Scale parameter: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Mathstat
Undid revision 562864011 by 112.135.252.35 (talk) rate was correct
en>Loraof
 
Line 1: Line 1:
{{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''&nbsp;&minus;&nbsp;''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]]

Latest revision as of 20:59, 1 September 2014


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.

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. 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.

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.

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 trapped because using not enough gems. However, this problem is now able to easily wind up resolved.

If you have any type of concerns regarding where and ways to utilize hack clash of clans - 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.

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.

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.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Scale_parameter&oldid=234208"