l

- iwanttogotostanford

l

- Stacey Warren - Expert brainly.com

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- katieb

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- iwanttogotostanford

ANSW CHOICES ARE:
1 over the quantity x minus 3 times the quantity x plus 4
1 over the quantity x minus 3 times the quantity x plus 2
1 over the quantity x plus 4 times the quantity x minus 5
1 over the quantity x plus 2 times the quantity x minus 5

- iwanttogotostanford

i think it is D but not sure

- mathmale

I will convert your verbal presentation into a mathematical definition. Your use of "over" tells me that you're dealing with rational fractions.
x minus 5 over x squared minus 3x minus 10 ⋅ x plus 4 over x squared plus x minus 12
Please verify whether or not the following is correct::\[\frac{ x-5 }{ x^2-3x-10 }+\frac{ 4 }{ x^2 }+x-12\]

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## More answers

- alekos

have you attempted the question?

- mathmale

It'd be well worth learning how to use Equation Editor to present expressions like this one clearly.

- iwanttogotostanford

correct but the x-12 on the second part of the equation is under the 4 withx^2

- mathmale

Next, look at this expression and determine how you might go about simplifying it.

- iwanttogotostanford

I attempted the question and i got D

- alekos

ok @mathmale, over to you

- iwanttogotostanford

@alekos is it D

- mathmale

\[\frac{ x-5 }{ x^2-3x-10 }+\frac{ 4 }{ x^2+x-12 }\]

- mathmale

Is this correct?
PLEASE: DO NOT DEAL IN ANSWERS. SHOW HOW YOU GOT YOUR RESULT. THAT'S WHAT COUNTS HERE.

- mathmale

Can these two rational fractions be simplified? Combined?

- alekos

mathmale will guide you

- iwanttogotostanford

no it doesn't i just need the answer right now i know how to do it but i have 100 other problems to do and i just need confirmation @alekos

- mathmale

I want to be helpful, but I don't deal in answers, at least not without seeing how you got your results. Could you please factor both denominators and then tell me what the LCD for the whole expression would be?

- iwanttogotostanford

x-3

- mathmale

I have factored the denom of the first rational function and have gotten two factors, both of which are different from yours. Try again, please. Factor x^2-3x-10.

- iwanttogotostanford

im not sure

- mathmale

then you're not ready to move on. That's my point.
You can factor x^2-3x-10 in multiple ways: simiple factoring, completing the square, synthetic division, graphing and more.
Any of these familiar to you?

- iwanttogotostanford

yes synthetic division

- mathmale

OK. Try using 5 as the divisor in synthetic division.
Here's the setup:

- mathmale

|dw:1449336095545:dw|

- mathmale

Edit my drawing to display your work.
If 5 is a root of the given algebraic function, then your synth. div. remainder will be zero.

- iwanttogotostanford

ok

- iwanttogotostanford

sorry but what are the 4 steps of synthetic division again? need a refresher

- mathmale

First, where did I get the numbers I typed out in my illustration, above?

- iwanttogotostanford

the 5 was the numerator of the equation and then I am not actually sure where you got the 1, 3, and 1 ! i thought we would use 1, -3, -10 or something of that sort

- mathmale

Then I've made a typo. 1, -3 and -10 are correct.
5 is not the numerator; rather, 5 is a possible root, and we are using synth div to check whether it actually is a root or not.

- iwanttogotostanford

oh ok

- mathmale

|dw:1449336394897:dw|

- mathmale

May I assume you've used the Draw utility before?

- iwanttogotostanford

yes I have will do

- iwanttogotostanford

|dw:1449336486924:dw|

- iwanttogotostanford

then i do 5 times 1

- mathmale

Yes, and put the result under the -3.

- mathmale

Plese show your work in the Draw utility.

- iwanttogotostanford

|dw:1449336543185:dw|

- iwanttogotostanford

that right so far?

