If $f(x)$ is a function with all coefficients $1$ or $-1$ and all roots real then degree of $f(x)$ can be If $f(x)$ is a real function with all roots real and $f(x)$ has all its coefficients $1$ or $-1$. (Some can be 1 and others can be -1 but no other value) . Then we have to find the constraint on degree of $f(x)$.  My attempt : Let $f(x)$ be   $a_{0}x^{n} +a_{1}x^{n-1} + ....+ a_{n-1}x +a_{n}$ So we have sum of roots(S) = $+1$ or $-1$ , Similarly product of roots taken two at once(P)= $+1$ or $-1$. So sum of squares of roots = $S^2 - 2P$ .This must be greater than 0.Solving this we get sum of squares of roots is equal to $3$. (Excluding the case when degree is$1$).But after this I couldn't arrive at any satisfactory result.  Note :: We have not been taught calculus yet.
Prove that polynomial of degree $4$ with real roots cannot have $\pm 1$ as coefficients (IITJEE) So I was going through  my 11th class package on Quadratic equations and I saw a question to prove that a polynomial of $4$th degree with all real roots cannot have $\pm 1$ as all its coefficients.   I tried proving it using calculus, by showing that at least one consecutive maxima  and minima will lie either above or below the x axis, but couldn't solve it using that.  I also tried using Descartes Rule of Signs but couldn't solve it with that too. Any help?
Do you recommend using semicolons after every statement in JavaScript? In many situations, JavaScript parsers will insert semicolons for you if you leave them out. My question is, do you leave them out?  If you're unfamiliar with the rules, there's a description of semicolon insertion on the . Here's the key point:     If the first through the nth tokens of a JavaScript program form are grammatically valid but the first through the n+1st tokens are not and there is a line break between the nth tokens and the n+1st tokens, then the parser tries to parse the program again after inserting a virtual semicolon token between the nth and the n+1st tokens.   That description may be incomplete, because it doesn't explain @Dreas's example. Anybody have a link to the complete rules, or see why the example gets a semicolon? (I tried it in JScript.NET.)   is related, but only talks about a specific scenario. As the title says. How do I get from  $\frac{n}{\varphi(n)}$ to $\sum\limits_{d \mid n} \frac{\mu^2(d)}{\varphi(d)}$?  I know that $$\frac{n}{\varphi(n)}=\frac{\sum\limits_{d \mid n} \varphi(d)}{\sum\limits_{d \mid n} \mu(d)\frac{n}{d}},$$ and I suspect it is down this road I should go. But I get totally confused by the sums and I have no clue on how to divide them and "combine" them into one again. Any hints or advice would be much appreciated. What special characters must be escaped in regular expressions? I am trying to write a VF page to list all the id and name columns from a custom object but found an error "List has more than 1 row" in the VF page.  Can anyone tell me what have I done wrong?  VF Page:  &lt;apex:page Controller="MP_controller" showHeader="false" sidebar="false"&gt;   &lt;apex:pageBlock&gt;      &lt;apex:pageBlockTable value="{! mp}" var="item"&gt;         &lt;apex:column value="{! item.id}"/&gt;         &lt;apex:column value="{! item.name}"/&gt;       &lt;/apex:pageBlockTable&gt;   &lt;/apex:pageBlock&gt; &lt;/apex:page&gt;   Apex controller  global with sharing class MP_controller {     public Merchant_Profile__c mp{get;set;}     public MP_controller(){         mp = [SELECT                  ID, name, Profile_account_name__c, Merchant_ID__c, Bank_Code__c, Country_Code__c,                 Currency_Code__c, Merchant_Level__c, Merchant_Type__c, MSO__c, Region__c,                 Retail_Products__c, Transaction_Mode__c, Created_Date__c, Date_Disabled__c, Date_Live__c,                 Merchant_Category_Code__c                FROM Merchant_Profile__c               WHERE Profile_Account_Name__c = '001O000000bQFnyIAG'];     } } How to reinstall Ubuntu keeping my data intact? Noticing that Newtonian gravity and electrostatics are equivalent, is there also a relationship between the general relativity and electrodynamics? In classical mechanics, we had Newton's law of gravity $F \propto \frac{Mm}{r^2}$.  