Download - 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.
WTAMU > Virtual Math Lab > Beginning Algebra
Last revised on July 22, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rightsreserved.
Accessibility | Accreditation | Compact with Texans | Contact Us | Form Policy |
House Bill 2504 | Legislative Appropriation Request
Link Policy and Privacy Statement | Online Institutional Resumes | OpenRecords/Public Information Act | Risk, Fraud and Misconduct Hotline
Site Map | State of Texas | Statewide Search | Texas Homeland Security |
University Organizational Chart
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West Texas A&M University | All Rights Reserved | Canyon, TX 79016 | 806-
651-0000
b b 2 c b g c c Title |
VML COLLEGE ALGEBRA INTERMEDIATE ALGEBRA BEGINNING
ALGEBRA GRE MATH THEA/ACCUPLACER
Beginning Algebra
Tutorial 4: Introduction to Variable Expressions and Equations
WTAMU > Virtual Math Lab > Beginning Algebra
Learning Objectives
After completing this tutorial, you should be able to:
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
In this problem, what is the base?
If you said 7, you are correct!
What is the exponent?
If you said 1, you are right!
Lets rewrite it as multiplication and see what we get for an answer:
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*Rewrite the base 7, one time in a product
Example 3: Evaluate
In this problem, what is the base?
If you said 1/3, you are correct!
What is the exponent?
If you said 2, you are right!
Lets rewrite it as multiplication and see what we get for an answer:
*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.
(answer/discussion to 2b)
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Practice Problem 3a: Evaluate the expression if x = 1, y = 2, and z = 3.
3a.
(answer/discussion to 3a)
Practice Problems 4a - 4b: Decide whether the given number is a solution of the
given equation.
4a. Is 0 a solution to ?
(answer/discussion to 4a)
4b. Is 8 a solution to ?
(answer/discussion to 4b)
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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.
(answer/discussion to 6a)
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6b. The quotient of a number and 9 is 1/3.
(answer/discussion to 6b)
Need Extra Help on these Topics?
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.
WTAMU > Virtual Math Lab > Beginning Algebra
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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Site Map | State of Texas | Statewide Search | Texas Homeland Security |
University Organizational Chart
West Texas A&M University | All Rights Reserved | Canyon, TX 79016 | 806-
651-0000
b b 2 c b g c c Title |
VML COLLEGE ALGEBRA INTERMEDIATE ALGEBRA BEGINNING
ALGEBRA GRE MATH THEA/ACCUPLACER
Beginning Algebra
Tutorial 8: Properties of Real Numbers
WTAMU > Virtual Math Lab > Beginning Algebra
Learning Objectives
After completing this tutorial, you should be able to:
Identify and use the addition and multiplication commutative properties.
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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
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 thatproblem. At the link you will find the answer as well as any steps that went into
finding that answer.
Practice Problems 1a - 1b: Use a commutative property to write an equivalent
expression.
1a. xy
(answer/discussion to 1a) 1b. .1 + 3x
(answer/discussion to 1b)
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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)
(answer/discussion to 2b)
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)
(answer/discussion to 3b)
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Practice Problems 4a - 4b: Write the opposite (additive inverse) and the reciprocal
(multiplicative inverse) of each number.
4a. -7
(answer/discussion to 4a)
4b. 3/5
(answer/discussion to 4b)
Need Extra Help on these Topics?
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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VML COLLEGE ALGEBRA INTERMEDIATE ALGEBRA BEGINNING
ALGEBRA GRE MATH THEA/ACCUPLACER
Beginning Algebra
Tutorial 30: Division of Polynomials
WTAMU > Virtual Math Lab > Beginning Algebra
Learning Objectives
After completing this tutorial, you should be able to:
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 .
Step 1: Set up the long division.
Step 2: Divide 1st term of divisor by first term of dividend to get first term of the
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:
Step 3: Take the term found in step 1 and multiply it times the divisor.
Scratch work:
Step 4: Subtract this from the line above.
Scratch work:
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Step 5: Repeat until done.
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 .
Step 1: Set up the long division.
Step 2: Divide 1st term of divisor by first term of dividend to get first term of the
quotient.
Scratch work:
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Step 3: Take the term found in step 1 and multiply it times the divisor.
Scratch work:
Step 4: Subtract this from the line above.
Scratch work:
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Step 5: Repeat until done.
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
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 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
finding that answer.
Practice Problems 1a - 1c: Divide.
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1a.
(answer/discussion to 1a) 1b.
(answer/discussion to 1b)
1c.
(answer/discussion to 1c)
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