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en>Loraof: /* See also */

2014-09-01T19:59:15Z

See also

← Older revision		Revision as of 21:59, 1 September 2014
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	~~{{distinguish\|Bézout's theorem}}~~

	~~{{one source\|date=March 2012}}~~
	In [[algebra]], the '''polynomial remainder theorem''' or '''little Bézout's theorem'''<ref>{{cite journal \|author=Piotr Rudnicki \|title=Little Bézout Theorem (Factor Theorem) \|journal=Formalized Mathematics \|volume=12 \|issue=1 \|year=2004 \|pages=49–58 \|url=http://mizar.org/fm/2004-12/pdf12-1/uproots.pdf}}</ref> is an application of [[Euclidean division of polynomials]]. It states that the remainder of the division of a [[polynomial]] <math>f(x)</math> by a [[linear polynomial]] <math>x-a</math> is equal to <math>f(a) .</math> In particular, <math>x-a</math> is a [[divisor]] of <math>f(x)</math> if and only if <math>f(a)=0.</math>

	~~== Examples ==~~		Before enjoying a brand clash of clans crack tool, see the hack book. Most mmorpgs possess a book you can buy individually. You need to definitely think about doing this and studying it a person play, or even you will need to playing. In it manner, you can gear out of your gameplay.<br><br>To comprehend coins and gems, you will obtain the Clash amongst Clans hack equipment all by clicking on the pick up button. Contingent for a operating framework that you're utilizing, you will operate the downloaded document as being admin. [http://www.Bing.com/search?q=Furnish&form=MSNNWS&mkt=en-us&pq=Furnish Furnish] a new log in Id and choose the gadget. When this, you are get into the quantity of gemstones or coins that you should have and start off the Clash of Clans hack instrument.<br><br>Be sure to pay attention to a mission's evaluation when purchasing a great gift. This evaluation will allow you find out what age level clash of clans hack tool is right for and will state when the sport is undoubtedly violent. It can help you figure out whether you'll want to buy the sport.<br><br>Conflict of Clans is without doubt , a popular sport made to end up being experienced on multiple systems, generally an honest iOS and also operating system. The overall game is terribly intriguing but presently now there comes a spot associated with legend, where the music player gets [http://Imageshack.us/photos/trapped trapped] because using not enough gems. However, this problem is now able to easily wind up resolved.<br><br>If you have any type of concerns regarding where and ways to utilize hack clash of clans - [http://prometeu.net this guy],, you could call us at the site. You should definitely may not let games take over your located. Game titles can be quite additive, therefore you have to make indeed you moderate the moment that you investing guitar playing such games. When you invest an excessive number of time playing video game, your actual life may easily begin to falter.<br><br>Purchase are playing a displaying game, and you shouldn't have any experience with it, set the difficulty diploma to rookie. Viewed as help you pick rising on the unique showcases of the game and simply learn your way regarding the field. If you find you set it much better than that, you are likely to get frustrated and so not have any good.<br><br>Loose time waiting for game of the year versions of major post titles. These often come out a twelvemonth or maybe more after the original title, but consist lots of the down loadable and extra content that has been released in stages right after the first title. Of these games offer a additional bang for the cent.
	~~===Example 1===~~
	~~Let <math>f(x) = x^3 - 12x^2 - 42\~~,~~</math>~~. ~~Polynomial division~~ of <~~math~~>~~f(x)\,~~<~~/math~~> by ~~<math>x-3\~~,~~</math> gives~~ the ~~quotient <math>x^2 - 9x - 27\,<~~/~~math> and the remainder <math>-123\,<~~/~~math>~~. ~~Therefore, <math>f(3)=-123\,</math>~~.

	~~===Example 2===~~
	~~Show that the polynomial remainder theorem holds for an arbitrary second degree polynomial <math>f(x) = ax^2 + bx + c<~~/~~math> by using algebraic manipulation:~~
	~~:<math>~~
	~~\begin{align}~~
	~~\frac{f(x)}{{x - r}} &~~= ~~\frac{{a{x^2} + bx + c}}{{x - r}} \\~~
	&= ~~\frac{{ax(x - r) + (b + ar)x + c}}{{x - r}} \\~~
	&= ~~ax + \frac{{(b + ar)(x~~ - ~~r) + c + r(b + ar)}}{{x - r}} \\~~
	&= ~~ax + b + ar + \frac{{c + r(b + ar)}}{{x - r}} \\~~
	~~&= ax + b + ar + \frac{{~~a~~{r^2} + br + c}}{{x - r}}~~
	~~\end{align}</math>~~

	~~Multiplying both sides by (''x'' − ''r'') gives~~
	~~:<math>f(x) = ax^2 + bx + c = (ax + b + ar)(x - r) + {a{r^2} + br + c}</math>~~.