- mathmale

|dw:1449336596714:dw|

- iwanttogotostanford

|dw:1449336647479:dw|

- mathmale

|dw:1449336738004:dw|

- iwanttogotostanford

|dw:1449336777011:dw|

- mathmale

|dw:1449336828122:dw|

- iwanttogotostanford

|dw:1449336907555:dw|

- mathmale

|dw:1449336969213:dw|

- mathmale

Well worth learning and practicing synth. div.

- iwanttogotostanford

yes very true, when else do i need or can i use synthetic division again?

- mathmale

The first rational function is now\[\frac{ x-5 }{ (x-5)(x+2) }\]

- iwanttogotostanford

then where do i go from there ?

- mathmale

You'll now need to use synth div again to factor the denom of the second rational fraction, which is \[\frac{ 4 }{ x^2+x-12 }\]

- iwanttogotostanford

oh ok can we go through that process together again please but not as slow this time?

- iwanttogotostanford

i want to really get the hang of synth division

- mathmale

Take a look at x^2+x-12. By looking at the last term (the constant 12), can you come up with several possible roots for this polynomial? What are some of the factors of -12?

- iwanttogotostanford

6 maybe? 24?

- iwanttogotostanford

4

- mathmale

Important: Look at the middle term (+1x). You need to find two possible factors of -12 that combine to give you a +1 result.

- mathmale

We're going to use your 4 and determine whether it's a root of x^2+x-12 or not.

- iwanttogotostanford

ok great

- mathmale

|dw:1449337384035:dw|

- iwanttogotostanford

|dw:1449337467865:dw| did i do this right so far?

- mathmale

Unfortunately, no. Your first coeff. underneath the division symbol is a '1' You must bring that '1
' downward and write it underneath the 1 and the horiz line as follows:

- mathmale

|dw:1449337601944:dw|

- iwanttogotostanford

|dw:1449337597519:dw|

- mathmale

|dw:1449337631108:dw|

- iwanttogotostanford

|dw:1449337723706:dw|

- iwanttogotostanford

@mathmale

- mathmale

|dw:1449337797442:dw|

- iwanttogotostanford

|dw:1449337832420:dw|

- mathmale

|dw:1449337865624:dw|

- iwanttogotostanford

|dw:1449337892477:dw|

- mathmale

|dw:1449337925131:dw|

- iwanttogotostanford

NO

- iwanttogotostanford

4 is not a root/solution

- mathmale

No, 4 is not a root of x^2+x-12.
If you go thru all these steps again, you'll find that -4 is a root. Therefore, (x+4) is a factor of x^2+x-12. The other factor is (x-3).
It is very important that you review our discussion and ensure that you can do all of this work by yourself.

- iwanttogotostanford

ok thank you what do i have to do next?

- iwanttogotostanford

so, would my final result be 1/(x-3)(x+4) ?

- mathmale

Then your original expression can be re-written as \[\frac{ x-5 }{ (x-5)(x+2) }+\frac{ 4 }{ (x-3)(x+4) }\]

- mathmale

what you wrote there was only the denominator of the 2nd fraction.

- iwanttogotostanford

oh my i do not know what to do next!

- mathmale

As y ou can see, the denominators of these two rational expressions are not equal. That means more work: You must identify the LCD of these 2 denominators.
Remember how to do that?
Note that (x-5)/(x-5) in the first rational function can be reduced.

- iwanttogotostanford

no, i do not remember how to do this

- mathmale

Looks like you ought to make a list of concepts you've forgotten and then carefully review that material. Finding LCDs is very important, a very basic skill.

- mathmale

(x-3)(x+4) is missing from the first denominator. (x-5)(x+2) is missing from the second denom.

- mathmale

to find the LCD, we multiply the (x-5)(x+2) denominator of the first rational function by (x-3)(x+4) from the 2nd rational function. If we mult. a denom., we must also mult the corresponding numerator by the same quantity.

