Find the value of $a$ and $b$ if $( x + 1 )$ and $( x -2 )$ are factors of $ax^3 - 4x^2 + bx - 12$. This is regarding polynomials.   Thank you
Find the value of $a$ and $b$ if $x+1$ and $x−2$ are factors of $ax^3−4x^2+bx−12$      Attemption $$ \begin{cases} f(-1) = -a - b - 16\\ f(2) = 8a + 2b - 28 \end{cases} $$ But then when I plug in $$b = 28 - 8a$$. Then I get $$-a -(28- 8a) - 16=0$$$$a = 34/7$$  In the book it says $a = 10$ and $b = -26$. So I am stumped. Please help
Prove that the product of two continuous functions is continuous I have some questions about a suspended user's earning and activity.   Can suspended users get reputation back after the suspension period? If a user is suspended for 7 days and he/she earns reputation during this period. Then will it be calculated after completing suspension period? If a suspended user has awarded a bounty to other user and the awarded bounty marked as illegal. Then will that bounty rolled back to their account?  Can suspended users get any alert from comments or chat room using @username method? Can suspended users post comments on his/her question/answer? If another user replies or ask a question on suspended user's post. Then, can a suspended user give response back? I've installed Xcode and MacPorts on a brand new mid-2013 MacBook Air, but MacPorts doesn't work, saying that it cannot find anything.  mba:~ cnst$ sudo port install wget Warning: The Command Line Tools for Xcode don't appear to be installed; most ports will likely fail to build. Warning: See http://guide.macports.org/chunked/installing.xcode.html for more information. ---&gt;  Computing dependencies for wgetError: Unable to execute port: can't read "build.cmd": Failed to locate 'make' in path: '/opt/local/bin:/opt/local/sbin:/bin:/sbin:/usr/bin:/usr/sbin' or at its MacPorts configuration time location, did you move it? mba:~ cnst$   And, indeed, there is no make or clang in the path, but it is definitely installed, though:  mba:~ cnst$ find /Applications -name "clang" -or -name "make" /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/clang /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/lib/clang /Applications/Xcode.app/Contents/Developer/usr/bin/make mba:~ cnst$   Why is there no make or clang in the path?  Why MacPorts doesn't work?  I notice that  mentions that you have to install something from ADC, Apple Command Line Developer Tools, but ADC doesn't work since last Thursday or Friday due to their security breach, so, the developer tools are nowhere to be found.  What's the proper solution here? get dynamic properties in javascript Trajectory of a photon around a Schwarzschild black hole? Consider a photon coming from the infinity in a unbounded orbit to a Schwarzschild black hole (Schwarzschild radius $r_{s}$) (see  for illustration). Its impact parameter is $b$ and its distance of closest approach is $r_{0}$ with $$b^2=\frac{r_{0}^{3}}{r_{0}-r_{s}}$$.  Then its trajectory in polar coordinates is defined by :  $$\frac{d\varphi}{dr} = \frac{1}{r^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{r}\right)\frac{1}{r^2}}}$$  Consequently : $$\varphi\left(r\right) = \int_{r_{0}}^{r} \frac{dp}{p^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{p}\right)\frac{1}{p^2}}}$$  and one can compute the total deviation using :  $\Delta\varphi = 2\times\left(\lim_{r\to+\infty}\varphi\left(r\right)-\frac{\pi}{2}\right)$  But my question is : how can I plot/draw the trajectory using the integral expression of $\varphi\left(r\right)$ ?    Because if I compute : $$f\left(r\right) = 2\times\left(\int_{r_{0}}^{r} \frac{dp}{p^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{p}\right)\frac{1}{p^2}}}-\frac{\pi}{2}\right)$$ I obtain $f\left(r_{0}\right) = -\pi$, and then $f$ increases up to zero, crosses zero, and tends to its positive value at infinity $\Delta\varphi$. It does not make sense for me and I do not understand how to compute the trajectory from that.    If I compute : $$g\left(r\right) = \int_{r_{0}}^{r} \frac{dp}{p^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{p}\right)\frac{1}{p^2}}}$$ it starts from $0$, and increase up to $\frac{\pi}{2}+\frac{\Delta\varphi}{2}$.    I would like to compute the trajectory in the $\left(x, y\right)$ plane, so how to use the values of $f\left(r\right)$ or $g\left(r\right)$ to compute the function $y\left(x\right)$ ? I want to make a power DC supply with voltage rating 0-200V. I want it to be a variable power supply with voltage from 1V to 200v. What should be my approach? Why do we need a diagonal matrix? Apart from simplifying matrix powers, why do want to diagonalize a matrix? Do they have any appealing application which can be used to motivate to study diagonal matrices.  Thanks for any answers. using System; using System.Security.Permissions; using Microsoft.SharePoint; using Microsoft.SharePoint.Security; using Microsoft.SharePoint.Utilities; using Microsoft.SharePoint.Workflow;  namespace Change_Title_Event.Change_Title {     /// &lt;summary&gt;     /// List Item Events     /// &lt;/summary&gt;     public class Change_Title : SPItemEventReceiver     {        /// &lt;summary&gt;        /// An item is being added.        /// &lt;/summary&gt;        public override void ItemAdding(SPItemEventProperties properties)        {            properties.AfterProperties["Title"] = "Kevin: Changed by event receiver at" + DateTime.Now.TimeOfDay.ToString();            base.ItemAdding(properties);        }       } }   Where simple example, which works perfect on my local Sharepoint 2010. It is sandboxed.  