:

Go to your instructors office for extra help during office hours or by appointment.

Use the WTAMU Virtual Math Lab (http://www.wtamu.edu/mathlab) as a

reference as you go through your class. Anytime you need to see some more

examples, want to go through some practice problems or want to take a practice

test on an algebra topic, it is just a click away.

See if your school has any tutors in math.

WTAMU provides the following FREE tutoring services for WT students:

Educational Services Tutoring

EST offers free one-on-one tutoring to all WT students in a variety of subjects

including math


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Located on campus: Student Success Center, 1st floor of Classroom Center

SMARTHINKING

SMARTHINKING is an online tutoring service that WT has contracted with to

provide free live one-on-one and offline web-based tutoring in a variety of subjects

including basic math, algebra, trigonometry, geometry, calculus I&II and stats for

WT students.

Located online: WT students can access this service by logging into and clickingon the SMARTHINKING link found on your WTClass homepage.

Online whiteboards equipped with math symbols and graphs are used to

communicate between the math e-structors and students. When posting a math

question to SMARTHINKING, make sure that you type in the directions, the

problem, how far you have gotten on the problem and your specific questions

about it.

For general information about SMARTHINKING go to their website at

http://www.smarthinking.com/

See if your school has a learning lab for math. Here at WTAMU, we have a Math

Lab located in Classroom Center 411. It is a place where WT students can work

on math homework and, as problems arise, get help. The workers will be unable to

sit with you one on one for long periods of time like a tutor, however they can help

you work on specific questions. Remember that they are not there to do your


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homework, but to answer specific questions that you have. There are also computer

programs, internet connections, and videos in there to help you.

Attend class full time:

Math is a sequential subject. That means that what you are learning today builds

on what you learned yesterday. Even problems based on a new math concept will

need some old skills to work them. (Think: Can you work problems with fractions

if you dont know the multiplication tables?)

Keep up with the homework:

It sounds simple but your time is limited, you have a job to go to, etc.. Think of

it this way: No homework, no learning. Homework helps you practice theapplications of math concepts. Its like learning how to drive: the longer you

practice, the better your driving skills become and the more confidence you will

have on the road. If you only read the drivers manual, youll never learn to drive

with confidence and skill. We suggest you try some of the unassigned problems,

too, for extra practice.

Try to understand the math problems:

When you work homework problems, ask yourself what you are looking for and

how you are going to get there. Dont just follow the example. Work the problem


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step-by-step until you know why you are doing what you are and have arrived at

the solution. If you follow the what, how, and whys, youll know what to do when

you see a similar problem later.

Use index cards to study tests:

Heres how you do that: When studying for a test, make sure you can understand

the problems on each math concept as well as work them. Then make the index

cards with problems on them. Mix the index cards (yes, shuffle the cards to mix

them up) and set the timer. Start working the problems in each card as it is dealt to

you. Oh, yeah, hide your textbook! This will simulate a math test taking

experience.

Ask questions in class:

Dont be ashamed to ask questions. The instructor WILL NOT make fun of you.

In fact, at least one other person may have the same question.

Ask questions outside of class:

OK, so like most people, you dont want to ask questions in class, OR you think

of a question too late. Then go to the instructors office and ask away.


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Check homework assignments:

Make sure that when you get your graded homework back you look over what

you got right as well as what you missed.

Pay attention in class:Math snowballs. If you dont stay alert to the instructors presentation, you may

miss important steps to learning concepts. Remember, todays information sets the

foundation for tomorrows work.

Dont talk in class:

If you have questions, please ask the instructor. The information you get from

classmates may be mathematically wrong! And if it isnt related to math info for

this class, save it for outside the classroom.

Read the math textbook and study guide:


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Yes, theres a reason why we ask you to spend all that money on them. If you

look carefully, you will see that your book contains pages with great examples,

explanations and definitions of terms. Take advantage of them.

Practice Problems

In all of the other tutorials at this Beginning Algebra website, we will have

practice problems with links to the answers for you to go through. Since this

tutorial did not have any math concepts there will be no practice problems for this

tutorial only.

We do suggest that you go back to the top and reread the tips on how to succeed ina math class and think about which one(s) will help you the most to be successful

in your math class.

Need Extra Help on these Topics?


