What is Fundamental theorem of algebra?
The '''fundamental theorem of [[algebra]]''' is a proven fact about [[polynomial]]s, sums of multiples of integer powers of one [[variable]]. It is based on [[mathematical analysis]], the study of [[real number]]s and [[Limit (mathematics)|limits]]. It was first proven by [[Germany|German]] [[mathematician]] [[Carl Friedrich Gauss]]. It says that for any [[polynomial]] <math>f(x)</math> with the degree <math>n</math>, where <math>n>0</math>, the polynomial equation <math>f(x)=0</math> must have at least one [[polynomial root|root]] <math>x</math>, and not more than <math>n</math> roots altogether. Some remarks: * the [[degree (mathematics)|degree]] <math>n</math> of a polynomial is the highest power of <math>x</math> that occurs in it * some of the roots may be [[complex number]]s * it is possible to 'count' a root <math>r</math> twice, if <math>r</math> is still a root of the polynomial <math>g(x) = f(x)/(x-r)</math>; if you will 'count' the roots in this way, then the polynomial <math>f(x)</math> with degree <math>n</math> has ''exactly'' <math>n</math> roots * it is not a theorem of pure algebra. It is not possible to prove this theorem without an element of analysis. This element has been reduced to the observation that, firstly, for polynomial functions <math>y=f(x)</math> of odd degree the pair of values <math>f(x)</math> and <math>f(-x)</math> has opposite positive and negative signs when <math>x</math> is large enough. And secondly, that any polynomial function <math>y=f(x)</math> on the real line that takes positive and negative values for <math>y</math> has to cross <math>x</math> axis. ==Related pages== *[[Fundamental theorem of arithmetic]] [[Category:Mathematical theorems]] [[Category:Algebra]] {{Math-stub}}