Polynomial equations of higher degree.

In Chapter 6, Section thm fundamental, we prove the Fundamental Theorem of Algebra, which states that if is a non constant polynomial over , then has a root.

Here are some examples of applications, based on the following theorems. Recall that a root of a polynomial is a number such that .

where , for every .

The ``if'' part is trivial.

For the ``only if'' part, suppose that . Then we have:

By section section algebraic form, all the terms have a common factor , whence the result.

An evident solution is 1 (why evident?).So we divide the polynomial on the left by . We have:

By the method of Section 6, we solve the equation , getting two solutions, namely and . In conclusion, the given cubic equation has three distinct solutions (as promised by 3.1): , and .

By Theorem 7.3, we know that is also a root of .

By Theorem 7.1, there exists a polynomial (of degree 2) such that . We have: . By the method of Section 6, we solve the equation . Finally the polynomial has four distinct complex roots: , , and . Remark that the third and the fourth solution are also conjugates.

The following corollary can be obtained either as a consequence of the Fundamental Theorem of Algebra thm fundamental, or as a consequence of the Intermediate Value Theorem in Calculus.

Let . This polynomial has no real root, but it has complex conjugate roots:

Moreover, by one of the methods described in subsection 6, we have:

and

It follows:

Thus:

Noah Dana-Picard 2007-12-24