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In most of the other tutorials at this Beginning Algebra website, we will have links

to other sources that help with the topics on its respective webpage. Since this

tutorial did not have any math concepts there will be no links.


Last revised on July 22, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rightsreserved.

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Beginning Algebra

Tutorial 4: Introduction to Variable Expressions and Equations


Learning Objectives


Evaluate an exponential expression.

Simplify an expression using the order of operations.


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Evaluate an expression.

Know when a number is solution to an equation or not.

Translate an english expression into a math expression.

Translate an english statement in to a math equation.

Introduction

This tutorial will go over some key definitions and phrases used when specifically

working with algebraic expressions as well as evaluating them. We will also

touch on the order of operations. It is very IMPORTANT that you understandsome of the math lingo that is used in an algebra class, otherwise it may all seem

Greek to you. Knowing the terms and concepts on this page will definitely help

you build an understanding of what a variable is and get you more comfortable

working with them. Variables are a HUGE part of algebra, so it is very important

for you to feel at ease around them in order to be successful in algebra. So let's get

going and help you get on the road to being variable savvy.


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Tutorial

Exponential Notation

An exponent tells you how many times that you write a base in a PRODUCT.

In other words, exponents are another way to write MULTIPLICATION.

Lets illustrate this concept by rewriting the product (4)(4)(4) using exponentialnotation:

In this example, 4 represents the base and 3 is the exponent. Since 4 was written

three times in a product, then our exponent is 3. We always write our exponent as

a smaller script found at the top right corner of the base.


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You can apply this idea in the other direction. Lets say you have it written in

exponential notation and you need to evaluate it. The exponent will tell you how

many times you write the base out in a product. For example if you had 7 as your

base and 2 as your exponent and you wanted to evaluate out you could write it out

like this:

Example 1: Evaluate

In this problem, what is the base?

If you said 5, you are correct!

What is the exponent?

If you said 4, you are right!

Lets rewrite it as multiplication and see what we get for an answer:


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*Rewrite the base 5, four times in a product

*Multiply

Example 2: Evaluate


If you said 7, you are correct!





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*Rewrite the base 7, one time in a product

Example 3: Evaluate


If you said 1/3, you are correct!




*Rewrite the base 1/3, two times in a product


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*Multiply

Note that when you have a 2 as an exponent, which is also known as squaring

the base. In this problem we could say that we are looking for 1/3 squared.

Order of Operations

Please Parenthesis or grouping symbols

Excuse Exponents (and radicals)

My Dear Multiplication/Division left to right

Aunt Sally Addition/Subtraction left to right

When you do have more than one mathematical operation, you need to use the

order of operations as listed above. You may have already heard of the saying

"Please Excuse My Dear Aunt Sally". It is just a way to help you remember the

order you need to go in when applying the order of operations.


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Example 4: Simplify .

*Multiply

*Add

*Subtract

Example 5: Simplify

*Inside ( )

*Exponent


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*Multiply

*Add

Example 6: Simplify .

Note that the absolute value symbol | | is a fancy grouping symbol. In terms of

the order of operations, it would be including on the first line with parenthesis.

So in this problem, the first thing we need to do is work the inside of the absolute

value. And then go from there.

*Inside | |


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*Exponent

*Add in num. and subtract in den.

Variable

A variable is a letter that represents a number.

Don't let the fact that it is a letter throw you. Since it represents a number, you treat

it just like you do a number when you do various mathematical operations

involving variables.

x is a very common variable that is used in algebra, but you can use any letter (a, b,

c, d, ....) to be a variable.


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Algebraic Expressions

An algebraic expression is a number, variable or combination of the two

connected by some mathematical operation like addition, subtraction,

multiplication, division, exponents, and/or roots.

2x + y, a/5, and 10 - r are all examples of algebraic expressions.

Evaluating an Expression

You evaluate an expression by replacing the variable with the given number and

performing the indicated operation.

Value of an Expression


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When you are asked to find the value of an expression, that means you are

looking for the result that you get when you evaluate the expression.

So keep in mind that vary means to change - a variable allows an expression to

take on different values, depending on the situation.

