Thread: polynomial division

Me being slow at math, I'm having trouble understanding the concept behind polynomial division, and was hoping someone could help clear me up on this!

Say we have:



From what I know, to divide, we divide like this:

which gives

We then multiply by which gives and so on...

Why do we divide by and then multiply the result by the divisor ?

Hope someone can help me understand this!

Thanks!

Hi, I guess I'm pretty dumb as no one has replied Hope someone can help me out on this! It still all seems totally arbitrary to me

Fisrt, when you write Do you mean ?

Hey thyristor,

I mean:

I don't really understand why we would divide 2x² by x. You can't just split up the denominator.
What I've learnt is that if is dividable by we know they have a common root, thus Since this is false is not dividable with

Hey thyristor,

Thanks for the reply!

Well, from what I worked out, the answer is

Verifying:



36 comes from in numerator cancelling out in denominator.
which gives us our original numerator...

Is this correct???

I also checked, the polynomial does not have any roots, but why does that make division invalid???

You did it right rgba. I didn't respond because I thought it would be difficult to write a decent example using TeX.

Hey Demen,

Ok cool, I'm just wondering why division works this way, and was hoping someone could help me out with an explanation?

Thanks!

No replies?

I guess maybe a decent explaination might be that polynomial division is really using the exact same rules for the numeric division you may have learned in elementary school. What you are really doing for both is taking apart the numerator a small piece of a time with the denominator. If you have to multiply by variables to acheive your answer, then that's what you have to do.

I think its kind of fun actually.

Hi Demen,

Thanks for the reply, and sorry for my late reply, been working on a VFX job the past month, and kinda been under lots of stress and pulling long nights

Originally Posted by rgba

I also checked, the polynomial does not have any roots, but why does that make division invalid???

If your polynomial could be divided by (x-5) without a remainder, it would be equal to (something)(x-5), and then it would necessarily be equal to zero for x=5. Which means 5 would be its root. Which it isn't, if your polynomial has no root.

Originally Posted by rgba

I'm having trouble understanding the concept behind polynomial division, and was hoping someone could help clear me up on this!

If I want to divide 28 by 9 I can see that it goes 3 times with 1 left over, a remainder of 1. There are a number of ways I can write this, but one way would be to say that .

With polynomial long division we are doing exactly the same thing. Except that instead of using integers as our numbers, we are using polynomials. So is just a number, and is also just a number. All the fancy manipulation is required because we don't actually know a value for either of those numbers so anything we write has to always be expressed in terms of .

It is often a good idea to remind yourself that binomials, trinomials, polynomials are just numbers; don't get fazed by them not looking like decimal numbers, they are just numbers.

So, if you want to truly understand why we do it the way we do, try doing it with just numbers.

Try this: .

Your answer will be .

When we easily see that and .

If you want to go deeper into this try Googling for the distributive property.

Hi there,

Sorry didn't see your reply!

That helps, only problem is I'm still not seeing the relationships (yes, I'm slow )

I always struggle with polynomials

I understand for the numerator, we have to find a polynomial, that when multiplied by the denominator produces the original numerator.

Maybe starting from something simpler:



Could you try explain a little more why this works? Why do we take and then multiply that result by .

Apologies for being so slow!

Whoa... Hold on a sec, are you still having issues just understanding the way division by polynomials works?

for a simple polynomial divided by, say what you want to do is line it up like in long division, but first, I'll show you some cool stuff about these beasts that may help you.

Now, this principle is what we use. we notice that for a polynomial, we can play with parenthesis and shift around things inside and out of those parenthesis, as long as it all still solves to the same thing. using this cool little fact, we can start some division.

That's an expaded process that you can go through to see how it all works. from there, you want to find a number to multiply the first term by such that the x will disappear in the division. We will have an equation and by division we get . With the division, you multiply the bottom term of the fraction by this ration, , and subtract from the top term
multiplying through we get:
This gives us the first term of the answer, . And going through the same steps above we will get: and you have the second term of the answer, and you are left with the remainder (if it doesn't solve out, meaning ) and that's the total answer. I know it's not really pretty, but it is just a general example, showing exactly what's going on. I hope this helps you out.

Thanks for the great reply! Much appreciated! Had a long day at work, so haven't had time to look at this properly, but will definitely look first thing when I wake up!

Looks really helpful!

