379 Summary Exercises on Polynomial Functions, Zeros, and Graphs We can now use the rational zeros theorem to determine that the possible rational zeros are {1, {2, {4, and {8. Based on Descartes’ rule of signs, we discard the negative rational zeros from this list and try to find a positive rational zero. We start by using synthetic division to check 4. Proposed zero 4)1 -3 6 -12 8 4 4 40 112 1 1 10 28 120 ƒ142 = 120 We find that 4 is not a zero. However, 4 70, and the numbers in the bottom row of the synthetic division are nonnegative. Thus, the boundedness theorem indicates that there are no zeros greater than 4. We can discard 8 as a possible rational zero and use synthetic division to show that 1 and 2 are zeros. 1)1 -3 6 -12 8 1 -2 4 -8 2)1 -2 4 -8 0 ƒ112 = 0 2 0 8 1 0 4 0 ƒ122 = 0 The polynomial now factors as ƒ1x2 = 1x - 121x - 221x2 + 42. We find the remaining two zeros using algebra to solve for x in the quadratic factor of the following equation. 1x - 121x - 221x2 + 42 = 0 x - 1 = 0 or x - 2 = 0 or x2 + 4 = 0 Zero-factor property x = 1 or x = 2 or x2 = -4 x = {2i Square root property The linear factored form of the polynomial is ƒ1x2 = 1x - 121x - 221x - 2i21x + 2i), and the corresponding zeros are 1, 2, 2i, and -2i. S Now Try Exercise 3. EXERCISES For each polynomial function, complete the following in order. (a) Use Descartes’ rule of signs to determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros. (b) Use the rational zeros theorem to determine the possible rational zeros. (c) Use synthetic division with the boundedness theorem where appropriate and/or factoring to find the rational zeros, if any. (d) Find all other complex zeros (both real and nonreal), if any. 1. ƒ1x2 = 6x3 - 41x2 + 26x + 24 2. ƒ1x2 = 2x3 - 5x2 - 4x + 3 3. ƒ1x2 = 3x4 - 5x3 + 14x2 - 20x + 8 4. ƒ1x2 = 2x4 - 3x3 + 16x2 - 27x - 18 5. ƒ1x2 = 6x4 - 5x3 - 11x2 + 10x - 2 6. ƒ1x2 = 5x4 + 8x3 - 19x2 - 24x + 12 7. ƒ1x2 = x5 - 6x4 + 16x3 - 24x2 + 16x (Hint: Factor out x first.) 8. ƒ1x2 = 2x4 + 8x3 - 7x2 - 42x - 9
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