For example, the area of a rectangle is length times width. Well, not every

rectangle is going to have the same length and width, so we can use an algebraic

expression with variables to represent the area and then plug in the appropriate

numbers to evaluate it. So if we let the length be the variable l and width be w, we

can use the expression lw. If a given rectangle has a length of 4 and width of 3, we

would evaluate the expression by replacing l with 4 and w with 3 and multiplying

to get a value of 4 times 3 or 12.

Lets step through some examples that help illustrate these ideas.

Example 7: Evaluate the expression when x = 4, y = 6, z = 8.


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Plugging in the corresponding value for each variable and then evaluating the

expression we get:

*Plug in 4 for x, 6 for y, and 8 for z

*Exponent

*Multiply*Add

*Subtract

Example 8: Evaluate the expression when x = 3, y = 5, and z = 7.


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Plugging in the corresponding value for each variable and then evaluating the

expression we get:

*Plug in 3 for x, 5 for y, and 7 for z

*Exponent

*Multiply

*Add

Equation

Two expressions set equal to each other.

Solution


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*Evaluate both sides

Is 2 a solution?

Since we got a TRUE statement (7 does in fact equal 7), then 2 is a solution to this

equation.

Example 10: Is 5 a solution of ?

Replacing x with 5 we get:

*Plug in 5 for x

*Evaluate both sides

Is 5 a solution?


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Since we got a FALSE statement (16 does not equal 14), then 5 is not a solution.

Translating an

English Phrase Into an

Algebraic Expression

Sometimes, you find yourself having to write out your own algebraic expression

based on the wording of a problem.

In that situation, you want to

read the problem carefully,

pick out key words and phrases and determine their equivalent mathematical

meaning,

replace any unknowns with a variable, and

put it all together in an algebraic expression.

The following are some key words and phrases and their translations:


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Addition: sum, plus, add to, more than, increased by, total

Subtraction: difference of, minus, subtracted from, less than, decreased by, less

Multiplication: product, times, multiply, twice, of

Division: quotient divide, into, ratio

Example 11: Write the phrase as an algebraic expression.

The sum of a number and 10.

In this example, we are not evaluating an expression, so we will not be coming

up with a value. However, we are wanting to rewrite it as an algebraic expression.


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It looks like the only reference to a mathematical operation is the word sum. So,

what operation will we have in this expression?

If you said addition, you are correct!!!

The phrase 'a number' indicates that it is an unknown number. There was no

specific value given to it. So we will replace the phrase 'a number' with the

variable x. We want to let our variable represent any number that is unknown

Putting everything together, we can translate the given english phrase with thefollowing algebraic expression:

The sum of a number and 10

*'sum' = +

*'a number' = variable x

Example 12: Write the phrase as an algebraic expression.

The product of 5 and a number.


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Again, we are wanting to rewrite this as an algebraic expression, not evaluate it.

This time, the phrase that correlates with our operation is 'product' - so what

operation will we be doing this time? If you said multiplication, you are right on.

Again, we have the phrase 'a number', which again is going to be replaced with a

variable, since we do not know what the number is.

Lets see what we get for this answer:

The product of 5 and a number

*'product' = multiplication

*'a number' = variable x


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Translating a Sentence into an Equation

Since an equation is two expressions set equal to each other, we will be using the

same mathematical translations we did above. The difference is we will have an

equal sign between the two expressions.

The following are some key words and phrases that translate into an equal sign (=):

Equal Sign (=) : equals, gives, is, yields, amounts to, is the same as

Example 13: Write the sentence as an equation. Let x represent the unknown

number.

The quotient of 3 and a number is .

Do you remember what quotient translates into? If you said division, you are

doing great.


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'Is' will be replaced by the symbol =.

Lets put together everything going left to right:

The quotient of 3 and a number is

Example 14: Write the sentence as an equation. Let x represent the unknown

number.

7 less than 3 times a number is the same as 0.

Do you remember what less than translates into? If you said subtraction, you are

doing great.

Do you remember what times translates into? If you said multiplication, you are

correct.


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'Is the same as' will be replaced by the symbol =.

Lets put together everything going left to right:

7 less than 3 times a number is the same as 0.

Practice ProblemsThese are practice problems to help bring you to the next level. It will allow you

to check and see if you have an understanding of these types of problems. Math

works just like anything else, if you want to get good at it, then you need to

practice it. Even the best athletes and musicians had help along the way and lots of

practice, practice, practice, to get good at their sport or instrument. In fact there is

no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and

then check your answer by clicking on the link for the answer/discussion for that

problem. At the link you will find the answer as well as any steps that went into

finding that answer.