But when I package this solution, it do not work on Sharepoint Online. I does not give any Errors either.  I think I am doing something wrong, something simple, but I can't simply find anything on Google about it.  Here is another who asks on the MSDN forum, but gets a very wierd reply, that I do not understand. Linux - Running The Same Command on Many Machines at Once I have about a dozen Linux boxes that I occasionally need to run the same command(s) on. Is there an easier way (or automated way) to do this asides from logging on to each machine and running the command, one at a time? It's not the same commands all the time and it's not at a preset time so it's not something ideal for tools like cron. Notify more frequently than once a day? Is there a way to subscribe to a question and be notified more frequently than once a day?  I'd like to be notified immediately on any posts to a question. >Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$. Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$.   I saw some proof of $\frac d {dx} x^n=nx^{n-1}$ using binomial theorem, which is only available for $n \in\mathbb N$. Do anyone have the proof of $\frac d {dx} x^n=nx^{n-1}$ for all real $n$? Thank you. How to check if a string starts with a specified string? I am stuck while calculating the peak current in the boost converter through the inductor.  Can you please check if I am right?  Ipeak=(Max Voltage/ R2) = 18/1000=0.018. So, my peak current through the inductor is just 18mA? Am I doing it correct. If this is correct, how will I account for the Vce drop inside the pin1 and pin2 of the IC. The darlington transistor inside the IC is just like a normal transistor. I have added two transistor outside the IC instead of using the darlington switch in the IC for thermal purposes.   Can you please help me with the calculation of Vswce and Ipeak. The calculation of Vswce is not mentioned in the datasheet. So, please help. And the calculation of Peak current is mentioned in the datasheet. But is my approach of calculating the peak current through the inductor right? Google isn't surfacing Events rich snippets in SERP when JSON-LD Structured Data is valid Google confirms our site has thousands of valid MusicEvent structured data objects presented using JSON-LD, as indicated in the Structured Data section of Webmaster Tools. However, rich snippets for the events do not show up in any search result pages. The Rich Cards tab in Webmaster Tools says:     We did not find any structured data for rich cards on your website.   The structured data is valid with no errors in the Structured Data Testing Tool. We do generate some warnings occasionally for missing endDate and price fields. Comparing our code to a survey of other sites whose rich snippets do appear, we are not doing anything unique in terms of our markup.  The data has been there for months, in fact over the months we've tried also using Microdata format. The Structured Data section indicates that many pages were crawled this week.  Besides SDTT validation, what else can I check to troubleshoot this issue? Equivalence Relation problem Let $S$ be the relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b)S(c,d)$ if and only if $ad=bc$. Prove this is an equivalence relation on $\mathbb{N} \times \mathbb{N}$.  I think I've found this to be reflexive and symmetric, but I'm stuck on transitivity. Can someone check my work so far and assist with testing transitivity?  Reflexive: Let $(x,y)S(x,y)$. Then $xy = yx$. So $S$ is reflexive. Symmetric: Suppose $(a,b)S(c,d)$. Then, $ad = bc$. Therefore, $da = cb$ and $cb = da$. Therefore, $(c,d)S(a,b)$. Thus, $S$ is symmetric.  Transitive: Suppose $(a,b)S(b,c)$. Then $ac = bb$.   This is where I'm stuck. Any ideas? PHP: How do I loop through every XML file in a directory? 2D random walk within a bounded area Fatal error: Class 'SelectQuery' not found in includes/database/database.inc on line 813 Fatal error: Class 'SelectQuery' not found in /home/content/71/11241671/html/drupal/includes/database/database.inc on line 813   We built a site in a subfolder called /drupal while building the site, and then moved the files to the root directory when we were finished but it gave the above error on a WSOD.  I have tried:    flushing the cache before moving  tried the registry rebuild truncating and replacing the registry and registry_file tables with a fresh install copy, and rebuilding the registry again rewrite base in htaccess setting a new base url in settings.php   But nothing has worked.  The second we put the files back into the /drupal folder, it works again. It also looks like the error mentions the file in the /drupal directory, which isnt there anymore.   This is on 7.37   Anyone have any ideas? Active versus passive transformations Some observations I made is for $\frac{a}{b}+\frac{b}{a}$, is either:   the denominator has to be one, the numerator has to be a multiple of the denominator or  the numerator and denominator have to be the same.    Obviously, with the given conditions case 3 is eliminated since  $a \neq b$.  For case 1, if $b=1$, then $a=1$ which is a contradiction to the given condition that says $a \neq b$.  For case 2, I would think of examples. For example if $b=2$ then a multiple is $4$, so $a=4$. Then we have:  $$\frac{a}{b}+\frac{b}{a}=\frac{4}{2}+\frac{2}{4}=\frac{10}{4}$$  which is not an integer, but how would i proceed to show this case for all integers.