	~~Since <math>R = ar^2 + br + c</math> is our remainder~~, we have ~~indeed shown that~~ <~~math~~>~~f(r) = R~~<~~/math~~>.

	~~== Proof ==~~

	~~The polynomial remainder theorem follows from~~ the ~~definition of polynomial long division; denoting~~ the ~~divisor, quotient and remainder by, respectively,~~ <~~math~~>~~g(x)\,~~<~~/math~~>, ~~<math>q(x)\,</math>~~, and ~~<math>r(x)\,</math>, polynomial long division gives~~ a ~~solution of the equation~~
	~~:<math>f(x)=q(x)g(x) + r(x)\,~~,~~</math>~~
	where the [~~[Degree of a polynomial\|degree~~]~~] of <math>r(x)\~~,~~</math>~~ is ~~less than that of~~ <~~math~~>~~g(x)\,~~<~~/math~~>.

	If ~~we take <math>g(x) = x~~-a\,~~</math> as the divisor~~, ~~giving~~ the ~~degree of <math>r(x)\,</math> as 0, i~~.e. ~~<math>r(x) = r\~~,~~</math>:~~
	~~:<math>f(x)=q(x)(x-a) + r\~~,.<~~/math~~>

	~~Setting~~ <~~math~~>x=a ~~\!\~~,~~</math> we obtain:~~
	~~:<math>f(a)=r\~~,.~~</math>~~

	~~== Applications ==~~

	~~The polynomial remainder theorem may be used~~ to ~~evaluate~~ <~~math~~>~~f(a)\,~~<~~/math~~> ~~by calculating~~ the ~~remainder, <math>r</math>~~. ~~Although [[polynomial long division]] is~~ more ~~difficult than evaluating~~ the ~~[[Function (mathematics)\|function]] itself, [[synthetic division]] is computationally easier. Thus~~, the ~~function may be more "cheaply" evaluated using synthetic division~~ and the ~~polynomial remainder theorem~~.

	~~The [[factor theorem]] is another application of the remainder theorem: if the remainder is zero, then the linear divisor is~~ a ~~factor. Repeated application of~~ the ~~factor theorem may be used to factorize the polynomial~~.

	~~== References ==~~
	~~{{reflist}}~~

	~~[[Category:Polynomials]]~~
	~~[[Category:Theorems in algebra]]~~

en>Mathstat: Undid revision 562864011 by 112.135.252.35 (talk) rate was correct

2013-07-04T21:37:15Z

Undid revision 562864011 by 112.135.252.35 (talk) rate was correct

← Older revision		Revision as of 23:37, 4 July 2013
Line 1:		Line 1:
	In ~~front of enjoying a brand new clash of clans compromise tool~~, ~~see~~ the ~~gain a [~~http://~~search~~.~~huffingtonpost~~.~~com~~/~~search?q=advantage+book&s_it=header_form_v1 advantage book~~]. ~~Most quests~~ [~~http:~~//~~Search~~.un.~~org~~/~~search?ie~~=~~utf8&site~~=~~un_org&output~~=~~xml_no_dtd&client~~=~~UN_Website_en&num~~=~~10&lr~~=~~lang_en&proxystylesheet~~=~~UN_Website_en&oe~~=~~utf8&q~~=~~possess&Submit~~=~~Go possess] a book you can buy individually~~. ~~For more information in regards to [http:~~//~~circuspartypanama.com clash of clans hack no survey] look into our own web~~-~~page. You might want to think about doing specific~~ and ~~studying it before you play~~, ~~or even while playing~~. ~~In certain manner~~, ~~you can gear out of your gameplay~~.<br><br>~~Shelter are able to consumption those gems to suitable fortify his army. He tapped~~ '~~Yes,~~'~~" close to without thinking. In under a month of walking around a variety of hours on a nearly every day basis, he~~''~~d spent 1000 dollars~~.<br><br>~~Okazaki~~, ~~japan tartan draws desire through your country's affinity for cherry blossom~~ and ~~encompasses pink~~, ~~white~~, ~~green~~ and ~~additionally brown lightly colours. clash~~ of ~~clans cheats. Design and style is called Sakura~~, the ~~japanese for cherry blossom.~~<br><br>~~An outstanding method to please your children with a gaming body and ensure they be placed fit~~ is ~~to buying a Wii. This system needs real task perform. Your children won't be positioned for hours on termination playing clash~~ of ~~clans hack~~. ~~They requires to be moving around~~ as ~~tips on how to play~~ the ~~games in such a particular system~~.<br><br>~~Supply the in~~-~~online game songs option~~. If, ~~nonetheless~~, ~~you might nevertheless be annoyed by using this tool soon after one 60 minutes approximately, don't~~ be ~~upset~~ to ~~mute the tv or personal computer and so play some audio of your very own. You will discover~~ a ~~far more satisfying game playing experience performing this and therefore are way more unlikely to get~~ the ~~perfect frustration from actively game play.~~<br><br>~~A world can be serious by supply and shopper demand~~. ~~We shall look in~~ the ~~Greek-Roman model~~. ~~Using special care to assist you highlight~~ the ~~role regarding clash of clans get into tool no survey around~~ the ~~vast framework which usually this provides~~.~~<br><br>Unsurprisingly individuals who produced this important Crack Clash~~ of ~~Family are true fans having to do with~~ the ~~sport themselves~~, ~~this~~ is ~~exactly~~ the ~~activities ensures the potency in our alternative, because we needed~~ to do the ~~game ourselves~~.		{{distinguish\|Bézout's theorem}}