- mathmale

\[\frac{ 5(x-3)(x+4) }{ (x-5)(x+2)(x-3)(x+4) }+\frac{ 4(x-5)(x+2) }{ (x-5)(x+2)(x-3)(x+4) }\]

- mathmale

See? the denominators are now equal. that's the whole point!

- mathmale

The main concepts we've covered here are 1) factoring the denominators, 2) finding the LCD of the 2 rational functions and 3) combining the two rational functions.

- iwanttogotostanford

ok!

- mathmale

yes, this is a lot of work, but it becomes easier with time and practice.

- iwanttogotostanford

ok, what do i do next to get the final conclusion

- mathmale

Please review our entire conversation and practice everything until you can do all the work yourself without help.

- iwanttogotostanford

ok

- iwanttogotostanford

are you a real person or a robot?

- mathmale

By the way, I'm a Stanford University graduate. I studied there 1962-1967. I'm very much a real person, 5'11" tall, overweight, smart, gray, with a stubble beard. ;)

- mathmale

;)

- iwanttogotostanford

ok so how can i get to the final conclusion?

- mathmale

What do YOU think?

- mathmale

3/5 - 1/5 = ?

- iwanttogotostanford

2/5

- mathmale

7/18 + 2/18 = ?

- iwanttogotostanford

9/18

- mathmale

Yes. So, you have to multiply out both numerators of those rational fractions, and then combine all the like terms you've produced.

- mathmale

This would mean, for example, multiplying out (x-5)(x-3)(x+4) and re-writing the product with powers of x in decreasing order.

- iwanttogotostanford

im not sure how to get the final answer

- iwanttogotostanford

its very confusing

- mathmale

Have to approach the numerator of the 2nd rational function in the same way.
I can't make my most recent suggestion any clearer than it already is. It just involves a lot of careful "grunt work."

- mathmale

If this problem is too confusing, pick an easier one to work on, one from the beginning of this section.

- iwanttogotostanford

I'm almost to the answer and i really need to get to the final answer, its urgent

- mathmale

Basic operation here is "combining fractions"
You've already shown that you know how to do that.

- mathmale

I'd love to go through problems with you and build up your knowledge and confidence, but I do not deal in answers and will not be providing one. Sorry.

- mathmale

type in another problem, the easiest you can find that you have not already attempted.

- iwanttogotostanford

ok well i can't improve unless i know I'm doing it right and getting the right answer

- mathmale

Please type in another problem, the easiest you can find and which y ou have not already done.

- iwanttogotostanford

@zepdrix @Jadedry i really need help finishing this one

- mathmale

Oh, great. Unfortunately, it appears you want others to do your work for you.
I'm sorry, but I don't operate that way. I will always contribute at least as much effort to enhance your learning as you contribute yourself. Little effort => little help. Please reconsider. Change your attitude. Recognize that you are responsible for your learning and no one else is. Then I'd be delighted to help you again.

- zepdrix

Wow you typed out the equation in words :) lol
that's confusing.
Was this the initial equation?\[\large\rm \frac{x-5}{x^2-3x-10}\cdot\frac{x+4}{x^2+x-12}\]

- iwanttogotostanford

yes! @zepdrix

- zepdrix

I don't feel like reading all the previous posts :3 too much words lol
So I'm not exactly sure where you left off.
But, I would start by factoring these denominators, ya?
\[\large\rm x^2-3x-10=(\qquad\quad)(\quad\qquad)\]

- iwanttogotostanford

yes!

- zepdrix

Remember how to factor?
You need two numbers which
`multiply` to -10, and
`add` to -3.

- iwanttogotostanford

-5 and 2 right

- zepdrix

Mmm ya that sounds right!\[\large\rm x^2-3x-10\quad=(x-5)(x+2)\]

- iwanttogotostanford

ok, great!

