A Readable Introduction to Real Mathematics by Daniel Rosenthal David Rosenthal & Peter Rosenthal

Author:Daniel Rosenthal, David Rosenthal & Peter Rosenthal
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Roots of other complex numbers can also be computed.

Example 9.2.12.

All of the solutions of the equation can be found as follows. First note that and the argument of 1 + i is . That is, . Suppose that and . Then . Therefore , so , and 3θ is or or . Therefore, θ itself can be , , or . This gives the three solutions of the equation : , , and .

9.3 The Fundamental Theorem of Algebra

One reason for introducing complex numbers was to provide a root for the polynomial x 2 + 1. There are many other polynomials that do not have any real roots. For example, if p(x) is any polynomial, then the polynomial obtained by writing out has no real roots, since its value is at least 1 for every value of x.

Does every such polynomial have a complex root? More generally, does every polynomial have a complex root? There is a trivial sense in which the answer to this question is “no,” since constant polynomials other than 0 clearly do not have any roots of any kind. For other polynomials, the answer is not so simple. It is a remarkable and very useful fact that every non-constant polynomial with real coefficients, or even with complex coefficients, has a complex root.

The Fundamental Theorem of Algebra 9.3.1.

Every non-constant polynomial with complex coefficients has a complex root.

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