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Practice Problems 1a - 1b: Evaluate.

1a.

(answer/discussion to 1a) 1b.

(answer/discussion to 1b)

Practice Problems 2a - 2b: Simplify each expression.

2a.

(answer/discussion to 2a)

2b.



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Practice Problem 3a: Evaluate the expression if x = 1, y = 2, and z = 3.

3a.


Practice Problems 4a - 4b: Decide whether the given number is a solution of the

given equation.

4a. Is 0 a solution to ?


4b. Is 8 a solution to ?



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Practice Problems 5a - 5b: Write each phrase as an algebraic expression. Let x

represent the unknown number.

5a. 9 less than 5 times a number.

(answer/discussion to 5a) 5b. The product of 12 and a number.(answer/discussion to 5b)

Practice Problems 6a - 6b: Write each sentence as an equation. Let x represent the

unknown number.

6a. The sum of 10 and 4 times a number is the same as 18.



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6b. The quotient of a number and 9 is 1/3.



The following are webpages that can assist you in the topics that were covered on

this page:

http://www.sosmath.com/algebra/fraction/frac3/frac39/frac39.html

This webpage goes over the order of operations.

http://www.purplemath.com/modules/translat.htm

This webpage helps with translating english into math.


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Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a

Math Class for some more suggestions.


Last revised on July 42, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights

reserved.

Accessibility | Accreditation | Compact with Texans | Contact Us | Form Policy |

House Bill 2504 | Legislative Appropriation Request

Link Policy and Privacy Statement | Online Institutional Resumes | Open

Records/Public Information Act | Risk, Fraud and Misconduct Hotline


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Identify and use the addition and multiplication associative properties.

Identify and use the distributive property.

Identify and use the addition and multiplication identity properties.

Identify and use the addition and multiplication inverse properties.

Introduction

It is important to be familiar with the properties in this tutorial. They lay the

foundation that you need to work with equations, functions, and formulas all of

which are covered in later tutorials, as well as, your algebra class. In some cases,it isn't very helpful to rewrite an expression, but in other cases it helps to write an

equivalent expression to be able to continue with a problem and solve it. An

equivalent expression is one that is written differently, but has the same value. The

properties on this page will get you up to speed as to how you can write

expressions in equivalent forms.


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Tutorial

The Commutative Properties of

Addition and Multiplication

a + b = b + a and ab = ba

The Commutative Property, in general, states that changing the ORDER of two

numbers either being added or multiplied, does NOT change the value of it.

The two sides are called equivalent expressions because they look different but

have the same value.

Example 1: Use the commutative property to write an equivalent expression to

2.5x + 3y.


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Using the commutative property of addition (where changing the order of a sum

does not change the value of it) we get

2.5x + 3y = 3y + 2.5x.

Example 2: Use the commutative property to write an equivalent expression to .

Using the commutative property of multiplication (where changing the order of a

product does not change the value of it), we get

The Associative Properties of

Addition and Multiplication

a + (b + c) = (a + b) + c and a(bc) = (ab)c


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The Associative property, in general, states that changing the GROUPING of

numbers that are either being added or multiplied does NOT change the value of it.

Again, the two sides are equivalent to each other.

At this point it is good to remind you that both the commutative and associative

properties do NOT work for subtraction or division.

Example 3: Use the associative property to write an equivalent expression to (a

+ 5b) + 2c.

Using the associative property of addition (where changing the grouping of a

sum does not change the value of it) we get

(a + 5b) + 2c = a + (5b + 2c).

Example 4: Use the associative property to write an equivalent expression to

(1.5x)y.


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Using the associative property of multiplication (where changing the grouping of

a product does not change the value of it) we get

(1.5x)y = 1.5(xy)

Distributive Properties

a(b + c) = ab + ac

or

(b + c)a = ba + ca

In other words, when you have a term being multiplied times two or more terms

that are being added (or subtracted) in a ( ), multiply the outside term times

EVERY term on the inside.

Remember terms are separated by + and -.

This idea can be extended to more than two terms in the ( ).


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Example 5: Use the distributive property to write 2(x - y) without parenthesis.

