The problem is to express the number  $$1\over x^5 + 2x^4 + 3x^3 + 3x^2 + 2$$  using only rational numbers in the denominator, knowing that $x^5 + 2 = -2(x + 1)(x^3 + x^2 + x)$.  (This is an example from a final exam of an algebra course from the past year (which is freely available to students).)  I know how to solve this problem, but I do not understand why this is possible. The trick comes from knowing that this number can be expressed using addition and multiplication of $x$ and rational numbers, ie. as  $$r_0 + r_1x + r_2x^2 + r_3x^3 + r_4x^4 + r_5x^5 + \dots$$  My solution follows, skip to questions if you are not interested.  From the polynomial equation it follows that $x^5$ equals $-2 - 2x - 4x^2 - 4x^3 - 2x^4$, therefore an expression containing $x$ with exponent 5 or greater can be rewritten as an expression with only $x^0$ to $x^4$.  So my number can be expressed as  $$ax^4 + bx^3 + cx^2 + dx + e$$  Putting an equals sign between these two expressions and multiplying by the denominator gives  $$1 = (x^5 + 2x^4 + 3x^3 + 3x^2 + 2)(ax^4 + bx^3 + cx^2 + dx + e)$$  I can multiply the right hand side and "normalize" it (by replacing $x^{\ge5}$) to get  $$1 = (e - f)x^4 + \dots + (2e - 2d)$$  From this formula I can create a system of linear equations, because the coefficient of $x^n$ has to be one if n equals zero, and zero otherwise. Solving this system will let me find the values of the coefficients.  Question 0: Why does this problem even have a solution of the form I used?  Question 1: Is there a solution like this for any expression $1 \over f(x)$, where f is a polynomial of $x$, knowing $x$ is a root of another polynomial? Will my method always work? Why?  Apologies if this is a duplicate, but everything I could find about rationalizing the denominator was about simple fractions like $1 \over 2 + \sqrt 6$.  I'd appreciate even just comments directing me towards what I can study because I am out of keywords.
This is an exercise in some textbooks.     Let $E$ be an algebraic extension of $F$. Suppose $R$ is ring that contains $F$ and is contained in $E$. Prove that $R$ is a field.   The trouble is really with the inverse of $r$, where $r\in R$. How to prove that $r^{-1}\in R$, in apparent lack of a characterization of $R$.  It occurred to me to use the smallest field containing $R$ ($R$ is easily shown to be an integral domain), that's the field of quotients, and proving that it's $R$ itself, but I don't really know how to proceed.  A not-too-weak, not-too-strong hint will be much appreciated.   Beware $ $ Readers seeking only hints should  beware that there is now a complete answer.
Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$ Recently, I ran across a product that seems interesting.  Does anyone know how to get to the closed form:  $$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$  I tried using the identity $\cos(x)=\frac{\sin(2x)}{2\sin(x)}$ in order to make it "telescope" in some fashion, but to no avail. But, then again, I may very well have overlooked something.  This gives the correct solution if $n$ is odd, but of course evaluates to $0$ if $n$ is even.   So, I tried taking that into account, but must have approached it wrong.   How can this be shown? Thanks everyone. SF Short Story about resetting the world every day Trying to identify a short story. I recall it was quite short, shorter than a novella. I read it within the last few years, but my impression is that it's from the second half of the twentieth century, maybe the sixties. I can't recall if I read it online, in an anthology, or collection. Set in America, a middle-aged man stumbles on a team (maybe like dwarves or gnomes) whose job it is to set up the world for the next day by removing all the previous day's &quot;stuff&quot;, like street furniture? (I know that makes no sense). He disrupts this process and somehow prevents progressing to the next day. I also remember the story was quite lighthearted with the team behaving in a very grumpy way like interrupted workmen. I thought the title was something like &quot;What happened to Tuesday&quot; or similar. For some reason, I thought it was by Keith Laumer, but a search doesn't reveal anything. The trope was similar to &quot;The Tunnel Under the World&quot; by Pohl, and &quot;The Adjustment Team&quot; by Dick, in that reality was constantly changed and controlled by an organisation of some kind. Baffling Elevation Profile out of Profile Tool Plugin When using the Profile Tool plugin I get a graph that looks like the 2nd derivative of what I want. That is, it looks like the rate of change of the slope of the elevation profile. I just want the elevation profile! The data is present for good 1' contour lines. Why is the Profile Tool interpreting the data in such a bizarre way? What am I missing? And what are the Min and Max units? I am quite new to Javascript. I have made a html script that includes services I published to ArcGIS for server. It all does exactly what I want and I made it by copying working scripts together and getting help from more experienced scripters. I want to finetune my service by adding a legend. Unfortunately I am not able to create a working script. I have added the parts of my script I assume that are necessary to create my legend. Does anybody see what I am doing wrong?        &lt;script&gt;         var map;         var wmsLayer;         var URL;         var legend;         require(["dojo/query", "esri/map", "esri/layers/WMSLayer", "esri/dijit/Legend", "esri/config", "dojo/domReady!"],         function (query, Map, WMSLayer, Legend, esriConfig) {          esriConfig.defaults.io.proxyUrl = "/proxy/proxy_gcx.php";            map = new Map("map", {               basemap: "topo",               center: [5.5, 52.485],               minZoom: 8,               maxZoom: 13,               zoom: 8,               slider: true           });            wmsLayer = new WMSLayer("URL", {               format: "png",               visibleLayers: ["31", "30", "29", "28"]           });            map.on("layer-add", function (results) {               //add the legend               legend = new Legend({                   map: map,                   layerInfos: [{                       layer: results,                       title: "test"                   }]               }, "legendDiv");               legend.startup();           });             map.addLayer(wmsLayer);       });     &lt;/script&gt; &lt;/head&gt;  &lt;body&gt;     &lt;div id="map"&gt;     &lt;/div&gt;     &lt;div id="legendDiv"&gt;     &lt;/div&gt; &lt;/body&gt;   EDIT: I have edited the code. I am still not done. It does give me a legend on my browser but it says there is no content in there. On my arcgis server manager there is a legend. If I want to make a backup of everything I have done since the fresh install of Ubuntu, what are the possible options? What all should I backup? I want to get all the settings that I changed, all the packages I installed, etc. Is it appropriate to ask questions about software licenses/licensing on ? Or is the appropriate place  ? Or is there no appropriate site currently and we should suggest such a site on ? Why are there both Static and Kinetic Friction? When dragging an object, there is a greater start-up force than the force it takes to keep it moving. Why is this? Why are there two different values for friction? Installing Elementary OS - will it automatically be added to the Grub menu? I currently have Windows 7 and Linux Mint installed.  When my PC starts up, it shows me the Grub menu, it shows an entry for Mint Linux and the Windows boot menu.  I am wondering, if I was to install Elementary OS, would it be automatically added to grub or do I have to put some command in the terminal?  I also want to ask, when I boot from my Elementary OS cd, and install it, just before it finishes, it shows some kind of error saying some error about cannot install to grub or something, I've tried installing to a separate partition and alongside "Them", but had no luck.  I'm really new to Linux, and want to keep my Mint Linux and Windows 7 because I have files on them. What is The Rule of Three? So I have Destiny on PS3. If I were to buy the Legendary Edition (Containing House of Wolves and Dark Below Expansion) will I be able to play with my characters from my original Destiny disc? May sound silly, but I need confirmation. How to troubleshoot booting to a black screen on VirtualBox I was using 13.04 as a guest OS on my VirtualBox 14.2.18. Host is Windows 8. After upgrade to 13.10 Ubuntu does not start. I can see a black screen and blinking cursor. I am pretty new to Ubuntu. I suspect a problem with graphics drivers. I can access tty and all files (/home) are there. Blank lines in an equation and some other similar function Why blank lines between equation makes error? there are a lot of commands in latex and almost for all of them these blank lines are not important,  but why in this case is important and is there any similar function? Where to upload too big files? I have a big HTML file that causes no errors in most of the browsers. However, when I open it with, let's say, Google Chrome, the program shuts down and I cannot trace why this is happening.   This HTML file is quite big and cannot be posted to a question in SO, but it should be ran as a whole as I cannot trace where the problem is. What is/are the most appropriate file upload website/s to upload and paste the link into the question? Prove the following inequality:   $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$   This should be solved using Taylor series.  I tried expanding the left to the 5th degree and the right site to the 3rd degree, but it didn't help.  Any tips? How do I rename all folders and files to lowercase on Linux? Can you set custom ringtones for contact groups? In iPhone, you can easily change the default ringtone (Settings → Sounds → Ringtone) and set a custom ringtone for individual people (Contacts → [some person] → Edit → Ringtone).   But can you set a custom tone for a group of contacts? (E.g. "family" or "workmates".)  I think that would be handy—and quite basic functionality, so to speak—but I couldn't find a way to do this through the iPhone UI. Is it possible at all (without jailbreaking)? I have a live USB made with the standard Ubuntu disk image creator for 16.04 LTS. In the current state, I can boot to this disk, but it still requires the original USB to run properly at all times.  Attempting to pull the drive and use the system generates a large amount of I/O errors, further suggesting that the USB drive is actually needed for things not already loaded into RAM (For example, Firefox worked without the USB, but I used it once before already. apt, on the other hand, did not.)  Is there a way to copy the entire Ubuntu live disk to RAM, so I can remove the drive and free up a USB slot for other things? How to prove this inequality $ x + \frac{1}{x} \geq 2 $ The  tag doesn't appear to be very useful.  It's used in a variety of ways (for reading, writing, showing, transforming, compiling, translating, obfuscating, obtaining, ... source code) on 1747 questions right now, and doesn't appear to be adding any classification value to the site.  Can we get rid of it? What does [&] mean before function?