Let $p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$ with   $a_0,a_1,\dots,a_n \in \Bbb R$. Prove that if $a_0a_n &lt; 0$, then $p$   has a positive root.   I was thinking of using the intermediate value theorem, but not quite sure how to formulate my proof and I also do not know how to show/why $p$ has a positive root.
Let: $P(x) = a_nx^n + a_{n-1}x^{n-1} + ..... + a_1x + a_0$ where $a_0a_n &lt; 0$ I have to prove that the Polynomial $P(x)$ has at least one positive root how can I prove it? Any ideas?
As far as I can tell downvoting is mostly useless.     For questions the only people who get downvoted are lower rep users, or incredibly stupid questions    most questions that are downvoted get closed anyway which should be sufficient downvoting doesn't have too much affect on your rep anyway a negative comment is more useful and more of a push too change or delete the question  For answers downvoting is often used incorrectly as well   people commonly competitively downvote (downvoting another answer to a question that you also answered) obviously not a good behavior downvoting (and upvoting) are contagious, meaning one downvote leads to a bunch of others even though downvotes don't effect rep that much they create a negative environment, no one wants their post downvoted, on the other hand criticizing comments, I think, are far more welcome      So basically in a nutshell a downvote serves no real purpose even if a post deserves it. 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Of course I can do it manually as in the MWE below, but was hoping that there might be an easy way to use the commented out syntax to accomplish this:  Code: \documentclass{article} \usepackage{xcolor} \usepackage{siunitx}  \newcommand*{\Color}[2][red]{\textcolor{#1}{#2}}%  \begin{document}  \num{99 976}  88\,9\Color{76} %\num{88 9\Color{76}}  77\,\Color{976} %\num{77 \Color{976}}  \end{document} What is the difference between "I've been to China" and "I've gone to China"? Both are in the present perfect, but one uses the verb 'to go' and the other 'to be'. Is there a rule for this?  Is there any chance that the differences between "I have been" vs. "I have gone", are differences in English dialect (I don't know if this is the right word).  Edit Could I say that North American English speakers use "I have gone", and maybe U.K speakers say "I've been"? Or vice versa. I am using a mirror modifier for modelling. 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So every now and then, a new bot is created, and usually you would ask a question using that bot or suggest enough edits to get the reputation necessary for that bot to operate. Instead of suggesting 100 edits, Undo has found a more efficient way, namely by . Now someone could frown on this action or not, actually it is the least intrusive way to get the bot the reputation necessary (I guess no one really likes a bot to suggest 100 edits just for the sake of earning enough reputation). The question is: is this actually the best way? Was this action allowed or should we look away because it is Smokey? Should we maybe have a special bot user status which derives the privileges from its master user? Anything else you want to share on this? Related: I have a Python script which takes as input a list of integers, which I need to work with four integers at a time.  Unfortunately, I don't have control of the input, or I'd have it passed in as a list of four-element tuples.  Currently, I'm iterating over it this way: for i in range(0, len(ints), 4):     # dummy op for example code     foo += ints[i] * ints[i + 1] + ints[i + 2] * ints[i + 3]  It looks a lot like &quot;C-think&quot;, though, which makes me suspect there's a more pythonic way of dealing with this situation.  The list is discarded after iterating, so it needn't be preserved.  Perhaps something like this would be better? while ints:     foo += ints[0] * ints[1] + ints[2] * ints[3]     ints[0:4] = []  Still doesn't quite &quot;feel&quot; right, though.  :-/ Related question: Supremum of a Continuous Function is Continuous Center of a free product Let $A$ and $B$ be two groups and let $G=A*B$ be their free product. Suppose both $A$ and $B$ are nontrivial. I want now to prove that $G$ is infinite and that $Z(G)$ is trivial.  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This would be fine if you could increase the width of the list as you can with the height, which would in theory separate the lines, but you can't.  Can we fix this, please?  UPDATE: I have found that  post is almost exactly the same as mine, but steady on with duplicate votes.   comment says that it will be fixed, but that question was asked more than five years ago, and in that time the problem has not been resolved.  UPDATE 2: It seems that  would help.  I changed a little CSS/HTML, and came up with this:  Although mine is a little cruder than it could be.  This is just an example.