For what natural numbers k and n can the polynomial $x^k+x^n$ be divided by $x^2-x+1$ Please explain the solution, or tell me what is it that I should learn/look up, in order to know how to solve such problems.
I am stuck at my exam practice here.      The remainder of the division of $x^3$ by $x^2-x+1$ is ..... and that of $x^{2007}$ by $x^2-x+1$ is .....   I tried the polynomial remainder theorem but I am not sure if I did it correctly.  By factor theorem definition, provided by ,     the remainder of the division of a polynomial $f(x)$ by a linear polynomial  $x-r$ is equal to $f(r)$.   So I attempted to find $r$ by factorizing $x^2-x+1$ first but I got the complex form $x=\frac{1\pm\sqrt{3}i}{2}=r$.   $f(r)$ is then $(\frac{1+\sqrt{3}i}{2})^3$ or $(\frac{1-\sqrt{3}i}{2})^3$ which do not sound right.  However, the answer key provided is $-1$ for the first question and also $-1$ for the second one. Please help.
Open an app on startup/after booting I need to force a selected application to start up each time my Android-based device is (re)started.  Is there anything I can use (configuration setting, external application etc.) to accomplish this task? Or is this kind of behavior limited to home screen managers only? How can I run scripts automatically when Ubuntu starts up so I don't have to run them manually after startup? Prove or Disprove the Existence of Solutions... Is there a way to disable the hotkeys? What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity? I tried and got this   $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an integer. $$\lim_{n\to\infty}n\sin(2\pi en!)=\lim_{n\to\infty}n\sin\left(2\pi n!\sum_{k=0}^n\frac{1}{k!}\right)=\lim_{n\to\infty}n\sin(2\pi m)=\lim_{n\to\infty}n\cdot0=0$$  Is it correct? How can I use Bengali script in an English document? The anime Elfen Lied is notorious for its one of a kind OP, featuring the mesmerizing theme song Lilium composed by MOKA and performed by Kumiko Noma, as well as sequences of art porn, a homage to Austrian symbolist painter Gustav Klimt. The idea to pose the characters as in Klimt's artworks came, as far as I know, from director Mamoru Kanbe, who continued this idea in the OP of Sora no Woto. (In my humble opinion, the OP of Sora no Woto is rather bland, and seriously, Klimt with Kalafina?)  This is a self-answered question and a modest attempt at unraveling the sources of and the symbolism represented in Elfen Lied's OP scenes. Other answers and/or suggestions are welcome. How to configure axis on pgfplots? How to configure axis on pgfplots? Similar for this.  \documentclass{article} \usepackage{tikz,pgfplots} \pagestyle{empty} \begin{document} \begin{tikzpicture}[&gt;=latex] %x axis \draw[-&gt;] (-5,0) -- (5,0) node[below] {$x$}; \foreach \x in {-4,...,-1,1,2,...,4} \draw[shift={(\x,0)}] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$}; %y axis \draw[-&gt;] (0,-5) -- (0,5) node[left] {$y$}; \foreach \y in {-4,...,-1,1,2,...,4} \draw[shift={(0,\y)}] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\y$}; \node[below left] at (0,0) {\footnotesize $0$}; \end{tikzpicture} \end{document} Bounded sequence which is not convergent, but differences of consecutive terms converge to zero I have a question that says "Show that there is a bounded sequence $x_n$ which is not convergent but has the property that $x_n - x_{n+1} \to 0$ as $n \to 0$.  What does this mean? Do I need to come up with an example or does the problem actually want me to prove such proposition?  By the way, I see that this sequence looks like Cauchy because of $x_n - x_{n+1} \to 0$ as $n \to 0$, but it is obviously not. In Volo's Guide to Monsters, there is a race in the bestiary called Grungs. Were these created for 5e specifically?  I'm pretty sure they were created for 5e, as they have no deities listed which is always a bit suspicious, but I'm just making sure. Radiation– white vs black house, hot or cool? In my book's thermodynamics chapter, it says that an      "object that radiates heat faster also absorbs heat faster. This means   that an object that is a more efficient radiator comes to equilibrium   with its environment more quickly. With this in mind, is it better to   paint your house black or white?"   I am confused which it would be. The book says white because      "in summer, you house is cooler than the environment and white   reflects away the heat. In winter your house is warmer than the   environment and white radiates away the heat" Google map only loading half of itself Today a couple of buffer overflows in NTP were announced , . It looks like updating my system to fix these issues is in order.   