8 - THEORY OF POLYNOMIALS

Abstract

This chapter discusses the theory of polynomials. To find the zeros of a polynomial, it will be necessary to divide the polynomial by a second polynomial. There is a procedure for polynomial division that parallels the long division process of arithmetic. The Fundamental Theorem of Algebra states that every polynomial P(x) of degree n ≥ 1 has at least one zero among the complex numbers. The zero guaranteed by this theorem may be a real number as the real numbers are a subset of the complex number system. The importance of the theorem is reflected in its title and it is necessary to create the complex numbers and that one need not create any other number system beyond the complex numbers to solve polynomial equations. Although the definition of a polynomial permits the coefficients to be complex numbers, there are limited examples to polynomials with real coefficients. Both the Linear Factor Theorem and the Fundamental Theorem of Algebra hold for polynomials with complex coefficients.