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Unformatted text preview: c) = 0. If c2 − 2 = 0 then c = ± 2. Since c is between 1 and 3, c is
positive, so c = 2.
Our primary use of the Intermediate Value Theorem is in the construction of sign diagrams, as in
Section 2.4, since it guarantees us that polynomial functions are always positive (+) or always negative (−) on intervals which do not contain any of its zeros. The general algorithm for polynomials
is given below. 186 Polynomial Functions
Steps for Constructing a Sign Diagram for a Polynomial Function
Suppose f is a polynomial function. 1. Find the zeros of f and place them on the number line with the number 0 above them.
2. Choose a real number, called a test value, in each of the intervals determined in step 1.
3. Determine the sign of f (x) for each test value in step 2, and write that sign above the
Example 3.1.5. Construct a sign diagram for f (x) = x3 (x − 3)2 (x + 2) x2 + 1 . Use it to give a
rough sketch of the graph of y = f (x).
Solution. First, we ﬁnd the zeros of f by solving x3 (...
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