I'm reading a proof in a linear algebra book. It mentions $$p(x) -p(c)= (x - c) h(x),$$ where $c$ is a constant, and $p(x)$ and $h(x)$ are polynomials.  Can we always factor $p(x) - p(c)$ in this way?   Please give a proof.
Let $F$ be a field. $(x-c)|(p(x)-p(c))$.   So if $x-c|p(x)=q(x)-\frac{r}{x-c}$ so for $x-c|p(x)-p(c)=s(x-c)-\frac{r}{x-c}$. Is that correct? Or is there a different way to go about proving this? I have proved that $x-c|p(x)$ has a remainder of $p(c)$ so It wuld make sense that $p(x)-p(c)$ would be $x-c$.
limit of Holder norms: $\sup\limits_{x\in [a,b]} f(x) = \lim\limits_{n\rightarrow\infty} \left(\int_a^b (f(x))^n \;dx\right)^{\frac{1}{n}}$ Inaccurate altitude GPS data with my Android? How to set scale across all blender objects and keyframes? Semigroups and solutions of equation Why doesn't backpropagation work when you initialize all the weight the same value (say 0.5), but works fine when given random numbers?   Shouldn't the algorithm calculate the error and work from there, despite the fact that the weights are initially the same? Which is correct: I'll be moving next month or I'll be shifting next month? For changing one's home from one place to another, I've heard people in western part of the world using the sentence:      I'll be moving next month.   In India, even in the English news channels, I've heard many people using the phrase:     I'll be shifting next month.   Which one is the correct usage? Or are both of them correct and the difference in usage is regional? How do I delete all Folders with a specific name (e.g. 128Kbps_Songs) without deleting its files  For example if I have directory "MP3_SONGS" with subdirectories "A", "B", "C" and each subdirectory has MANY directory Contains Film Name "FILM_NAME1", "FILM_NAME2", "FILM_NAME3" in it, In That Each Film NAme I have specific Folder NAme (128Kbps_Songs) in this "128Kbps_Songs" Folder I have That Film mp3 Songs Files  how can I remove the Folder "128Kbps_Songs", in Each Film_Name directory and Have to get these mp3 songs in that FILM_NAME..   ( Have to Delete 128Kbps_Songs Folder in all FILM_NAME without deleting the mp3 Songs in it, have to get the mp3 in FILM_NAME Directory..  MP3_SONGS (Parent Directory) ------ A , B, C  A (Sub Directory1) ------ FILM_NAME1 , FILM_NAME2 B (Sub Directory2) ------ FILM_NAME3 , FILM_NAME4 c (Sub Directory3) ------ FILM_NAME5 , FILM_NAME6  FILM_NAME1 (Directory) ---- 128Kbps_Songs (Directory) ------ FILM_NAME1 (MP3 Files) FILM_NAME2 (Directory) ---- 128Kbps_Songs (Directory) ------ FILM_NAME2 (MP3 Files)  FILM_NAME3 (Directory) ---- 128Kbps_Songs (Directory) ------ FILM_NAME3 (MP3 Files) FILM_NAME4 (Directory) ---- 128Kbps_Songs (Directory) ------ FILM_NAME4 (MP3 Files)  FILM_NAME5 (Directory) ---- 128Kbps_Songs (Directory) ------ FILM_NAME5 (MP3 Files) FILM_NAME6 (Directory) ---- 128Kbps_Songs (Directory) ------ FILM_NAME6 (MP3 Files)  Need Files in: FILM_NAME1 (Directory)  ------ FILM_NAME1 (MP3 Files) FILM_NAME2 (Directory)  ------ FILM_NAME2 (MP3 Files)  FILM_NAME3 (Directory)  ------ FILM_NAME3 (MP3 Files) FILM_NAME4 (Directory)  ------ FILM_NAME4 (MP3 Files)  FILM_NAME5 (Directory)  ------ FILM_NAME5 (MP3 Files) FILM_NAME6 (Directory)  ------ FILM_NAME6 (MP3 Files) I have a definition:  \newcommand{\arnold}{Arnold Schwarzenegger}   When I refer to it by:  \arnold is a   it is rendered as:   Arnold Schwarzeneggeris a    In order to have a space in front of "is" I would need to write  \arnold\ is a   Is there some other shorter way? What kind of construct is 'would that that were all'? There is a place in Bram Stoker’s Dracula where I can’t quite parse the grammar:     8 May. — I began to fear as I wrote in this book that I was getting too   diffuse. But now I am glad that I went into detail from the first, for   there is something so strange about this place and all in it that I cannot   but feel uneasy. I wish I were safe out of it, or that I had never come. It   may be that this strange night existence is telling on me, but would that   that were all!   I understand the meaning of the text in bold (it’s equivalent to if only that was all!), but I don’t understand the grammar being used. What kind of construct is this? In Texas Hold'em poker, there are some cases where a particular card, often called kicker, acts as a tiebreaker between players to determine who wins the pot, or if the pot has to be shared.  For some combinations such as three or four of a kind, there is no doubt: the kicker is the 5th card of the chosen combination, and determines the winner.  Example     Alice has 7 and 10   Bob has 7 and K   On the board are 7, 7, 2, 3, 4   In that case, both have a three of a kind. But because Bob has a king and Alice only a 10, he wins.  Most of poker websites explaining the rules clearly mention the role of the kicker applying to three of a kind and double pairs. But much fewer say something about colors, straights and full houses, and I found many contradicting answers. I also asked the question to some used online poker players and they also gave me contradicting answers.  I have tried to search for official tournament rules, but most of them only explain what happens with bad behaviors, bad deals, incorrect or confusing betting, showing cards when you shouldn't, acting when it's not your turn, etc. without mentioning subtleties about combinations at all.  To simplify my question, I will take three examples; I think it's better to start with examples before getting to the general answer if one exists. So, what's happening in the following 3 examples? Could you give a more generalized answer?  