Because of this, all laws of classical electrostatics applied to classical gravity if we assumed that all charges attracted each other due to Coulomb's law being analogous.  We can "tweak" classical electrostatics to fit gravity.  In modern physics, does the reverse work?  Can we "tweak" General Relativity to accurately describe electrostatics or even electromagnetism? How do I stop sharing call history between two iPhones? My husband and I both just upgraded to iOS 8.3.  Now I notice that if I have a missed call it shows up on his notice screen and all my calls show up in his recent call history (and vice versa).  Both phones don't ring and they don't need to be near each other or connected by Wi-Fi.  We do share an iTunes account.  How do I stop sharing call history between the two phones? Given a Möbius transformation that maps one pair of concentric circles to another pair of concentric circles, why is the ratio of the radii preserved through the map?   I thought about how Möbius transformations are compositions of rotations, scaling, inversion, and translation, and that intuitively, these types of maps shouldn't change the ratio of radii between two circles.   Would it be correct to just say that if $\frac{r_1}{r_2}$ is the ratio of radii between the two circles, then   1) The radii are invariant under translation, $z \mapsto z+a$, so $\frac{r_1}{r_2}$ stays the same  2) Under scaling by a factor $z \mapsto az$, $\frac{ar_1}{ar_2} = \frac{r_1}{r_2}$   3) Under inversion, $z \mapsto \frac{1}{z}$, $\frac{1/r_1}{1/r_2} = \frac{r_2}{r_1}$  Or is there a different/better way to think about this problem? Story about protagonist finding a world that has died with a museum I asked a question on Stack Overflow: . I want to award my bounty to  by user  However, my Stack Overflow account is suspended recently for 7 days, and the suspension still has 5 more days to end.And I know I can't vote, ask, answer or comment as pointed on  by  with the sentence below.  You will be unable to vote, ask, answer, or comment.  I couldn't find anything about not being able to accept an answer or award bounty so I tried to accept the mentioned answer above and award my bounty to it and I got a  Suspended users cannot vote  rejection which can be seen below with a screenshot.  I know this  If you do not award your bounty within 7 days, the highest voted answer created after the bounty started with at least 2 upvotes will be awarded half the bounty amount.  from . None of the answer given to my question is upvoted and the answer I mentioned is not the oldest one after the bounty started so it won't be awarded by community. What can I do? (except being a good guy and not get suspended :)) When I do upgrade it gets stuck here:  user@plato:~# sudo apt-get upgrade Reading package lists... Done Building dependency tree        Reading state information... Done Calculating upgrade... Done The following packages will be upgraded:   bind9-host coreutils cpio curl dnsutils libbind9-90 libc-bin libc-dev-bin   libc6 libc6-dev libcgmanager0 libcurl3 libcurl3-gnutls libdns100   libdrm-intel1 libdrm-nouveau2 libdrm-radeon1 libdrm2 libisc95 libisccc90   libisccfg90 libjasper1 liblwres90 libplymouth2 libsepol1 libssl-dev   libssl-doc libssl1.0.0 linux-libc-dev mime-support multiarch-support ntp   ntpdate openssl plymouth plymouth-theme-ubuntu-text python3-distupgrade   ubuntu-release-upgrader-core unzip 39 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Need to get 16.0 MB of archives. After this operation, 12.3 kB of additional disk space will be used. Do you want to continue? [Y/n] y 0% [Connecting to us.archive.ubuntu.com (2001:67c:1562::15)] ░   When I do update it gets stuck here:  user@plato:~# apt-get update 0% [Connecting to us.archive.ubuntu.com (2001:67c:1562::14)] [Connecting to sec░   I'm able to ping us.archive.ubuntu.com just fine:  user@plato:~# ping us.archive.ubuntu.com PING us.archive.ubuntu.com (91.189.91.23) 56(84) bytes of data. 64 bytes from economy.canonical.com (91.189.91.23): icmp_seq=1 ttl=54 time=81.5 ms 64 bytes from economy.canonical.com (91.189.91.23): icmp_seq=2 ttl=54 time=81.5 ms 64 bytes from economy.canonical.com (91.189.91.23): icmp_seq=3 ttl=54 time=81.5 ms 64 bytes from economy.canonical.com (91.189.91.23): icmp_seq=4 ttl=54 time=81.4 ms ^C --- us.archive.ubuntu.com ping statistics --- 4 packets transmitted, 4 received, 0% packet loss, time 3004ms rtt min/avg/max/mdev = 81.425/81.528/81.582/0.355 ms   What can i do to fix this?    