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**Unformatted text preview: **e which are
often sharper3 bounds than Cauchy’s Bound.
Theorem 3.11. Upper and Lower Bounds: Suppose f is a polynomial of degree n with n ≥ 1.
• If c > 0 is synthetically divided into f and all of the numbers in the ﬁnal line of the division
tableau have the same signs, then c is an upper bound for the real zeros of f . That is, there
are no real zeros greater than c.
• If c < 0 is synthetically divided into f and the numbers in the ﬁnal line of the division
tableau alternate signs, then c is a lower bound for the real zeros of f . That is, there are
no real zeros less than c.
NOTE: If the number 0 occurs in the ﬁnal line of the division tableau in either of the above
cases, it can be treated as (+) or (−) as needed.
3 That is, better, or more accurate. 3.3 Real Zeros of Polynomials 213 The Upper and Lower Bounds Theorem works because of Theorem 3.4. For the upper bound part of
the theorem, suppose c > 0 is divided into f and the resulting line in the division tableau contains,
for example, all nonnegative numbers. This means f (x) = (x − c)q (x) + r, where the coeﬃci...

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