## The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is one of the most important theorems in algebra, maybe the most important. It states that the number of roots of a polynomial of degree is exactly This means also that the number of roots of the polynomial equation is also This is because if is a root then is a factor, so that the number of factors is equal to the number of roots.

There are several points to remember:

1. The roots may be complex or real. The fundamental theorem only states the existence of the roots. It does not state their nature.

2. If all roots are real then the graph of will intersect the – axis times – once for each root.

3. If has roots, then it will have at least turning points, because there has to be a tuning point between any two consecutive roots, and there may be other turning points, where besides.

There is also a very useful theorem which help to locate the roots:

If all the coefficients of are real then if is a complex root of so is the complex conjugate . This often means that a polynomial can be factorised into a product of linear factors, with real but not necessarily rational roots, and quadratic factors.

The n Roots of Unity

There is a very aesthetic illustration of the Fundamental Theorem. The equation has n roots, which all lie evenly spaced on a unit circle at the origin when drawn on an Argand diagram. These are called the roots of unity. The diagram shows the five roots of unity in orange and the 16 roots of unity in blue. Note that 1 is always a root.  