What I have tried so far.  I've tried following directions on this blog:   To force apt-get to use IPv4 exclusively   Create a file in /etc/apt/apt.conf.d/99force-ipv4 Add this line to newly created file Acquire::ForceIPv4 false; Apt-get will now use IPv4 exclusively Calculate sums of inverses of binomial coefficients How to calculate the sum of sequence $$\frac{1}{\binom{n}{1}}+\frac{1}{\binom{n}{2}}+\frac{1}{\binom{n}{3}}+\cdots+\frac{1}{\binom{n}{n}}=？$$ How about its limit? While there be any issue applying for a new passport before stamping? Bizarre spacing with 4 $\star$s Why do we assume $a+b=b+a$ in a Ring with 1. Also, is it true with Rings without 1? I was wondering why do some people use redundant axioms in definitions?  If you just expand $(a+1)(b+1)=(a+1)b+a+1=ab+b+a+1$ $(a+1)(b+1)=a(b+1)+b+1=ab+a+b+1$. Hence, $ab+a+b+1=ab+b+a+1$, then cancel ab and 1. Then, you get it's commutative for free.   Why do we then assume it?   Also, is it even stronger. If we drop the condition that 1 is in R, then can we still deduce a+b=b+a? As clearly the argument just done falls apart if you haven't got 1 in R. How do you disable browser autocomplete on web form field / input tags? Prevent VLC from starting a new instance every time a media file is opened Sell your home and invest in growth stock mutual fund I purchased a home and I made an agreement with a guy who is living in that home and paying me monthly rent. Since I live with my brother I don't have to live in my purchased home.  The rent I'm getting is only 10% of total cost of home while stock fund industry (mutual fund) is paying 20% minimum i.e. I'm getting 10% loss every year and that is a big amount.  I'm not really sure if my home also increase it's value by 10% every year. But I don't think it will be.  Now I'm thinking of selling that home and investing all of my money in a growth mutual fund. Any advice? The (countably infinite) wedge sum of circles is quotient of  disjoint countable union of circles $\amalg S_i$, with points $x_i\in S_i$ identified to a single point, while the infinite earring H is the topological space defined by the union of circles in the Euclidean plane $\mathbb{R}^2$ with center $(1/n, 0)$ and radius $1/n$ for $n = 1, 2, 3, ...$. In the definition of countably infinite wedge sum of circles, it is not specified the size of circles, the points which to be identified to a single point etc. So we can take disjoint union of circles of radius $1/n$ and identify a point from each to a common single point to get the infinite earring. I couldn't understood the difference between these two topological spaces. Can one explain more precisely the difference between these two spaces? Correcting projection of 360° content onto a sphere - distortion at the poles I'm using the Autopano Giga 4.4 and Autopano Video Pro 2.6 to stitch and create spherical content, but I've recently moved into wanting to create 360° tours via Unity. I know this is partly a developers question, but if anyone out there can help me with the right specs for exporting a spherical image or video from Giga to use to create the image for the inside of the sphere on Unity, I'd appreciate it!  Experimenting with Premiere and FCPX to edit highlight reels of spherical content, I've had seamless experiences created. For some reason in Unity, the images appear to be stretched around the sphere wrong and it's creates issues on the top and bottom of the image:    The image shown is the bottom of the sphere. I'm using the Android platform to hopefully turn these tours into apps when completed. Thank you for any help!