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.