|
|
| Line 1: |
Line 1: |
| In [[algebra]], the '''factor theorem''' is a theorem linking factors and [[Zero (complex analysis)|zeros]] of a [[polynomial]]. It is a [[special case]] of the [[polynomial remainder theorem]].<ref>{{citation|first=Michael|last=Sullivan|title=Algebra and Trigonometry|page=381|publisher=Prentice Hall|year=1996|isbn=0-13-370149-2}}.</ref>
| | <br><br>Luke is actually a superstar within the building as well as [http://lukebryantickets.pyhgy.com justin bieber concert] career expansion initial second to his 3rd studio recording, & , is the confirmation. He broken to the picture in 2001 regarding his unique blend of straight down-house convenience, video star wonderful appears and lines, is defined t in the significant way. The new record Top about the land graph and #2 in the pop maps, producing it the 2nd top very first during those times of 2008 for any region artist. <br><br>The son of a , is aware of persistence and determination are key elements in terms of a prosperous occupation- . [http://www.banburycrossonline.com buy concert tickets] His very first recording, Keep Me, generated the very best hits “All My Girlfriends “Country and [http://www.senatorwonderling.com luke bryan past tour dates] Say” Person,” while his hard work, Doin’ Factor, located the performer-a few direct No. 2 single people: In addition Getting in touch with Is often a Good Factor.”<br><br>In the tumble of 2001, Tour: Luke And which had an impressive list of , which include Urban. “It’s much like you’re receiving a acceptance to visit to the next level, claims individuals performers that had been an element of the Concert tourmore than in to a larger sized amount of artists.” It covered as among the best organized tours in luke bryan tickets indianapolis ([http://lukebryantickets.flicense.com lukebryantickets.flicense.com]) its 15-year background.<br><br>Here is my web-site - [http://minioasis.com miley cyrus concert tickets] |
| | |
| The factor theorem states that a polynomial <math>f(x)</math> has a factor <math>(x - k)</math> [[if and only if]] <math>f(k)=0</math> (i.e. <math>k</math> is a root).<ref>{{citation|first1=V K|last1=Sehgal|first2=Sonal|last2=Gupta|title=Longman ICSE Mathematics Class 10|page=119|publisher=Dorling Kindersley (India)|isbn=978-81-317-2816-1}}.</ref>
| |
| | |
| ==Factorization of polynomials==
| |
| {{Main|Factorization of polynomials}}
| |
| Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent.
| |
| | |
| The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact, thus producing a lower degree polynomial whose zeros may be easier to find. Abstractly, the method is as follows:<ref>{{citation|first=R. K.|last=Bansal|title=Comprehensive Mathematics IX|page=142|publisher=Laxmi Publications|isbn=81-7008-629-9}}.</ref> | |
| # "Guess" a zero <math>a</math> of the polynomial <math>f</math>. (In general, this can be ''very hard'', but math textbook problems that involve solving a polynomial equation are often designed so that some roots are easy to discover.)
| |
| # Use the factor theorem to conclude that <math>(x-a)</math> is a factor of <math>f(x)</math>.
| |
| # Compute the polynomial <math> g(x) = f(x) \big/ (x-a) </math>, for example using [[polynomial long division]] or [[synthetic division]].
| |
| # Conclude that any root <math>x \neq a</math> of <math>f(x)=0</math> is a root of <math>g(x)=0</math>. Since the [[polynomial degree]] of <math>g</math> is one less than that of <math>f</math>, it is "simpler" to find the remaining zeros by studying <math>g</math>.
| |
| | |
| ===Example===
| |
| Find the factors at
| |
| : <math>x^3 + 7x^2 + 8x + 2.</math>
| |
| | |
| To do this you would use trial and error to find the first x value that causes the expression to equal zero. To find out if <math>(x - 1)</math> is a factor, substitute <math>x = 1</math> into the polynomial above:
| |
| : <math>x^3 + 7x^2 + 8x + 2 = (1)^3 + 7(1)^2 + 8(1) + 2</math>
| |
| : <math>= 1 + 7 + 8 + 2</math>
| |
| : <math>= 18.</math>
| |
| | |
| As this is equal to 18 and not 0 this means <math>(x - 1)</math> is not a factor of <math>x^3 + 7x^2 + 8x + 2</math>. So, we next try <math>(x + 1)</math> (substituting <math>x = -1</math> into the polynomial):
| |
| : <math>(-1)^3 + 7(-1)^2 + 8(-1) + 2.</math>
| |
| | |
| This is equal to <math>0</math>. Therefore <math>x-(-1)</math>, which is to say <math>x+1</math>, is a factor, and <math>-1</math> is a [[Root of a function|root]] of <math>x^3 + 7x^2 + 8x + 2.</math>
| |
| | |
| The next two roots can be found by algebraically dividing <math>x^3 + 7x^2 + 8x + 2</math> by <math>(x+1)</math> to get a quadratic, which can be solved directly, by the factor theorem or by the [[quadratic equation]].
| |
| | |
| : <math>{x^3 + 7x^2 + 8x + 2 \over x + 1} = x^2 + 6x + 2</math> | |
| | |
| and therefore <math>(x+1)</math> and <math>x^2 + 6x + 2</math> are the factors of <math>x^3 + 7x^2 + 8x + 2.</math>
| |
| | |
| ==Formal version==
| |
| Let <math>f</math> be a one-variable polynomial with coefficients in a commutative ring <math>R</math>, and let <math>a \in R</math>. Then <math>f(a) = 0</math> if and only if <math>f(x)=(x-a)g(x)</math> for some polynomial <math>g</math>. In this case, <math>g</math> is determined uniquely.
| |
| | |
| As for the problem of algorithmically finding all roots, if <math>f</math> is given and <math>a</math> is known, then <math>g</math> can be computed by [[polynomial long division]]; then one can compute the remaining roots of <math>f</math>, including repeated roots, by factoring <math>g</math>.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| {{DEFAULTSORT:Factor Theorem}}
| |
| [[Category:Polynomials]]
| |
| [[Category:Theorems in algebra]]
| |
Luke is actually a superstar within the building as well as justin bieber concert career expansion initial second to his 3rd studio recording, & , is the confirmation. He broken to the picture in 2001 regarding his unique blend of straight down-house convenience, video star wonderful appears and lines, is defined t in the significant way. The new record Top about the land graph and #2 in the pop maps, producing it the 2nd top very first during those times of 2008 for any region artist.
The son of a , is aware of persistence and determination are key elements in terms of a prosperous occupation- . buy concert tickets His very first recording, Keep Me, generated the very best hits “All My Girlfriends “Country and luke bryan past tour dates Say” Person,” while his hard work, Doin’ Factor, located the performer-a few direct No. 2 single people: In addition Getting in touch with Is often a Good Factor.”
In the tumble of 2001, Tour: Luke And which had an impressive list of , which include Urban. “It’s much like you’re receiving a acceptance to visit to the next level, claims individuals performers that had been an element of the Concert tourmore than in to a larger sized amount of artists.” It covered as among the best organized tours in luke bryan tickets indianapolis (lukebryantickets.flicense.com) its 15-year background.
Here is my web-site - miley cyrus concert tickets