# 3.2 Rational expressions: rational expressions

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<para>This module is from<link document="col10614">Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr.</para><para>A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.</para><para>The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.</para><para>Objectives of this module: be able to recognize a rational expression, be familiar with the equality and negative properties of fractions.</para>

## Overview

• Rational Expressions
• Zero-Factor Property
• The Equality Property of Fractions
• The Negative Property of Fractions

## Rational expressions

In arithmetic it is noted that a fraction is a quotient of two whole numbers. The expression $\frac{a}{b}$ , where $a$ and $b$ are any two whole numbers and $b\ne 0$ , is called a fraction. The top number, $a$ , is called the numerator, and the bottom number, $b$ , is called the denominator.

## Simple algebraic fraction

We define a simple algebraic fraction in a similar manner. Rather than restrict­ing ourselves only to numbers, we use polynomials for the numerator and denomi­nator. Another term for a simple algebraic fraction is a rational expression . A rational expression is an expression of the form $\frac{P}{Q}$ , where $P$ and $Q$ are both polyno­mials and $Q$ never represents the zero polynomial.

## Rational expression

A rational expression is an algebraic expression that can be written as the quotient of two polynomials.

Examples 1–4 are rational expressions:

$\frac{x+9}{x-7}$ is a rational expression: $P$ is $x+9$ and $Q$ is $x-7$ .

$\frac{{x}^{3}+5{x}^{2}-12x+1}{{x}^{4}-10}$ is a rational expression: $P$ is ${x}^{3}+5{x}^{2}-12x+1$ and $Q$ is ${x}^{4}-10$ .

$\frac{3}{8}$ is a rational expression: $P$ is 3 and $Q$ is 8.

$4x-5$ is a rational expression since $4x-5$ can be written as $\frac{4x-5}{1}$ : $P$ is $4x-5$ and $Q$ is 1.

$\frac{\sqrt{5{x}^{2}-8}}{2x-1}$ is not a rational expression since $\sqrt{5{x}^{2}-8}$ is not a polynomial.

In the rational expression $\frac{P}{Q}$ , $P$ is called the numerator and $Q$ is called the denominator.

## Domain of a rational expression

Since division by zero is not defined, we must be careful to note the values for which the rational expression is valid. The collection of values for which the rational expression is defined is called the domain of the rational expression. (Recall our study of the domain of an equation in Section [link] .)

## Finding the domain of a rational expression

To find the domain of a rational expression we must ask, "What values, if any, of the variable will make the denominator zero?" To find these values, we set the denominator equal to zero and solve. If any zero-producing values are obtained, they are not included in the domain. All other real numbers are included in the domain (unless some have been excluded for particular situational reasons).