Multiplying every term on the inside of the ( ) by 2 we get:

*Distribute 2 to EVERY term inside ( )

Example 6: Use the distributive property to write - (5x + 7) without parenthesis.

*A - outside a ( ) is the same as times (-1)

*Distribute the (-1) to EVERY term inside ( )

*Multiply


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Basically, when you have a negative sign in front of a ( ), like this example, you

can think of it as taking a -1 times the ( ). What you end up doing in the end is

taking the opposite of every term in the ( ).

Example 7: Use the distributive property to find the product

3(2a + 3b + 4c).

As mentioned above, you can extend the distributive property to as many terms

as are inside the ( ). The basic idea is that you multiply the outside term times

EVERY term on the inside.

*Distribute the 3 to EVERY term

*Multiply


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Identity Properties

Addition

The additive identity is 0

a + 0 = 0 + a = a

In other words, when you add 0 to any number, you end up with that number as a

result.

Multiplication

Multiplication identity is 1

a(1) = 1(a) = a


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And when you multiply any number by 1, you wind up with that number as your

answer.

The Inverse Properties

Additive Inverse (or negative)

For each real number a, there is a unique real number,

denoted -a, such that

a + (-a) = 0.

In other words, when you add a number to its additive inverse, the result is 0.

Other terms that are synonymous with additive inverse are negative and opposite.


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Multiplicative Inverse

(or reciprocal)

For each real number a, except 0, there is a unique real number such that

In other words, when you multiply a number by its multiplicative inverse the

result is 1. A more common term used to indicate a multiplicative inverse is the

reciprocal. A multiplicative inverse or reciprocal of a real number a (except 0) is

found by "flipping" a upside down. The numerator of a becomes the denominator

of the reciprocal of a and the denominator of a becomes the numerator of the

reciprocal of a.

These two inverses will come in big time handy when you go to solve equations

later on. Keep them in your memory bank until that time.

Example 8: Write the opposite (or additive inverse) and reciprocal (or

multiplicative inverse) of -3.


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The opposite of -3 is 3, since -3 + 3 = 0.

The reciprocal of -3 is -1/3, since -3(-1/3) = 1.

When you take the reciprocal, the sign of the original number stays intact.

Remember that you need a number that when you multiply times the given number

you get 1. If you change the sign when you take the reciprocal, you would get a -1,

instead of 1, and that is a no no.

Example 9: Write the opposite (or additive inverse) and reciprocal (or

multiplicative inverse) of 1/5.

The opposite of 1/5 is -1/5, since 1/5 + (-1/5) = 0.

The reciprocal of 1/5 is 5, since 5(1/5) = 1.


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Practice Problems

These are practice problems to help bring you to the next level. It will allow you

to check and see if you have an understanding of these types of problems. Math





To get the most out of these, you should work the problem out on your own and

then check your answer by clicking on the link for the answer/discussion for thatproblem. At the link you will find the answer as well as any steps that went into


Practice Problems 1a - 1b: Use a commutative property to write an equivalent

expression.

1a. xy

(answer/discussion to 1a) 1b. .1 + 3x



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Practice Problems 2a - 2b: Use an associative property to write an equivalent

expression.

2a. (a + b) + 1.5

(answer/discussion to 2a)2b. 5(xy)


Practice Problems 3a - 3b: Use the distributive property to find the product.

3a. -2(x - 5)

(answer/discussion to 3a) 3b. 7(5a + 4b + 3c)



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Practice Problems 4a - 4b: Write the opposite (additive inverse) and the reciprocal

(multiplicative inverse) of each number.

4a. -7


4b. 3/5



The following are webpages that can assist you in the topics that were covered on

this page:

http://www.mathleague.com/help/wholenumbers/wholenumbers.htm#commutative

property

This webpage helps with the commutative property.


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http://www.mathleague.com/help/wholenumbers/wholenumbers.htm#associativepr

operty

This webpage helps with the associative property.

http://www.mathleague.com/help/wholenumbers/wholenumbers.htm#distributivepr

operty

This webpage helps with the distributive property.

http://home.earthlink.net/~djbach/basic.html#anchor904011

This webpage goes over the commutative, associative, and distributive properties.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a

Math Class for some more suggestions.

e 3 ) Title |


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Beginning Algebra

Tutorial 30: Division of Polynomials


Learning Objectives


Divide a polynomial by a monomial.