- zepdrix

\[\large\rm \frac{x-5}{\color{orangered}{x^2-3x-10}}\cdot\frac{x+4}{x^2+x-12}\]So the expression becomes,\[\large\rm \frac{x-5}{\color{orangered}{(x-5)(x+2)}}\cdot\frac{x+4}{x^2+x-12}\]

- zepdrix

How bout the other denominator? :d
Any ideas?

- iwanttogotostanford

ok! and divide?

- mathmale

Unfortunately, your presentation of the original problem in words did not adequately indicate that you were MULTIPLYING. The problem is vastly easier if that 's the case and you have no need to worry about LCDs. Please learn how to present mathematical expressions in mathematical terms, not in words unless absolutely necessary.

- zepdrix

It's usually a good idea to provide a picture of your problem if you're able to :)
Otherwise it can lead to confusion and wasted time.
That was unfortunate :( Hmm

- anonymous

He used a dot to indicate that the fractions were indeed multiplied.

- iwanttogotostanford

ok, how do i get to the final conclusion?

- zepdrix

\[\large\rm \frac{x-5}{(x-5)(x+2)}\cdot\frac{x+4}{\color{orangered}{x^2+x-12}}\]Let's factor this other denominator before doing any division (cancelling).

- zepdrix

\[\large\rm x^2+x-12\quad=(\qquad\quad)(\qquad\quad)\]Oh it looks like you guys did this one before as well :)
The numbers were -3 and 4 ya?

- iwanttogotostanford

x-3 and x+4

- zepdrix

ok cool\[\large\rm \frac{x-5}{(x-5)(x+2)}\cdot\frac{x+4}{\color{orangered}{(x-3)(x+4)}}\]

- iwanttogotostanford

noww what

- zepdrix

If you have any factors that are the same in the numerator and denominator,
they can divide out evenly.
We usually use the word "cancel" to describe this.
Maybe put brackets around the numerators so it's easier to match them up.\[\large\rm \frac{(x-5)}{(x-5)(x+2)}\cdot\frac{(x+4)}{(x-3)(x+4)}\]

- zepdrix

Hmm, so in that first fraction, we have an (x-5) in the top and bottom, ya?\[\large\rm \frac{\cancel{(x-5)}}{\cancel{(x-5)}(x+2)}\cdot\frac{(x+4)}{(x-3)(x+4)}\]What is left in the numerator when we do this?

- iwanttogotostanford

just a 2?

- iwanttogotostanford

@zepdrix

- zepdrix

The numerator, the top.
Why would it be a 2? >.< lol

- iwanttogotostanford

im not sure! I thought there would be no numerator left

- iwanttogotostanford

or jjust x+4

- zepdrix

Ok good, the numerator is sort of "disappearing", so it's a lil confusing.
When you divide things that are equivalent, you're not left with 0,
you're left with 1.
Examples:
4/4=1
x/x=1
So we'll leave a 1 on top when we cancel out the factors,\[\large\rm \frac{1}{(x+2)}\cdot\frac{(x+4)}{(x-3)(x+4)}\]

- iwanttogotostanford

yes, agreed

- zepdrix

And then how bout the other fraction?
What can you do with that guy? :d

- iwanttogotostanford

x+4 would cancel out

- zepdrix

\[\large\rm \frac{1}{(x+2)}\cdot\frac{1}{(x-3)}\]Ok great.
Now let's bring the fractions together by multiplication.\[\large\rm =\frac{1\cdot1}{(x+2)\cdot(x-3)}\quad=\frac{1}{(x+2)(x-3)}\]And depending on how your teacher/assignment wants the final answer,
we can either leave it like this,
or multiply out the brackets in the denominator.

- zepdrix

Oh you had answer choices :) My bad

- iwanttogotostanford

ok great! so it would be this: 1 over the quantity x minus 3 times the quantity x plus 2

- zepdrix

yay good job \c:/

- iwanttogotostanford

yess! ok thank you

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