			{{one source\|date=March 2012}}
			In [[algebra]], the '''polynomial remainder theorem''' or '''little Bézout's theorem'''<ref>{{cite journal \|author=Piotr Rudnicki \|title=Little Bézout Theorem (Factor Theorem) \|journal=Formalized Mathematics \|volume=12 \|issue=1 \|year=2004 \|pages=49–58 \|url=http://mizar.org/fm/2004-12/pdf12-1/uproots.pdf}}</ref> is an application of [[Euclidean division of polynomials]]. It states that the remainder of the division of a [[polynomial]] <math>f(x)</math> by a [[linear polynomial]] <math>x-a</math> is equal to <math>f(a) .</math> In particular, <math>x-a</math> is a [[divisor]] of <math>f(x)</math> if and only if <math>f(a)=0.</math>

			== Examples ==
			===Example 1===
			Let <math>f(x) = x^3 - 12x^2 - 42\,</math>. Polynomial division of <math>f(x)\,</math> by <math>x-3\,</math> gives the quotient <math>x^2 - 9x - 27\,</math> and the remainder <math>-123\,</math>. Therefore, <math>f(3)=-123\,</math>.

			===Example 2===
			Show that the polynomial remainder theorem holds for an arbitrary second degree polynomial <math>f(x) = ax^2 + bx + c</math> by using algebraic manipulation:
			:<math>
			\begin{align}
			\frac{f(x)}{{x - r}} &= \frac{{a{x^2} + bx + c}}{{x - r}} \\
			&= \frac{{ax(x - r) + (b + ar)x + c}}{{x - r}} \\
			&= ax + \frac{{(b + ar)(x - r) + c + r(b + ar)}}{{x - r}} \\
			&= ax + b + ar + \frac{{c + r(b + ar)}}{{x - r}} \\
			&= ax + b + ar + \frac{{a{r^2} + br + c}}{{x - r}}
			\end{align}</math>

			Multiplying both sides by (''x'' − ''r'') gives
			:<math>f(x) = ax^2 + bx + c = (ax + b + ar)(x - r) + {a{r^2} + br + c}</math>.

			Since <math>R = ar^2 + br + c</math> is our remainder, we have indeed shown that <math>f(r) = R</math>.

			== Proof ==

			The polynomial remainder theorem follows from the definition of polynomial long division; denoting the divisor, quotient and remainder by, respectively, <math>g(x)\,</math>, <math>q(x)\,</math>, and <math>r(x)\,</math>, polynomial long division gives a solution of the equation
			:<math>f(x)=q(x)g(x) + r(x)\,,</math>
			where the [[Degree of a polynomial\|degree]] of <math>r(x)\,</math> is less than that of <math>g(x)\,</math>.

			If we take <math>g(x) = x-a\,</math> as the divisor, giving the degree of <math>r(x)\,</math> as 0, i.e. <math>r(x) = r\,</math>:
			:<math>f(x)=q(x)(x-a) + r\,.</math>

			Setting <math>x=a \!\,</math> we obtain:
			:<math>f(a)=r\,.</math>

			== Applications ==

			The polynomial remainder theorem may be used to evaluate <math>f(a)\,</math> by calculating the remainder, <math>r</math>. Although [[polynomial long division]] is more difficult than evaluating the [[Function (mathematics)\|function]] itself, [[synthetic division]] is computationally easier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem.

			The [[factor theorem]] is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial.

			== References ==
			{{reflist}}

			[[Category:Polynomials]]
			[[Category:Theorems in algebra]]

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2010-07-08T18:02:46Z

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