378 CHAPTER 3 Polynomial and Rational Functions 102. Swing of a Pendulum Grandfather clocks use pendulums to keep accurate time. The relationship between the length of a pendulum L and the time T for one complete oscillation can be expressed by the equation L = kTn, where k is a constant and n is a positive integer to be determined. The data in the table were taken for different lengths of pendulums. L (ft) T (sec) L (ft) T (sec) 1.0 1.11 3.0 1.92 1.5 1.36 3.5 2.08 2.0 1.57 4.0 2.22 2.5 1.76 (a) As the length of the pendulum increases, what happens to T ? (b) Use the data to approximate k and determine the best value for n. (c) Using the values of k and n from part (b), predict T for a pendulum having length 5 ft. Round to the nearest hundredth. (d) If the length L of a pendulum doubles, what happens to the period T ? We use all of the theorems for finding complex zeros of polynomial functions in the next example. Summary Exercises on Polynomial Functions, Zeros, and Graphs EXAMPLE Finding All Zeros of a Polynomial Function Find all zeros of ƒ1x2 = x4 - 3x3 + 6x2 - 12x + 8. SOLUTION We consider the number of positive zeros by observing the variations in signs for ƒ1x2. ƒ1x2 = +x4 - 3x3 + 6x2 - 12x + 8 1 2 3 4 Because ƒ1x2 has four sign changes, we can use Descartes’ rule of signs to determine that there are four, two, or zero positive real zeros. For negative zeros, we consider the variations in signs for ƒ1-x2. ƒ1-x2 = 1-x24 - 31-x23 + 61-x22 - 121-x2 + 8 ƒ1-x2 = x4 + 3x3 + 6x2 + 12x + 8 Because ƒ1-x2 has no sign changes, there are no negative real zeros. The function has degree 4, so it has a maximum of four zeros with possibilities summarized in the table. Positive Negative Nonreal Complex 4 0 0 2 0 2 0 0 4
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