Hi there,

I think I'm getting somewhere, but the details are still hazy. As much as I appreciate your reply, it's still not helping me as I hoped. I think I need to work from a much simpler problem.

I've written a whole bunch of stuff on paper, trying different approaches to understanding the problem.

Just thought about this:



so I tried this:

which is also the same as the above, only the numbers are broken apart. I tried using the

method of long division for polynomials. It didn't work. I got .

I thought about this:

- it appears that you cannot split the dividend () - it has to somehow be kept

a whole in relation to the numerator (). I think I'm seeing something, but I'm not quite sure what it is yet.

I guess we can treat as though it were some binomial . I'm trying to use numbers (no

variables) as a simpler approach to understanding this?

I can see:

- there is some factor that when multiplied by gives . I.e. , so .

That is . But how do we work this o6ut with the divisor as opposed to the divisor ?

I think the general question to ask is how divides when we don't take but rather try to think of as .

I imagine that once I understand it as above, it will make alot more sense to me as a whole?

Originally Posted by Arcane_Mathematician

Whoa... Hold on a sec, are you still having issues just understanding the way division by polynomials works?

Yes, very embarrassing for me But I'm trying, I'm trying!

I try to visualize stuff, heres my attempt at visualizing the math:



Maybe not a good idea to try that way? Maybe it's more of an abstract thing?

EDIT: I've also had a look at the remainder theorem, but that doesn't really help me

Originally Posted by rgba

so I tried this:

which is also the same as the above, only the numbers are broken apart. I tried using the

method of long division for polynomials. It didn't work. I got .

That works fine.

Originally Posted by MagiMaster

Originally Posted by rgba

so I tried this:

which is also the same as the above, only the numbers are broken apart. I tried using the

method of long division for polynomials. It didn't work. I got .

That works fine.

Ah yes of course! Stupid me.

Ok, I guess I'll try from there... Is there any geometric trick you could use to visualize it?

EDIT: I still struggle with the process of division.

I don't understand why the process works like this given:



1) Divide 10 by 2 which gives 5.
2) Multiply 5 and (2 + 3) and subtract that from 10. Why does this work?

I'm simply not understanding the process. I've tried to visualize it by drawing pictures and stuff. I just don' get it. Ok, maybe I do have a better idea now?

Aaaaaaaaaaah. I must be pretty bloody thick!

I'm going to read this thread again.

I can see that the quotient expands to

I just need to understand why this expansion works the way it does...

Maybe I can get a little closer...

EDIT: Some sort of geometric series. It looks like there is a pattern?

Another try at this:

We have a sum . This is our divisor.

We have a dividend, .

Take . Call this an intermediate result. (?)

Now that we have our intermediate result, we need to take the other part of the divisor the in into consideration, while keeping in mind the in .

has divided which gives us .

So we then take into consideration (our intermediate result with respect to the in , and subtract this from . This effectively cancels out the in . This is the dividend of our remainder.

I still don't see it?

I apologise if people are getting fed up with my questions. I'm honestly trying, and I hope someone can drive the point home!

My main problem is I can see the procedure, I don't see why it works though.

Originally Posted by rgba

method of long division for polynomials. It didn't work. I got .

That's the right answer. -15/(2+3)=-15/5=-3 and 5+(-3)=2. You were right.

Let me see if I can possibly make this a bit simpler.

(4x^2 + 10x + 2)/(x - 1)

Are you wondering how we figure out the first step which is to multiply the denominator by 4x?

you want the leading terms to cancel out when you subtract them 4x^2 - x does not cancel so multiply the entire denominator by 4x. Now when we subtract the leading terms 4x^2 - 4x^2 = 0. This simplifies the first part of it and we know that we at least have 4x completely moved out side of the fraction.

I hope this made sense. If not just ignore it, it's only the way I think of it working.

Originally Posted by rgbd

I also checked, the polynomial 2x^2 - 3x + 1 does not have any roots,

Say what?

It isn't really relevant to the question asked, but that's an odd thing to say. The root at 1 is visible by inspection, 1/2 is also visible with a second's guessing, you should have a standard formula handy for the two roots of a quadratic equation if they aren't obvious, and any graphing calculator will show you axis intercepts.

Sometimes people get the hang of this by reviewing familiar long division procedures with numbers, only first writing he numbers out in a visually similar form using powers of 10 with the letter "x" standing for 10.

Like: write as

and then just do the division longhand in parallel using the two notations side by side.

