en>Michael Hardy at 15:37, 9 August 2013

2013-08-09T15:37:31Z

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	~~Friends call her Felicidad~~ and ~~her spouse doesn~~'~~t like it~~ at ~~all~~. ~~Managing individuals~~ is ~~how she tends to make money~~ and ~~she will not change it whenever soon~~. ~~Alabama has always been his house~~ and ~~his family loves it~~. ~~The preferred hobby~~ for ~~him~~ and ~~his children~~ is ~~to generate and now he~~ is ~~attempting to earn money with it~~.<br><br>~~My website [http:~~//~~www~~.~~kaonthana~~.~~com~~/~~index~~.~~php?mod~~=~~users&action~~=~~view&id=5336 http:~~//~~www~~.~~kaonthana.com~~]		{{Unreferenced\|date=December 2009}}
			'''Polynomial function theorems for zeros''' are a set of [[theorem]]s aiming to find (or determine the nature) of the [[zero (cm)\|Remainder theorem]]
			* [[Factor theorem]]
			* [[Descartes' rule of signs]]
			* [[Gauss–Lucas theorem]]
			* [[Rational root theorem\|Rational zeros theorem]]
			* [[Extreme value theorem\|Bounds on zeros theorem]] also known as the '''boundedness theorem'''
			* [[Intermediate value theorem]]
			* [[Complex conjugate root theorem]]
			* [[Properties of polynomial roots]]

			==Background==
			A [[polynomial function]] is a [[function (mathematics)\|function]] of the form
			:<math> p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0 ,\,</math>
			where <math> a_i\, (i = 0, 1, 2, ..., n) </math> are [[complex number]]s and <math> a_n \ne 0 </math>.

			If <math> p(z) = a_n z^n + a_{n-1} z^{n-1} + ... + a_2 z^2 + a_1 z + a_0 = 0</math>, then <math>z</math> is called a '''zero''' of <math>p(x)</math>. If <math>z</math> is real, then <math>z</math> is a '''real zero''' of <math>p(x)</math>; if <math>z</math> is imaginary, the <math>z</math> is a '''complex zero''' of <math>p(x)</math>, although complex zeros include both real and imaginary zeros.

			The [[fundamental theorem of algebra]] states that every polynomial function of degree <math> n \ge 1 </math> has at least one complex zero. It follows that every polynomial function of degree <math> n \ge 1 </math> has exactly <math> n </math>complex zeros, not necessarily distinct.

			* If the degree of the polynomial function is 1, i.e., <math> p(x) = a_1 x + a_0 \,</math>, then its (only) zero is <math>\frac{-a_0}{a_1}</math>.
			* If the degree of the polynomial function is 2, i.e., <math> p(x) = a_2 x^2 + a_1 x + a_0 \, </math>, then its two zeros (not necessarily distinct) are <math>\frac{-a_1 + \sqrt{{a_1}^2 - 4 a_2 a_0}}{2 a_2} </math> and <math>\frac{-a_1 - \sqrt{{a_1}^2 - 4 a_2 a_0}}{2 a_2} </math>.

			A degree one polynomial is also known as a [[linear function]], whereas a degree two polynomial is also known as a [[quadratic function]] and its two zeros are merely a direct result of the [[quadratic formula]]. However, difficulty rises when the degree of the polynomial, ''n'', is higher than 2. It is true that there is a cubic formula for a cubic function (a degree three polynomial) and there is a quartic formula for a quartic function (a degree four polynomial), but they are very complicated. To make matters worse, there is no general formula for a polynomial function of degree 5 or higher (see [[Abel–Ruffini theorem]]).

			==The theorems==
			===Remainder theorem===
			The remainder theorem states that if <math>p(x)</math> is divided by <math>x - c</math>, then the remainder is <math>p(c)</math>.
			<br> For example, when <math>p(x) = x^3 + 2x - 3</math> is divided by <math>x - 2</math>, the remainder (if we don't care about the quotient) will be <math>p(2) = 2^3 + 2(2) - 3 = 9</math>. When <math>p(x)</math> is divided by <math>x + 1</math>, the remainder is <math>p(-1) = (-1)^3 + 2(-1) - 3 = -6</math>. However, this theorem is most useful when the remainder is 0 since it will yield a zero of <math>p(x)</math>. For example, <math>p(x)</math> is divided by <math>x - 1</math>, the remainder is <math>p(1) = (1)^3 + 2(1) - 3 = 0</math>, so 1 is a zero of <math>p(x)</math> (by the definition of zero of a polynomial function).

			{{DEFAULTSORT:Polynomial Function Theorems For Zeros}}
			[[Category:Theorems in complex analysis]]


			[[de:Nullstelle]]

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Friends call her Felicidad and her spouse doesn't like it at all. Managing individuals is how she tends to make money and she will not change it whenever soon. Alabama has always been his house and his family loves it. The preferred hobby for him and his children is to generate and now he is attempting to earn money with it.<br><br>My website [http://www.kaonthana.com/index.php?mod=users&action=view&id=5336 http://www.kaonthana.com]

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en>Michael Hardy at 15:37, 9 August 2013

en>Luckas-bot: r2.5.2) (robot Adding: ru:Кривая Персея