which comes first emotion or thinking
Emotions
Keerthi
true
james
either could come first. arguing for the supremacy of one over the other may not hold, ceteris paribus.
Oyetoro
Depends on which part of brain is dominated in individual.
Gaurav
thinking. it's a thought that provokes an emotion
Free
I often saw cat in my dreams nd thats why i m afraid of cats.. wherever i see cat i just escape from that place what is this nd why this happens
usually whatever you see during the day or what impacted you in the slightest, will appear in your dream.
The
What kind of dream? Why you afraid of cat because you dream about it? Bad dream?
Yoselin
i saw cat nd lizard nd oftenly they attack me nd i could not save my self nd then i wakeup nd feels fear
Tasneem
from childhood i like cata nd play with it but now because of that dreams i dont like cats nd lizards
Tasneem
some times you feels uncomfortable to cat have around you ,but still you should take out from fear..
Kanzul
yes i want to take out this fear but i cannot face cat wherever i see cat i feel fear
Tasneem
you know your weakness, that is it..you should see the videos on cats ..it will be enough for you.. after that you see the variation in your mind illusions
Kanzul
if it is more fear, you take Consultancy from psychiatrist..
Kanzul
thats the problem i can't see cats videos i dont wanna see cats or lizards
Tasneem
what you think? that cat attack on you..😅
Kanzul
is there any past memory relate with this?
Kanzul
yes i think that cat will attack me but i just saw cats in dreams attacking me
Tasneem
can you understand hindi?
Kanzul
urdu
Tasneem
اچھا ہے..آپ نے کبھی علم نفسیات اور خواب کی تعبیرات کی کتابیں پڑی ہے..اپنے آپ کو درست کرنے کے لیے یہ سب سے بہترین نسخہ ہے..
Kanzul
the cat is your mind reflecting your Self. you are not afraid of cats, the cat is just a symbol for the fear you have of life and death. Cat is life, cat eats lizard, cat brings death. You are either the cat or the lizard, in your subconscious mind.
Will
Will
it may have bitten in your childhood stage,and you still remember it
hamed
Where is semantic memory u mentioned the name but didn't disscuss the topic..?
If u have want to make happy ur babby not by blood but by heart by giving all ur time to them but ur mother wants u to gave that time on urself and ur studies and that makes her happy and mom doesnot know that i accept her as my babby she thought she is just my friend
Iqra
Actually she is 4 years younger than me but i love her and said my baabby to her what should i do now?
Iqra
If anyone can help me plz do it i am super depressed
Iqra
Hlo Iqra...
IAS
Hi!
Iqra
?
Iqra
If u r in real conjunction wd him say ur mom fluently that I wana make him mine I feel depressed
IAS
Mom has best hearing power as to understood her/his baby
IAS
But my mom dont accept that she think i am just runing my life
Iqra
U have to b very clear wd ur mom
IAS
Where from u
IAS
Well i didn't tell about that but i said i love her she is like my soul sister and i love to gave her time but she said if u want to meat her itsok but dont talk on phone because u consume alot time on her and thats not fair
Iqra
Bhai i am from earth
Iqra
How can I suggest u it WL take alot time
IAS
Actually she dont like her because i gave 3 to 4 hours on phone to her
Iqra
Where frm u r?
IAS
Okay thank u so much bhai i think i got my answer
Iqra
whats your relation with her ? where you meet her nd why your mom forbid you from talking with her
Tasneem
Ur most welcome sister
IAS
She live in my neighborhood we are frnds from childhood but she is 4 yeras younger than me thats why i called her my bachaa and i gave alot of time to her because she have no sister brother and also no frnd and my mom dont like that
Iqra
Because 3 to 4 hours on phone she dont like usage of phone so much no one did accept me infront her thats why she is against her she dont hate her she just said meet her face to face not on phone
Iqra
Okay thank u so much all of u now i am feeling good and got my answe also
Iqra
yeah your mom said right but dont talk too face to face sometimes things that are above limits leads to problems
Tasneem
welcome😊
Tasneem
Yeah u are right inshallah everything will be alright soon
Iqra
Thank u
Iqra
inshaa allah..
Tasneem
i have a question?
Melissa
Yea
Iqra
Iqra
what is one strategy that would promote positive risk taking behavior?
Melissa
I dont know
Iqra
Why do I get upset when my girlfriend doesn't tell me she loves me back in front of her friends?
Do not depend on her, or don't hope just take care of her ..it doesn't matter that how much she love you..matter that how much you love her..
Kanzul
Because you're insecure, read "The Rational Male" by Rollo Tomassi if you want to make sure that you and your girlfriend are/remain satisfied in your relationship
Leon
and don't listen to Kanzul, don't be a pushover
Leon
may be you are suspicios if she stay around her male friends
Kanzul
Thank you both, yeah umm it makes sense that I'm insecure and yeah I mean I do love her alot the thing is it's not with her Male friends it's in general.
Apollo
But also what is considered a pushover so I can avoid that Leon
Apollo
what is this that a persom constantly saw dream from very early age till now nd saw same dream what is this
Candy
how
Tasneem
Candy
