Basic Technical Mathematics, 11th Edition

15.2 The Roots of an Equation

• The Fundamental Theorem of Algebra • Linear Factors • Complex Roots • Remaining Quadratic Factor

In this section, we present certain theorems that are useful in determining the number of roots in the equation $f(x)\hspace{0.17em}=\hspace{0.17em}0$ and the nature of some of these roots. In dealing with polynomial equations of higher degree, it is helpful to have as much of this kind of information as we can find before actually solving for all of the roots, including any possible complex roots. A graph does not show the complex roots an equation may have, but we can verify the real roots, as we show by use of a graphing calculator in some of the examples that follow.

The first of these theorems is so important that it is called the fundamental ...