# 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 .

Theorem 1.7.1   Let be a polynomial in one complex variable , with complex coefficients. The complex number is a root of if, and only if, there exists a polynomial such that .

Proof. We denote

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.

Example 1.7.2   Solve the equation .

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 .

Theorem 1.7.3   Let be a polynomial in one complex variable , with real coefficients. If the complex number is a root of , then its complex conjugate (v.s. 1.5) is also a root of .

Example 1.7.4   Let . We see easily that .

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.

Corollary 1.7.5   A polynomial of odd degree over the reals has at least one real root.

Proof. Let be a polynomial of odd degree with real coefficients. By the Fundamental Theorem of Algebra, it has exactly complex roots, counted with multiplicity. As the coefficients are real, the roots are organized by pairs of conjugate complex numbers. Suppose that is a root of . If is real, we are done; otherwise, is another root of . Take now another root . If If is real, we are done; otherwise, is another root of . It is not important to know whether or . Iterate this process until we discover the first (maybe the only) real root of . As roots are organized by pairs, the maximum number of roots (either distinct or not) which can be involved in the process is even, and it is equal to . The last root cannot be equal to a previous one, as in such a case its conjugate should appear also now, and this is impossible. Thus, must be equal to its conjugate, i.e. is a real number.

Remark 1.7.6   A consequence of these theorems is a method for factorizing polynomials of higher degree in one real variable, despite the fact that they have no real root. Let us see an example.

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