SECTION 4.3 The Real Zeros of a Polynomial Function 225 7 Use the Intermediate Value Theorem The next result, called the Intermediate Value Theorem , is based on the fact that the graph of a polynomial function is continuous; that is, it contains no “holes” or “gaps.” Using ZERO (or ROOT), the zero between 0 and 1 is found to be 0.20 (rounded to two decimal places), and the zeros 4− and 1− are confirmed. Zooming in shows that there are in fact two distinct zeros near 3. See Figure 31(c). The two remaining zeros are 3.23 and 3.37, each of which is rounded to two decimal places. Now Work PROBLEM 63 THEOREM Intermediate Value Theorem Let f denote a polynomial function. If a b < and if f a( ) and f b( ) are of opposite sign, there is at least one real zero of f between a and b . Although the proof of this result requires advanced methods in calculus, it is easy to “see” why the result is true. Look at Figure 32. Figure 32 If f is a polynomial function and f a( ) and f b( ) are of opposite sign, then there is at least one real zero between a and b . x a f(a) f(b) y Zero b f (a) f (b) y 5 f(x) (b) x a f (b) f (b) f(a) y Zeros b f(a) y 5 f(x) (a) The Intermediate Value Theorem together with the TABLE feature of a graphing utility provides a basis for finding zeros. Using the Intermediate Value Theorem and a Graphing Utility to Locate Zeros Find the positive real zero of f x x x 1 5 3 ( ) = − − correct to two decimal places. EXAMPLE 11 Solution Because f 1 1 0 ( ) = − < and f 2 23 0, ( ) = > we know from the Intermediate Value Theorem that the zero lies between 1 and 2. Divide the interval 1, 2 [ ] into 10 equal subintervals, and use the TABLE feature of a TI-84 Plus CE to evaluate f at the endpoints of the intervals. See Table 8. We can conclude that the zero is between 1.2 and 1.3 since f 1.2 0 ( ) < and f 1.3 0 ( ) > . Now divide the interval 1.2, 1.3 [ ] into 10 equal subintervals and evaluate f at each endpoint. See Table 9. The zero lies between 1.23 and 1.24, and so, correct to two decimal places, the zero is 1.23. Now Work PROBLEM 89 There are many other numerical techniques for approximating the zeros of a polynomial function. The one outlined in Example 11 (a variation of the bisection method ) has the advantages that it will always work, it can be programmed rather easily on a computer, and each time it is used another decimal place of accuracy is achieved. See Problem 120 for the bisection method, which places the zero in a succession of intervals, with each new interval being half the length of the preceding one. Table 8 Table 9
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