DQW

8 = 2*4 Multiply out (x - 2y)(x^2 + 2xy + 2y^2) and see if you get x^3 - 8y^3. You will not ! Simply because (-2y)(2y^2) = -4y^3 instead of -8y^3, which is = (-2y)(4y^2)

Buoyancy is nothing but a force that results from gravity. It is simply the effect of the differential hydrostatic pressure of a fluid in a gravitational field. In cosmology, you do not see buoyant forces because interstellar space has virtually zero density.

Recall how the general complex number [imath]z =r \cdot e^{i \phi} [/imath] is plotted on the argand plane. It is a point that is a distance 'r' from the origin, at an angle [imath]\phi [/imath] from the positive real (x) axis. Now look at your roots. They all have the same modulus = |a|^{1/n}. So they must all lie on a circle of radius r = |a|^{1/n}. In addition, note that their arguments divide [imath]2 \pi [/imath] into n equal parts. The first root has argument [imath] \theta /n [/imath]. The next root's argument is [imath] \theta /n + (2 \pi/n) [/imath]. The one following this has an argument [imath]\theta /n + 2 \cdot (2 \pi/n) [/imath] and so on... So these points form a regular polygon of n sides, centered on the origin. You can visualize the points as the ends of position vectors sticking out radially from the origin.The symmetry of the arragement of these vectors requires that their resultant be a zero vector. So, the sum of the roots is also zero. In the case of even values of n there's a "more direct" argument that can be given. For every point at angle [imath] \phi [/imath] there is a diametrically opposite (antipodal) point at angle [imath] \pi + \phi [/imath]. These points are of the form x + iy and -x -iy, so naturally, their sum = 0. Similarly, each of the roots can be paired up with its antipode and added to give zero. The total sum must hence be zero.

x^3 – 8y^3 is of the form a^3 – b^3, with a=x, b=2y. Also b^3 comes out of the factors by multiplying b*b^2. And, in this case, b=2y and b^2 = 4y^2. So, that's where the 4 comes from. PS : There's a typo in your final factorization. The last term is 4y^2, not 4y.

Looks good to me.

I didn't think RFID would get out past a metallic chassis. Maybe in the high frequency range, perhaps ? Externet, the idea is simply to use a radio transponder beeping out a specific signature. I believe that's essentially what RFID involves but I'm not really cartain.

For part (b), you will not understand what it says unless you read what your textbook has to say about this (or the notes from when your teacher taught it in class) and look at some worked examples to see how it's done.

Anthropos, when the plane is described as "smooth" that means there's no friction. What's the other important force you're forgetting about ? What kind of force (actually it's a pair of forces) do you always have between two bodies that are pressing against each other ?

When you arrive at a remainder of x^2 - x, you must stop, because the order of the remainder polynomial is smaller the the order of the divisor. In fact, if you look at your next step, you will find your mistake. How did you multiply (x^3 + 2x + 1) by (-x) and get (x^2 - 2x - x) ?

I can't begin to list the number of misconceptions of science you've revealed here. Darwinian Evolution is a scientific theory - it is objective, falsifiable and predictive. In science, a theory is not something weak - it's the best you can get. Quantum Mechanics is a theory. It is not a fact. And it is this theory that is used to build the computer that you're presently reading this on. ID is neither falsifiable nor predictive. It does not fall under the definition of science. Hence should not be taught in science class. Period.

Is this addressed at me ? I am not an administrator and take no part in any administrative decisions on this forum. Unless you specifically ask me a question, I shall not respond to your posts henceforth. Admins : Please feel free to delete these last two posts of mine. I'm sorry this thread has digressed thus. I apologize. We should get the thread back on track if we wish to do any justice to the OP.

In that case, how are your calculus fundamentals ? Stewart's book is a good place to start. http://www.amazon.com/exec/obidos/search-handle-url/ref=pd_sim_b_2/102-6482689-5502555?%5Fencoding=UTF8&search-type=ss&index=books&field-author=James%20Stewart

Yes it is. No, I don't ! It was very much 'proof like' !

Of course, this is beyond what the question wants you to do, but by spending another 15 minutes on the geometrical interpretation is valuable. If you think you'd want to do this, ask me and I'll guide you.

Good ! And now that you're actually done with the problem, comes the important part : The Geometry !

Say' date=' what's [imath']e^{2 \pi i} = [/imath] ?

Just one tiny error in line 2 : check the last term in that expansion (the term for m = n).

Nearly there...isn't there a term in that last expression that you can actually evaluate and plug in for ? PS : You have no idea how much nicer it looks to see all the steps in there, following logically from each other !

You underestimate the power of free trade. With NK being the unpredictable menace that it is, what's preventing Japan and SK from going nuclear ? It's definitely not an inability to learn the physics.

It would help us to know what your current level of (math/physics) preparation is. Also, are you asking for something that'll keep you busy for the next 3 weeks ? With a short time frame it is better to pick one or two useful areas and learn them well rather than do a half-hearted job of several things. What I would recommend strongly is revising basic math and physics. Only if you are confident in your fundamentals should you move to new stuff. All the best for your college term.

Shouldn't that be e^{2i pi/n} ?

It should ! Write down the result you would get from summing a geometric series. What is the first term, and what's the common ratio ?

It's not a big deal. Everyone slips up once in a while - I've done it more times than I'd care to remember ! Besides, the reason you thought mesons held quarks together is probably from some vague memory of Yukawa's prediction of the pion as the mediator of the strong force.

Your first line : "Because ..." is incorrect. There is no reason for those things to be equal. This would mean that all the roots were the same and hence their sum would simply be n times the value of any one root. Also, it has no bearing on what follows. Your last 2 lines are also incorrect. How are you left with a "1" in the sum after taking out exp(i*theta/n) ? What happened to all the terms ? I suggest you continue from the previous line, and procedd without skipping steps. Write each exponential term as a product of two terms - the constant exponential and the varying one. Then take the constant outside the sum .... etc.

The big picture