dayshavoo
Candy
thank you
Tasneem
😊
Tasneem
samething keeps happening,repeat
Candy
why do I have dreams of my boyfriend boss
Because you love him and his memories. Or that you are reminded of him very often.
Jobin
There are some explanation why we have dream. It can be just your brain picking random memory in your brain when you sleep (eventhough we sleep our brain still working) and this random memory your brain pick out become your dream
Kelly
👍
Jobin
or this boss left an impression in you, but it doesn't necessarily mean you have a romantic feelings towards this person. Maybe this boss just do some random thing and it left an impression in your memory and you still think about it subconsciously when you sleep and it manifested itself in ur dream
Kelly
That is perfectly normal and no cause for worry. You might not want to tell your boyfriend though ;)
Leon
What do you think is the reason for your friends or the family members coming into the dreams especially wet dreams? And after that what's the feeling of the person?
Jobin
how to get rid of fearful avoidant issues
Do you think that seminary student needs psychological studies?
I think it's best for high schoolers.. we all should be aware of what we are
Alexsus
Yes definitely, especially in an age like this were more and more I've observed students not knowing what they truly are and not knowing how to deal with psychological issues. Plus, it gives them the advantage of knowing how to deal with other people with different ideas and issues.
Tav
Absolutely. How would you be able to make any serious decisions in your life, when you are a mess and don't understand your feelings, motivation and reasoning behind this or that issue. Moreover, I think whether you are a seminary student or not, you should also be taught critical thinking skills.
Sofia
Thank you all
Jobin
whats is schizophrenia
long-term mental disorder of a type involving a breakdown in the relation between thought, emotion, and behavior, leading to faulty perception, inappropriate actions and feelings, withdrawal from reality and personal relationships into fantasy and delusion, and a sense of mental fragmentation.
Amber
What is burn out syndrome?
Jobin
Emotional and physical exhaustion which appears after a long period of stress.
Avram
what is social conflict
a conversation with someone & having conflict with .
Candy
Soory Guys hope you ar doing well Im in strong desire to get this kind of book INTRODUCTORY PSYCHOLOGY but ther cost is too and I'm not able to buy it If you see this announcement please I need soft copy even few please
jacobo
Anybody who is able to access this book just help even soft copy to saves me Im so soory
jacobo
Deen social conflict means misunderstand occurred among the people in a certain social context can be caused by poor governance
jacobo
what is top-notch?
excellent? the best?
Travis
the very best.
Snuffles
best of something. above the expectation
Jem
something like expert level
Evil
zenith
Syed
Hi! what is introspection?
jacobo
the ultimate best.. of excellent quality
Melissa
honor and respect
Randall
Understanding
Candy
Hi! This book INTRODUCTORY PSYCHOLOGY I'm not able to buy anybody can help me even soft copy
jacobo
i want some pdf books who can help me
Leon
is there any website where I can get a true diagnosis of a personality disorder I think I may possess? I can't seek therapy due to insurance complications
Well most websites that offer actual medical diagnoses cost money or they will connect you to a doctor for some sort of meeting.
Elise
yeah I've been looking for a decent site for around a year now and that's all Ive found thank you
Bailey
Yeah it is quite hard to find a reliable site.
Elise
diagnosis couldn't get from a websites. you have to refer a doctors for a consultation.
patience
hi
Surya
Kanzul
Compare and contrast situational influences and dispositional influences and give an example of each. Explain how situational influences and dispositional influences might explain inappropriate behavior.
what is an ethical consideration that is what a search for my research is seeking out the truth what lies in the darkness and it's hidden by the light
Rev
Ethical consideration includes the risk of doing your research
Mary
***cirt.gcu.edu/research/developmentresources/tutorials/ethics
Amanda
as well as the research can not have lasting affects on the subjects.
Amanda
Is this live
Little
We as researchers should not harm the subjects of our research, be it the population under study or the selected sample. I think ethical consideration requires that we disclose as much as is practically possible about the requirements of the research to all concerned.
Abel
It is ethically sound also to to acknowledge others' findings wherever we reference them in our research because we are to build and grow the body of knowledge by building on the research others have done. This is my contribution.
Abel
What is CBT?
cognitive behavioral therapy
Joni
When they use this CBT?
zahir
what is cbt
Timothy
mostly used to help treat anxiety and depression related diagnosis. But it's used in other types of mental health issues.
Joni
cognitive behavioral therapy. As stated earlier in this discussion.
Joni
Thank yoi
zahir
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