How can I find out if they have been fixed in the Ubuntu repositories, such that if I were to run:   sudo apt-get update sudo apt-get upgrade   then the fix would get installed and the vulnerability closed?   Edit: The selected answer specifically addresses the question of how to identify if a given CVE has been fixed or not, "Does Ubuntu generally post timely security updates?"  is certainly related but not identical Is it possible to redirect connections to a specific IP/port to an external IP/port?  Example:   Server A has the external IP xxx.xxx.xxx.xxx Server B has the external IP yyy.yyy.yyy.yyy User sends a request to server A on port 2106, I want to redirect it to server B at port 2106. How do you make a deep copy of an object? It's a bit difficult to implement a deep object copy function. What steps you take to ensure the original object and the cloned one share no reference? I often use Cycles' built in Glass BSDF shader to achieve realistic refraction effects, but in many cases, it seems that Cycles' glass caustics creates an unfortunately large number of fireflies (pixels that are abnormally bright relative to their surroundings). In some models of mine, even when I render for 3000 or more samples/pixel, my final image is still covered in fireflies. While I can improve this in many cases by increasing the total amount of light available in the scene, this loses a lot of the potency of the refractive effects I'm looking for, especially in scenes with low light levels.  So my question is, how can I reduce the amount of fireflies while using Cycles' caustics without sacrificing the quality of the refractive effects?  Note: I am aware that Cycles' caustics will perform much better once Cycles has full support for bidirectional path-tracing, but it looks unlikely that that will become , unfortunately. How to customize MacBooks' keyboard? I spilled a juice on my MacBook-Pro keyboard. Consequently, some keys like 2, esc and tab don't work anymore on my keyboard.   Is there anyway to change those disabled keys with combination of some other keys, like Fn,Ctrl, Option+ something, using an app or a command? How do I remove an account from Google multi-login? I have three accounts among the multiple accounts I have for Google's products. One of them belongs to a friend who accidentally made his account one of mine when he was using my PC.  How do I remove his account from my Google multiple accounts? Logging out from his account doesn't work. His account still displays under the user dropdown even though I can't log in to it. How do you close windows/tabs in 2.8? I've tried a lot of stuff, but always end up opening up more. Understanding a probability paradox Three prisoners are informed by their jailer that one of them has been chosen at random to be executed and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information because he already knows that at least one of the two will go free. The jailer refuses to answer the question, pointing out that if A knew which of his fellow prisoners were to be set free, then his own probability of being executed would rise from 1 3 to 1 2 because he would then be one of two prisoners. What do you think of the jailer’s reasoning?  If the jailer refuses to say anything, then the probability that prisoner $A$ is excecuted is $\frac{1}{3}$. If the jailer says to prisoner $A$ that prisoner $B$ will walk free, then $2$ prisoners remain to be considered, $A$ and $C$. One dies, one does not. Heads or tails essentially. $\frac{1}{2}$ ought to be the conditional probability that $A$ dies given that $B$ walks free no? Apparently not though, allegedly the correct answer is still $\frac{1}{3}$. Even my attempt to calculate the correct answer yielded the result $\frac{1}{2}$.   Let $A_D$ and $C_D$ respectively denote the event of $A$ and $C$ dying. Let $B_F$ denote the event that $B$ walks free. Assume that the jailer tells prisoner $A$ that prisoner $B$ will walk free.   Here's my attempt.  $$P(A_D\mid B_F)=\frac{P(A_D\cap B_F)}{P(B_F)}=\frac{P(A_D\cap B_F)}{P((B_F\cap A_D)\cup (B_F\cap C_D))}=\frac{P(A_D)P(B_F\mid A_D)}{P(A_D)P(B_F\mid A_D)+P(C_D)P(B_F\mid C_D)}=\frac{\frac{1}{3}\times 1}{\frac{1}{3}\times 1+\frac{1}{3}\times 1}=\frac{1}{2}$$  What am I doing wrong?  Edit 1: Intuitively I am still troubled but I understand now that $B_F$ may occur even though the jailer does not necessarally say $B$.   Edit 2: I suppose that it makes some sense if the faiths of the prisoners had already been decided before prisoner $A$ asked the jailer the question. If the jailer decides to reveal one of the others who will walk free then that must've been the case.