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Stitz-Zeager_College_Algebra_e-book

# This has 3 variations in sign hence f has either 3

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Unformatted text preview: o see what happens. 1 2 4 −4 −11 12 −3 ↓ 2 −1 −6 3 1 4 −2 −12 60 2 ↓ 2 0 −6 √ 34 √0 −12 0 ↓ 43 12 √ √ −3 4 4 √3 0 ↓ −4 3 4 0 √ √ 2 This gives us 4x4 − 4x3 − 11x2 + 12x − 3 = x − 1 x− 3 x− − 3 2 with the constant in front 1 p(x) = 4 x − 2 2 x− √ 3 (4), or, when written √ x− − 3 We have shown that p is a product of its leading term times linear factors of the form (x − c) where c are zeros of p. It may surprise and delight the reader that, in theory, all polynomials can be reduced to this kind of factorization. We leave that discussion to Section 3.4, because the zeros may not be real numbers. Our ﬁnal theorem in the section gives us an upper bound on the number of real zeros. Theorem 3.7. Suppose f is a polynomial of degree n, n ≥ 1. Then f has at most n real zeros, counting multiplicities. Theorem 3.7 is a consequence of the Factor Theorem and polynomial multiplication. Every zero c of f gives us a factor of the form (x − c) for...
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