Fundamental theorem of algebra

The fundamental theorem of algebra is a proven fact about polynomials, sums of multiples of integer powers of one variable. It is based on mathematical analysis, the study of real numbers and limits. It was first proven by German mathematician Carl Friedrich Gauss. It says that for any polynomial  with the degree , where , the polynomial equation  must have at least one root , and not more than  roots altogether.

Some remarks: 
 the degree  of a polynomial is the highest power of  that occurs in it 
 some of the roots may be complex numbers
 it is possible to 'count' a root  twice, if  is still a root of the polynomial ; if you will 'count' the roots in this way, then the polynomial  with degree  has exactly  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  of odd degree the pair of values  and  has opposite positive and negative signs when  is large enough. And secondly, that any polynomial function  on the real line that takes positive and negative values for  has to cross  axis.

Related pages
Fundamental theorem of arithmetic

Category:Mathematical theorems
Category:Algebra