380 CHAPTER 3 Polynomial and Rational Functions 9. ƒ1x2 = 8x4 + 8x3 - x - 1 (Hint: Factor the polynomial.) 10. ƒ1x2 = 2x5 + 5x4 - 9x3 - 11x2 + 19x - 6 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) Find the rational zeros, if any. (d) Find all other real zeros, if any. (e) Find any other complex zeros (that is, zeros that are not real), if any. (f) Find the x-intercepts of the graph, if any. (g) Find the y-intercept of the graph. (h) Use synthetic division to find ƒ142, and give the coordinates of the corresponding point on the graph. (i) Determine the end behavior of the graph. (j) Sketch the graph. 11. ƒ1x2 = x4 + 3x3 - 3x2 - 11x - 6 12. ƒ1x2 = -2x5 + 5x4 + 34x3 - 30x2 - 84x + 45 13. ƒ1x2 = 2x5 - 10x4 + x3 - 5x2 - x + 5 14. ƒ1x2 = 3x4 - 4x3 - 22x2 + 15x + 18 15. ƒ1x2 = -2x4 - x3 + x + 2 16. ƒ1x2 = 4x5 + 8x4 + 9x3 + 27x2 + 27x (Hint: Factor out x first.) 17. ƒ1x2 = 3x4 - 14x2 - 5 (Hint: Factor the polynomial.) 18. ƒ1x2 = -x5 - x4 + 10x3 + 10x2 - 9x - 9 19. ƒ1x2 = -3x4 + 22x3 - 55x2 + 52x - 12 20. For the polynomial functions in Exercises 11–19 that have irrational zeros, find approximations to the nearest thousandth. 3.5 Rational Functions: Graphs, Applications, and Models ■ The Reciprocal Function f 1x2 =1 x ■ The Function f 1x2 = 1 x2 ■ Asymptotes ■ Graphing Techniques ■ Rational Models A rational expression is a fraction that is the quotient of two polynomials. A rational function is defined by a quotient of two polynomial functions. Rational Function A rational function ƒ is a function of the form ƒ1x2 = p1x2 q1x2 , where p1x2 and q1x2 are polynomial functions, with q1x2 ≠0.
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