Divide a polynomial by a polynomial using long division.


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Introduction

In this tutorial we revisit something that you may not have seen since grade school:

long division. In this tutorial we are dividing polynomials, but it follows the same

steps and thought process as when you apply it numbers. Let's forge ahead.

Tutorial

Divide

Polynomial Monomial

Step 1: Use distributive property to write every term of the numerator over the

monomial in the denominator.

If you need a review on the distributive property, go to Tutorial 8: Properties of

Real Numbers.


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Step 2: Simplify the fractions.

If you need a review on simplifying fractions, go to Tutorial 3: Fractions.

Example 1: Divide .

Step 1: Use distributive property to write every term of the numerator over the

monomial in the denominator

AND

Step 2: Simplify the fractions.


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*Divide EVERY term by 2x

*Simplify each term

Divide

Polynomial Polynomial

Using Long Division

Step 1: Set up the long division.


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The divisor (what you are dividing by) goes on the outside of the box. The

dividend (what you are dividing into) goes on the inside of the box.

When you write out the dividend, make sure that you insert 0's for any missing

terms. For example, if you had the polynomial , the first term has

degree 4, then the next highest degree is 1. It is missing degrees 3 and 2. So if we

were to put it inside a division box, we would write it like this:

This will allow you to line up like terms when you go through the problem.

Step 2: Divide 1st term of divisor by first term of dividend to get first term of the

quotient.

The quotient (answer) is written above the division box.

Make sure that you line up the first term of the quotient with the term of thedividend that has the same degree.


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Step 3: Take the term found in step 1 and multiply it times the divisor.

Make sure that you line up all terms of this step with the term of the dividend

that has the same degree.

Step 4: Subtract this from the line above.

Make sure that you subtract EVERY term found in step 3, not just the first one.

Step 5: Repeat until done.

Step 6: Write out the answer.

Your answer is the quotient that you ended up with on the top of the divisionbox.

If you have a remainder, write it over the divisor in your final answer.


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Example 2: Divide .



quotient.

Note that the "scratch work" that you see at the right of the long division shows

you how that step is filled in. It shows you the "behind the scenes" of how each

part comes about.


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Scratch work:


Scratch work:


Scratch work:


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We keep going until we can not divide anymore. It looks like we can go one

more time on this problem.

We just follow the the same steps 2 - 4 as shown above. Our "new divisor" is thelast line 8x + 1.

Step 2 (repeated): Divide 1st term of divisor by first term of dividend to get first

term of the quotient.

Scratch work:

Step 3 (repeated): Take the term found in step 1 and multiply it times the

divisor.


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Scratch work:

Step 4 (repeated): Subtract this from the line above.

Scratch work:

Step 6: Write out the answer.


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Example 3: Divide .



quotient.

Scratch work:


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Scratch work:


Scratch work:


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We keep going until we can not divide anymore.

We just follow the the same steps 2 - 4 as shown above. Our "new divisor" is

always going to be the last line that was found in step 4.

Step 2 (repeated): Divide 1st term of divisor by first term of dividend to get first

term of the quotient.

AND

Step 3 (repeated): Take the term found in step 1 and multiply it times the divisor.

AND

Step 4 (repeated): Subtract this from the line above.

The following is the scratch work (or behind the scenes if you will) for the rest

of the problem. You can see everything put together following the scratch work

under "putting it all together". This is just to show you how the different pieces


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came about in the final answer. When you work a problem like this, you don't

necessarily have to write it out like this. You can have it look like the final product

shown after this scratch work.

Scratch work for steps 2, 3, and 4

for the last three terms of the quotient

2nd term:

3rd term:


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4th term:

Putting it all together:

Step 6: Write out the answer


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Practice Problems

These are practice problems to help bring you to the next level. It will allow youto check and see if you have an understanding of these types of problems. Math





To get the most out of these, you should work the problem out on your own andthen check your answer by clicking on the link for the answer/discussion for that

problem. At the link you will find the answer as well as any steps that went into


Practice Problems 1a - 1c: Divide.


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1a.

(answer/discussion to 1a) 1b.


1c.

(answer/discussion to 1c)


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Algebra Beginn

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