Stitz-Zeager_College_Algebra_e-book

# This division shows 1 is a zero descartes rule of

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Unformatted text preview: on 3.2, we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section presents results which will help us determine good candidates to test using synthetic division. There are two approaches to the topic of ﬁnding the real zeros of a polynomial. The ﬁrst approach (which is gaining popularity) is to use a little bit of mathematics followed by a good use of technology like graphing calculators. The second approach (for purists) makes good use of mathematical machinery (theorems) only. For completeness, we include the two approaches but in separate subsections.1 Both approaches beneﬁt from the following two theorems, the ﬁrst of which is due to the famous mathematician Augustin Cauchy. It gives us an interval on which all of the real zeros of a polynomial can be found. Theorem 3.8. Cauchy’s Bound: Suppose f (x) = an xn +an−1 xn−1 +. . .+a1 x+a0 is a polynomial a0 a1 of degree n with n ≥ 1. Let M be the largest of the numbers: ||an|| , ||an|| , . . . , |a|n−|1 | . Then all the an real zeros of f lie in in the inte...
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