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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
of degree n with n ≥ 1. Let M be the largest of the numbers: ||an|| , ||an|| , . . . , |a|n−|1 | . Then all the
real zeros of f lie in in the inte...
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