Hey guys! Many thanks for the replies! Sorry I haven't replied sooner, been swamped with work! Finally its done!

I get it now, I was trying too hard to visualize it, then I thought about it in abstract terms and it really made sense. I guess my mistake was trying to associate some "pictures" with the math, it doesn't really work all the time for me.

And as a result of a hangover I figured it out! Woke up one morning, bloody head was shot. Was lying in a stupor most of the day, and while doing that I started thinking about it and it all clicked.

Pretty simple actually :P

Thanks again guys!

Originally Posted by rgba

Me being slow at math, I'm having trouble understanding the concept behind polynomial division, and was hoping someone could help clear me up on this!

Say we have:



From what I know, to divide, we divide like this:

which gives

We then multiply by which gives and so on...

Why do we divide by and then multiply the result by the divisor ?

Hope someone can help me understand this!

Thanks!

This is how I see it, you divide by x because that is what is in the denominator, if it was x^2 + 5 rather than x+5 you would divide by x^2.

That will tell you how many times that power of x will go into the denominator.

When you know how many times it goes into the denominator you then to take that many times away from the top. Now rememeber you are dividing by x-5 not just x so you multiply the x-5 by the result and take that away.

That makes the biggest term in x on the top disappear, then you repeat the process with what is left

Division is basically subraction. You deal with the biggest terms first.
You have to take the multiple of the whole of the denominator off otherwise you are not dividing by the denominator, rather sumething else, and obviously if you divide by something else, well you won't get the right asnwer will you?

Originally Posted by rgba

Could you try explain a little more why this works? Why do we take and then multiply that result by .

Because the object of the first division is to find out how many times x+7 goes in to the dividend and we only need to make sure the answer is not too large. x in to 2x squared is all that is needed to determine this. The 7 is irrelevant becuase it is a lower order term; 7 times x to the zero.

But more fundamentally, your question is out of place, odd as that may sound to you.
Why is a question for religion or metaphysics.
Why is a squared plus b squared = c squared for a right triangle ?
Why is the perimeter [circumference] of a circle = Pi times the diameter ?
No one knows.
But we can prove it is true and learn HOW to use it.

Originally Posted by rgba

Me being slow at math, I'm having trouble understanding the concept behind polynomial division, and was hoping someone could help clear me up on this!

Say we have:



From what I know, to divide, we divide like this:

which gives

We then multiply by which gives and so on...

Why do we divide by and then multiply the result by the divisor ?

Hope someone can help me understand this!

Thanks!

It seems to me as if you'd like to learn how to solve rational expressions. You do do so by factoring. For example, in your aforementioned scenario you could factor the expression to the expression , which is also equivalent to the expression since multipication and division are equivalent operations. Let's try a rational expression that can be simplified to a greater degre, such as . Notice that the numerator of the expression is the difference of two squares, that is . We can simplify such an expression, that is to say into the expression . Therefore, our numberator is simplified to , and as a result our new expression is . Now in our denominator, we can notice that and have a common factor, that is . Ergo, our denominator can be simplified to the following expression . We now have the rational expression . As can quite obviously be seen, the in both the numerator and denominator can be cancelled to yield the expression
You could also simplify this expression into , but the previous fraction is fine as well.

Being able to solve rational expressions is highly important; if you are uanble to do so than you will have a very hard time when you begin studying calculus, as limits can often require several cancellations. In order to be good at it than you must be efficient at factoring, which you can do by doing factoring exercises. Try completing the exercise at this link:

http://www.mash.dept.shef.ac.uk/Reso...al_equatns.pdf

After you get a good grasp of factoring than you can move onto simplifying rational expressions; try doing the exercises at this link to get good at it:

http://college.cengage.com/mathemati...endix_a/a4.pdf

Feel free to write back if you have any other questions.

Hi there!

Sorry didn't see your replies. Didn't get email notifications.

Originally Posted by paulfr2

But more fundamentally, your question is out of place, odd as that may sound to you.
Why is a question for religion or metaphysics.
Why is a squared plus b squared = c squared for a right triangle ?
Why is the perimeter [circumference] of a circle = Pi times the diameter ?
No one knows.
But we can prove it is true and learn HOW to use it.

True, I did consider that

Ellatha:

I understand how to solve rational expressions. Sure, not a problem. I just had a hard time seeing how polynomial division worked!

Thanks for those links, look very useful, I will definitely read them

Cheers!