How do I factorize this thing? $x^4+x^2+1$ I tried to solve the integral $\int{\frac{1}{x^4+x^2+1}}$ and after trying some substitutions that did not work, I plugged the integral into an integral calculator and it turns out that $x^4+x^2+1$ can be written as $(x^2-x+1)(x^2+x+1)$ and at first I thought it was factorized using the formula $(a-b)(a+b) = a^2 - b^2$ where $a = x^2$ and $b = x+1$ but after trying that I realized it is wrong because $b = -(x-1)$ in the first parantheses and in the second $b = x+1$.  So what is the intuition behind this factorization?
By viewing the polynomials as a difference of two squares, factorise the following polynomial: $$x^4+x^2+1.$$ I searched but couldn't find a way to solve this  Edit: By using Hans Lundmark hint, I get:  $$(x^2+1)^2-x^2$$   Is it fully factorized?
How to disable the Windows8 volume On-Screen-Display? Here is a picture of what i am talking about.    I am at my wits end with that annoying thing, please somebody help me. How to obtain a confidence interval for a percentile? $L^p$ integrable but not $L^q$ integrable Why wouldn't the `which` command work for `cd`? I can't find the executable for `cd` either! I tried which cd and it didn't give a path but instead returned the exit code 1 (checked with echo $?). The coreutil cd itself is working, so the executable should be there, right? I also ran a find for cd, but there was no executable file shown. How is it implemented then?  Update:  I don't know if I should ask this in another post but since I think it is good here, I'm expanding (?) the post... So the answer was actually quite simple, there's no executable for that — because it's a builtin — But I've found some builtins (bash shell in Fedora) have the executable files! So builtin -> no executable isn't right I suppose? Maybe an answer explaining what builtins actually are (builtin commands?), which actually is the matter here, rather than focusing more on cd... Some good links posted previously indicate that builtins are not programs... so what are they? How do they work? Are they just functions or threads of the shell? Shell Type:  &gt;echo $SHELL /bin/ksh   Cronjob :   15 * * * * /bin/ksh /wls_domains/resMGT/logs/bea/wlr3queuetransaction.sh &gt; /wls_domains/resMGT/logs/bea/data/script.log  The script is as below:  ##log files and lookup file "wlr3queue.txt" are in the same path of the script ##/wls_domains/resMGT/logs/bea/wlr3queuetransaction.sh  Script :  #!/bin/ksh  dt=`date +%Y-%m-%d` xy=`date|awk '{print $4}'|awk -F":" '{print $1}'` yz=`expr $xy - 1` if [ $yz -le 9 ] then yz=0$yz fi  if [ $xy -gt 0 ] then  for i in `cat wlr3queue.txt` do  sum=0  count=`gzgrep -i "$dt" admin_resMGT_access.log* | grep "$yz:[0-5][0-9]" | grep "$i" | wc -l`  if [ $count -gt 0 ] then name=`echo $i | cut -d'/' -f3`  gzgrep -i "$dt" admin_resMGT_access.log* | grep "$yz:[0-5][0-9]" | grep "$i" | awk '{print $8}' &gt; consumed_time.txt  for j in `cat consumed_time.txt` do j_ms=`echo "$j * 1000" | bc` echo $j_ms &gt;&gt; consumed_ms_time.txt done  for k in `cat consumed_ms_time.txt` do sum=`echo "$sum + $k" | bc` done  avg=`echo "scale=2; ($sum/$count) "| bc`  min=`cat consumed_ms_time.txt | sort -n | head -1`  max=`cat consumed_ms_time.txt | sort -n | tail -1`  else  avg=0  min=0  max=0  fi ( echo $yz","$count","$avg","$min","$max )&gt;&gt;/wls_domains/resMGT/logs/bea/data/$name$dt.csv   rm -f consumed_ms_time.txt  done  else  echo "script won't execute at this hour" &gt; temp.txt  fi   I have executed the following commands and the script ran successfully.  ./wlr3queuetransaction.sh sh -x wlr3queuetransaction.sh /bin/ksh /wls_domains/resMGT/logs/bea/wlr3queuetransaction.sh  /bin/ksh/ wlr3queuetransaction.sh.   How to debug? What to do? Proving that the general linear group is a differentiable manifold We know that the the general linear group is defined as the set $\{A\in M_n(R): \det A \neq 0\}$. I have a homework on how to prove that it is a smooth manifold. So far my only idea is that we can think of each matrix, say $A$, in that group as an $n^2-$dimensional vector. So i guess that every neighborhood of $A$ is homeomorphic to an open ball in $\mathbb{R}^{n^2}$ (However, i don't know how to prove this.)  Now, I'm asking for help if anyone could give me a hint on how to prove that the general linear group is a smooth manifold since I really don't have an idea on how to do this. (By the way, honestly, I don't really understand what a $C^{\infty}-$smooth structure means which is essential to the definition of a smooth manifold.) Your help will be greatly appreciated. :) Should first and last name be formatted? Given a file like this:  1,768,12,46576457,7898 1,123,435,134,146 2,345,6756856,12312,1311 5,234,567465,12341,1341 1,3245,4356345,2442,13 9,423,2342,121,463 9,989,342,121,1212   I would like to list all rows (in bash terminal) such that the value in column 1 appears at least twice (in column 1). The result should be  1,768,12,46576457,7898 1,123,435,134,146 1,3245,4356345,2442,13 9,423,2342,121,463 9,989,342,121,1212 What to do when visa admission expires during lockdown? If $G$ is a group and $G'$ is generated by $\{xyx^{-1}y^{-1}|x,y\in G\}$, then $G'\trianglelefteq G$ and  $G/G'$ is Abelian.  