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Unformatted text preview: a general
polynomial has any complex zeros at all? We have many examples of polynomials with no real
7 Trust us on this.
See Section 9.3. 222 Polynomial Functions zeros. Can there be polynomials with no zeros whatsoever? The answer to that last question is
“No.” and the theorem which provides that answer is The Fundamental Theorem of Algebra.
Theorem 3.13. The Fundamental Theorem of Algebra: Suppose f is a polynomial function
with complex number coeﬃcients of degree n ≥ 1, then f has at least one complex zero.
The Fundamental Theorem of Algebra is an example of an ‘existence’ theorem in mathematics. Like
the Intermediate Value Theorem, Theorem 3.1, the Fundamental Theorem of Algebra guarantees
the existence of at least one zero, but gives us no algorithm to use in ﬁnding it. In fact, as we
mentioned in Section 3.3, there are polynomials whose real zeros, though they exist, cannot be
expressed using the ‘usual’ combinations of arithmetic symbols, and must be approximate...
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