Example 1 - Flushes     Alice has 2&clubs; and 3&clubs;   Bob has 4&clubs; and 5&clubs;   On the board are 6&clubs;, 8&clubs;, 10&clubs;, 2&hearts;, 5&diams;   Contradicting answer 1: it's a tie, because the highest card included in the flush is the 10, which everybody chooses to include in their 5 showdown cards. Contradicting answer 2: Bob wins, because he has the greatest private card that is part of the flush  Example 2 - Straight     Alice has 6 and K   Bob has 6 and J   On the board are 4, 5, 7, 8, 10   Contradicting answer 1: it's a tie, because the greatest card in the straight is the 8 for both Alice and Bob Contradicting answer 2: Alice wins, because she owns an extra king, compared to the jack of Bob  Example 3 - Full house     Alice has 3 and 7   Bob has 3 and 6   On the board are 3, 3, 2, 2, 5   Contradicting answer 1: it's a tie, because one is supposed to choose only five cards to make a combination, and a full house is already five cards. There couldn't be any kicker, and thus their showdowns are strictly equals. Contradicting answer 2: Alice wins because of her extra 7, compared to the 6 of Bob  Thank you very much for your answers.  Note: I'm unable to post next to you, so I edit my own post; strange not be able to answer to an answer.  Ok, So if I summarize what you are saying :   In the flush case, Bob wins because at some point, their hand differs. Technically, I can continue comparing the highest private card to decide who wins. I had it correct. In the straight case, if the highest card of the straight is public, then it's technically always a tie, no matter what the players had as second private card (asuming that only one of the two cards was part of the straight). In a full house if both players have the same triplet and the same pair, it's always a tie, no matter what the players had as second private card (assuming again that only one of the two was part of the full house).   Please confirm me that what I summarized is correct.  Thank you for your answer. We are provided with a string A (may contain repeated characters as well ). One needs to make all possible permutations of A , and find out how many pairs of strings chosen from these permutations are not similar.  Two strings are similar if we can make both of them exactly same by carrying out at most 1 swap in each of them .  Example :- if we are provided with string abcd  , then for permutation abcd , the permutations which are not similar to it are bcda , bdac , cadb , cdba , dabc , dcab . Similar is the case for other permuations as well. So overall number of desired pairs are 24*6=144 .  I tried solving it for each possible pair by approach mentioned here     But as the permutations can be large enough , I need a better approach of solving this. An inequality for arctan$x$ Why it is true that $|\arctan x-\arctan y|\leq |x-y|$? I have tried to use $\tan x$ but I didn't conclude to something. What determines the direction of precession of a gyroscope? I understand how torque mathematically causes a change to the direction of angular momentum, thus precessing the .  However, the direction, either clockwise or counterclockwise, of this precession seems to me a bit arbitrary beyond mathematical definitions. (What if torque were defined by a "left hand rule", for example?).  What's the fundamental physical reason that the precession of a gyroscope would be directed in a certain direction? What happens to the energy when waves perfectly cancel each other (destructive interference)? It appears that the energy "disappear" but the law of conservation of energy states that it can't be destructed. My guess is that the kinetic energy is transformed into potential energy. Or maybe it depends on the context of the waves where do the energy goes? Can someone elaborate on that or correct me if I'm wrong? Android: No icon for Notification How do  I create a desktop wallpaper slideshow with my own set of images?   With the images coming from a pictures folder, or pre-selected With the images displayed in order, or randomly With the images changing every few minutes, or after random periods of time I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra).  Mathematics is a system of axioms which you choose yourself for a set of undefined entities, such that those entities satisfy certain basic rules you laid down in the first place on your own.  Now using these laid-down rules and a set of other rules for a subject called logic which was established similarly, you define certain quantities and name them using the undefined entities and then go on to prove certain statements called theorems.  Now what is a proof exactly? Suppose in an exam, I am asked to prove Pythagoras' theorem. Then I prove it using only one certain system of axioms and logic. It isn't proved in all the axiom-systems in which it could possibly hold true, and what stops me from making another set of axioms that have Pythagoras' theorem as an axiom, and then just state in my system/exam "this is an axiom, hence can't be proven".  EDIT : How is the term "wrong" defined in mathematics then ? You can say that proving fermat's last theorem using the number-theory axioms was a difficult task but then it can be taken as an axiom in another set of axioms .  Is mathematics as rigorous and as thought-through as it is believed and expected to be ? It seems to me that there many loopholes in problems as well as the subject in-itself, but there is a false backbone of rigour that seems true until you start questioning the very fundamentals . Find a plane that passes through a point and is perpendicular to 2 planes Find an equation of a plane that passes through $p(1,5,1)$ and is perpendicular to planes $2x+y-2z = 2$ and $x+3z=4$.  I basically need the 2 other points to make the vector and perform the cross product.  Since $ax+by+cz+d=0$ is the form of a plane. Can I obtain the points as $(a,b,c)$ of the 2 planes given? Also, how can I set them up to obtain the points? Why does QgsSpatialIndex.nearestNeighbor(point, 1) sometimes return 2 lines? I'm trying to get the closest line to a point using PyQGIS in a plugin. The active code file is this:   When I request the closest line (only 1)  nearestIds = self.spIndex.nearestNeighbor(self.feat2.geometry().centroid().asPoint(),1)   I sometimes get a list with two lines and the first one is not the closest but as far as I can see the second one is. Why is this?   Does QgsSpatialIndex use only boundingboxes for lines? How do I get the real closest line always, not only most of the time? Ubuntu halts after few seconds of starting and nothing works command, virtual terminal, GUI