I have no idea how to solve this question. I'm trying to show that given $p(x) = a_0 + a_1 x + .. + a_n x^{n}$ and further assuming that $a_n \neq 0$ and n is odd, there exists x such that p(x)=0.  I'm guessing that Rolle's theorem might come into play somewhere, since I know that by Rolle's theorem if $f:[a,b] \rightarrow \mathbb{R}$ is continuous on [a,b] and differentiable on (a,b) and f(a) = f(b) then $\exists c \in (ab)$ s.t f'(c)=0. Other than this hunch, I have no idea where to begin solving this.  Help would be much appreciated!
I want to prove that every real polynomial of odd degree has at least one real root, using  the intermediate value theorem.  Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in \mathbb{R}$ and $n \in \mathbb{N}$.  By the fundamental theorem of algebra I know that $P(x)$ has exactly $2n+1$ complex roots, so  $P(x) = (x+r_1)(x+r_2) . . . (x+r_{2n+1})$ for each $r_i \in \mathbb{C}$  I do not know how to complete this but I do know that, at some point, I probably have to show that each root with imaginary part non zero has to come in conjugate pairs, and since $2n+1$ is odd there is at least $1$ root that is imaginary part $0$ and thus real.
Firefox - Websites with custom fonts not displaying I am having trouble with the following problem:   $f: N\times N\rightarrow N$ and $f(i,j)=2^{i-1}(2j-1)$. Prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically equivalent.   Work: I am told that Euclid's Lemma will be useful to proving injection and also that for any positive integer $n$, $n$ is a product of prime numbers.   For injection, I set $f(a,b)=f(x,y)\rightarrow 2^{a-1}(2b-1)=2^{x-1}(2y-1)$.  Then I distributed across to get $2^ab-2^{a-1}=2^xy-2^{x-1} \rightarrow 2^ab-2^xy=2^{a-1}-2^{x-1} \rightarrow 2(2^{a-1}b-2^{x-1}y)=2^{a-1}-2^{x-1}$.   From here, I am unsure how to apply Euclid's Lemma to arrive at the injection result.  I am even more unsure how to prove surjection. What is a good introductory book on quantum mechanics? Why do we use the base form of verbs instead of the past form in past tense questions? How can I run ssh-add automatically, without a password prompt? I want to communicate between several computers on my network (static Ethernet), through SSH. In order to do that I need to run ssh-add every time I log in on a specific machine, how can I do it so that it's set up once and it doesn't ask me for the passphrase every time I log in or reboot my machine?  I know that there is a way that you should add some lines to the bash_profile file, but I still need to type the password every time I reboot/log in to a specific machine.  if [ -z "$SSH_AUTH_SOCK" ] ; then     eval `ssh-agent -s`     ssh-add fi Single word which means "keep moving forward" I need a single word which means "keep moving forward". What are my options?  I need word as   persevere Be persistent, refuse to stop  cognizant (sometimes followed by 'of') having or showing knowledge or understanding or realization or perception  [] keep moving forward  I want to make meaningful name for firm  please help me Jordan normal form and invertible matrix of generalized eigenvectors proof Struggling to find a place to start with this proof- just began learning about Jordan normal.  Given a 2-by-2 matrix $A$ and a Jordan normal form matrix $J_{\lambda}$, there exists a matrix $S = [v1, v2]$ s.t. $$S^{-1}AS = J_{\lambda}$$ That is, $J_{\lambda}$ is similar to $A$.  WTS: $$(A - \lambda I)v_{1} = 0 $$ $$(A - \lambda I)v_{2} = v_{1}$$    I was going to try to start with the similarity condition and work into producing the two statements below.  Transform the similarity condition to: $$(A = SJ_{\lambda}S^{-1}) * v_{1}$$  Rewrite our first statement to prove as: $$Av_{1} = \lambda I v_{1}$$   Substitute: $$\lambda I v_{1} = SJ_{\lambda}S^{-1}v_{1}$$  But I've been continually getting stuck.  Tried a few ways other than this.  What intuition am I missing? Case insensitive 'Contains(string)' Which is stronger, a rope without knots or a rope with knots? Suppose I have two identical ropes, one of which I manually tie some knots in. If I use them to hang clothes, which one is more durable?      Personally I think that rope with knots will be more durable, but I can't come up with a satisfactory reason. What is the equivalent of Persian idiom "When the reed blooms"? In Persian, we say "When the  blooms" when we want to  express that an event:   Never happens. (This is only the opinion of the speaker so it's not a fact) It's very unlikely to happen. It's going to take so long to actually happen. (It's so long that you'd be better off giving up expecting it)  3.1. We also use it when someone is always postponing something they have to do so you never actually see it happen.   For example:      Kyle: When are you going to pay my money back?      Cartman: When the  blooms.   