At first, I thought this is easy because I thought $g[x,y]g^{-1}=[gxg^{-1},gyg^{-1}]$. But it was a mistake. I could not prove that it is also subgroup of G and $G/G'$ is abelian. I need help. I installed Ubuntu yesterday, so I'm trying to get used to it.  I'm really close to my router, but my wifi signal gets too low. Sometimes I'm in Facebook or Youtube and the signal is high, almost full, but it is usually in one or two bars, and sometimes none.  I can use Internet normally, I can watch videos and stuff, but the problem is when I try to update something or Ubuntu is downloading stuff, the downloads get stuck or get cancelled telling me that I should check my connection.  How can I fix this? I should be getting a really good signal, I'm really close to my router, my phone has full signal, and a notebook with Windows 7 has full signal here too.  Sorry my bad English and I hope you can help me.  $ lsusb Bus 009 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 004 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 008 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 007 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 003 Device 002: ID 0bda:8179 Realtek Semiconductor Corp.  Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 006 Device 002: ID 0458:003a KYE Systems Corp. (Mouse Systems) NetScroll+ Mini Traveler / Genius NetScroll 120 Bus 006 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 005 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub A matrix and its transpose have the same set of eigenvalues/other version: $A$ and $A^T$ have the same spectrum Change date format [YYYYMMDD] to [MM/DD/YYYY] I generated a report that formats date as 20160509 and my Google Sheets locale is already set to United States. I've played around with various options within Format > Number > Date Format but I can't change the dates to MM/DD/YYYY format without doing a simple find and replace. How do I accomplish this? As I found solutions on making site-to-site VPN between Azure and AWS using a software VPN server, I'm asking myself if it's possible to connect the VPN gateways (Azure VNET / AWS VPC) directly.  Is there a limitation that forbids it ? How to check if PHP array is associative or sequential? PHP treats all arrays as associative, so there aren't any built in functions. Can anyone recommend a fairly efficient way to check if an array contains only numeric keys? Basically, I want to be able to differentiate between this: $sequentialArray = [     'apple', 'orange', 'tomato', 'carrot' ];  and this: $assocArray = [     'fruit1' =&gt; 'apple',     'fruit2' =&gt; 'orange',     'veg1' =&gt; 'tomato',     'veg2' =&gt; 'carrot' ]; How to reduce vertical space in matrix? I want to reduce the size of the vertical space in matrix environment and have a result something like this     But at present I manage something like below     with the following code    $\begin{bmatrix}     00 &amp; 08 &amp; 10 &amp; 18 &amp; 20 &amp; 28 &amp; 30 &amp; 38 \\     01 &amp; 09 &amp; 11 &amp; 19 &amp; 21 &amp; 29 &amp; 31 &amp; 39 \\     02 &amp; 0a &amp; 12 &amp; 1a &amp; 22 &amp; 2a &amp; 32 &amp; 3a \\     03 &amp; 0b &amp; 13 &amp; 1b &amp; 23 &amp; 2b &amp; 33 &amp; 3b \\     04 &amp; 0c &amp; 14 &amp; 1c &amp; 24 &amp; 2c &amp; 34 &amp; 3c \\     05 &amp; 0d &amp; 15 &amp; 1d &amp; 25 &amp; 2d &amp; 35 &amp; 3d \\     06 &amp; 0e &amp; 16 &amp; 1e &amp; 26 &amp; 2e &amp; 36 &amp; 3e \\     07 &amp; 0f &amp; 17 &amp; 1f &amp; 27 &amp; 2f &amp; 37 &amp; 3f   \end{bmatrix}$   I can change the font using \mathsf{}, but other than that, how I reduce the vertical space between the rows? I did find things like  to increase vertical space between rows in matrix, but not something that can decrease the space. Is there a higher register word that means to criticize something highly specific and inconsequential? Any synonym I can think of seems a little too vernacular, e.g. "to split hairs", "to nitpick". What's a more professional/academic way of saying this? I'd prefer a single word if one exists. After upgrading to Ubuntu 17.10, Ubuntu Dock icons of some of my apps are not appearing. They appear properly in the dash. I verified the *.desktop files for the applications in /usr/share/applications and ~/.local/share/applications.  E.g. dock icon is not displaying for "Terminator". Check out the attached image.    This is how the icon for the same application appears in dash search:    Content of /usr/share/applications/terminator.desktop:  [Desktop Entry] Name=Terminator Comment=Multiple terminals in one window TryExec=terminator Exec=terminator Icon=terminator Type=Application Categories=GNOME;GTK;Utility;TerminalEmulator;System; StartupNotify=true X-Ubuntu-Gettext-Domain=terminator X-Ayatana-Desktop-Shortcuts=NewWindow; Keywords=terminal;shell;prompt;command;commandline; [NewWindow Shortcut Group] Name=Open a New Window Exec=terminator TargetEnvironment=Unity   Any pointers to debug this issue will be helpful. Is it true that for each set $M$ a given injective function $f: M \rightarrow M$ is surjective, too?  Can someone explain why it is true or not and give an example? Adjective meaning "argumentative for argument's sake"?