Unformatted text preview: rval [−(M + 1), M + 1].
The proof of this fact is not easily explained within the conﬁnes of this text. This paper contains
the result and gives references to its proof. Like many of the results in this section, Cauchy’s Bound
is best understood with an example.
Example 3.3.1. Let f (x) = 2x4 + 4x3 − x2 − 6x − 3. Determine an interval which contains all of
the real zeros of f .
Solution. Cauchy’s Bound says to take the absolute value of each of the non-leading coeﬃcients
of f , namely, 4, 1, 6 and 3, and divide them by the absolute value of the leading coeﬃcient, 2.
Doing so produces the list of numbers 2, 2 , 3, and 2 . Next, we take the largest of these values, 3,
as our value M in the theorem and add one to it to get 4. The real zeros of f are guaranteed to lie
in the interval [−4, 4].
Whereas the previous result tells us where we can ﬁnd the real zeros of a polynomial, the next
theorem gives us a list of possible real zeros.
Theorem 3.9. Rational Zeros Theorem: Suppose f (x) = an xn + an−1 xn−1 + . . ....
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