Now I don't know if I got the plant's name right because there are some other versions like:        In any event, they have mentioned this plant in the idiom because apparently some versions of it actually never bloom. Others believe it's because farmers harvest them before they get the chance to bloom (e.g to make music instruments), so it never happens. I'd love to know the equivalent idioms or expressions for this.  PS: I only know "When pigs fly" but I'm not a fan of it because the word pig sounds a bit rude to me and partly because it implies a strong impossibility which doesn't work for the #3 use. What can cause an answer to get deleted without information about who actually deleted it? Usually when answer gets deleted directly (i.e. not as side effect of its parent question deleted) it says "deleted by [name here] [time stamp]". Plus, it always leave a revision.  However, I just stumbled over  on Stack Overflow:        Weird, don't you think? Looking in the , nothing strange in there and nothing strange in  as well.  Finally, the last straw... :        Another thought was maybe some automagical conversion to comment, but no such comment exists.  Can anyone shed some light over this phenomenon? Origin of "Fits [x] to a T"? The above phrase is something I've known for as long as I can remember, though I don't know from where. What is its origin and usage? Example: &quot;While Android products aren't quite what I need, iOS devices fit my needs to a T.&quot; I want to run a command line that opens a new chrome window with another profile. I tested the following commands: Preconditions:  I have two profiles: Profile 1 and Profile 2  Chrome is already running with Profile 1   open -a &quot;Google Chrome.app&quot; --args --profile-directory=&quot;Profile 2&quot; The above command does nothing other than bringing focus to the currently running chrome with Profile 1 I also tried: open -a &quot;Google Chrome.app&quot; --args --profile-directory=&quot;Profile 2&quot; --new-window The effect is the same. I want to run a command line that opens a new chrome window with Profile 2 while there is already a Chrome window running with Profile 1 Writing letters of recommendation for more than one student to the same department Suppose multiple students have asked me for letters of recommendation to the same graduate department in the US. Am I expected to explicitly compare the students to one another? Am I supposed to do this in both letters (assuming they haven't been submitted yet)? Do I need to explicitly state a preference for one over the other, or can I just get away with listing relative strengths and weaknesses (for instance, "X is stronger academically, but Y has more research experience and is a stronger programmer")? Can players enchant items? I know that the DM can give you enchanted items, but can players spend gold and downtime to enchant their own? Map matrix element to string Will something accelerate forever if a constant force is applied to it on a frictionless surface? If so how? I don't understand the fact that let's say hypothetically I was pushing a block on a frictionless surface, and I kept applying the same force on the object how it would keep accelerating! Wouldn't there have to be a certain velocity where it would stop getting faster? If not then what is actually acting on the object to cause it to accelerate infinitely? Let $\alpha$ be an irrational real number and let $a_j$ be a sequence of rational numbers converging to $\alpha$. Suppose that each $a_j$ is a fraction expressed in lowest terms: $a_j = \alpha_j / \beta_j$. Prove that the $\beta_j$ tend to $\infty$  Attempt: AFSOC that $\beta_j$ does not tend to infinity, then it is bounded by some $M$. We can also find an interval $\alpha \in (k, k+1)$. Let $\epsilon &gt; 0$, $\exists N_0$ such that for all $N &gt; N_0$ we have $|a_N - \alpha|  &lt; \epsilon$. So $|\frac{p_N}{q_N} - \alpha| &lt; \epsilon$. We try to use the fact that $q$ is bounded but fail to derive a contradiction. I do not know how to use the fact that $\alpha$ is between two integers, although it might not be relevant at all.  Ideas? I am working on vector auto-regression (VARs) and impulse response function (IRFs) estimation based on panel data with 33 individuals over 77 quarters.  How should this type of situation be analyzed?  What algorithm's exist for this purpose?  I would prefer to conduct these analyses in R, so if anyone is familiar with R code or a package designed for this purpose that they could suggest, that would be especially helpful. How to watch Youtube fullscreen on one monitor of a dual-monitor display? How to watch Youtube fullscreen on one monitor of a dual-monitor display, in Linux?  If I do it in Google Chrome, as soon as I do anything at the other monitor, Chrome jumps back to normal window.  Edit: added that OS is Linux.