The Number of Zeros A polynomial of degree n has at most n real zeros

# The number of zeros a polynomial of degree n has at

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The Number of Zeros A polynomial of degree n has at most n real zeros. Descartes’rule of signs determines the possible combinations of positive and negative real zeros. Upper and lower bounds help narrow the search for zeros. How to Find Zeros Rational zero theorem : List possible rational zeros: Factors of constant, a 0 Factors of leading coefficient, a n SECTION 4.4 SUMMARY EXERCISES SECTION 4.4 In Exercises 1–6, find the following values by using synthetic division. Check by substituting the value into the function. f ( x ) 3 x 4 2 x 3 7 x 2 8 g ( x ) 2 x 3 x 2 1 1. f (1) 2. f ( 1) 3. g (1) 4. g ( 1) 5. f ( 2) 6. g (2) In Exercises 7–10, determine whether the number given is a zero of the polynomial. 7. 7, P ( x ) x 3 2 x 2 29 x 42 8. 2, P ( x ) x 3 2 x 2 29 x 42 9. 3, P ( x ) x 3 x 2 8 x 12 10. 1, P ( x ) x 3 x 2 8 x 12 SKILLS 4.4 The Real Zeros of a Polynomial Function 433 In Exercises 11–20, given a real zero of the polynomial, determine all other real zeros, and write the polynomial in terms of a product of linear and/or irreducible quadratic factors. Polynomial Zero Polynomial Zero 11. P ( x ) x 3 13 x 12 1 12. P ( x ) x 3 3 x 2 10 x 24 3 13. P ( x ) 2 x 3 x 2 13 x 6 14. P ( x ) 3 x 3 14 x 2 7 x 4 15. P ( x ) x 4 2 x 3 11 x 2 8 x 60 3, 5 16. P ( x ) x 4 x 3 7 x 2 9 x 18 1, 2 17. P ( x ) x 4 5 x 2 10 x 6 1, 3 18. P ( x ) x 4 4 x 3 x 2 6 x 40 4, 2 19. P ( x ) x 4 6 x 3 13 x 2 12 x 4 2 (multiplicity 2) 20. P ( x ) x 4 4 x 3 2 x 2 12 x 9 1 (multiplicity 2) In Exercises 21–28, use the rational zero theorem to list the possible rational zeros. 21. P ( x ) x 4 3 x 2 8 x 4 22. P ( x ) x 4 2 x 3 5 x 4 23. P ( x ) x 5 14 x 3 x 2 15 x 12 24. P ( x ) x 5 x 3 x 2 4 x 9 25. P ( x ) 2 x 6 7 x 4 x 3 2 x 8 26. P ( x ) 3 x 5 2 x 4 5 x 3 x 10 27. P ( x ) 5 x 5 3 x 4 x 3 x 20 28. P ( x ) 4 x 6 7 x 4 4 x 3 x 21 In Exercises 29–32, list the possible rational zeros, and test to determine all rational zeros. 29. P ( x ) x 4 2 x 3 9 x 2 2 x 8 30. P ( x ) x 4 2 x 3 4 x 2 2 x 3 31. P ( x ) 2 x 3 9 x 2 10 x 3 32. P ( x ) 3 x 3 5 x 2 26 x 8 In Exercises 33–44, use Descartes’ rule of signs to determine the possible number of positive real zeros and negative real zeros. 33. P ( x ) x 4 32 34. P ( x ) x 4 32 35. P ( x ) x 5 1 36. P ( x ) x 5 1 37. P ( x ) x 5 3 x 3 x 2 38. P ( x ) x 4 2 x 2 9 39. P ( x ) 9 x 7 2 x 5 x 3 x 40. P ( x ) 16 x 7 3 x 4 2 x 1 41. P ( x ) x 6 16 x 4 2 x 2 7 42. P ( x ) 7 x 6 5 x 4 x 2 2 x 1 43. P ( x ) 3 x 4 2 x 3 4 x 2 x 11 44. P ( x ) 2 x 4 3 x 3 7 x 2 3 x 2 For each polynomial in Exercises 45–62: (a) use Descartes’ rule of signs to determine the possible combinations of positive real zeros and negative real zeros; (b) use the rational zero test to determine possible rational zeros; (c) test for rational zeros; and (d) factor as a product of linear and/or irreducible quadratic factors. 45. P ( x ) x 3 6 x 2 11 x 6 46. P ( x ) x 3 6 x 2 11 x 6 47. P ( x ) x 3 7 x 2 x 7 48. P ( x ) x 3 5 x 2 4 x 20 49. P ( x ) x 4 6 x 3 3 x 2 10 x 50. P ( x ) x 4 x 3 14 x 2 24 x 51. P ( x ) x 4 7 x 3 27 x 2 47 x 26 52. P ( x ) x 4 5 x 3 5 x 2 25 x 26 53. P ( x ) 10 x 3 7 x 2 4 x 1 54. P ( x ) 12 x 3 13 x 2 2 x 1 55. P ( x ) 6 x 3 17 x 2 x 10 56. P ( x ) 6 x 3 x 2 5 x 2 57. P ( x ) x 4 2 x 3 5 x 2 8 x 4 58. P ( x ) x 4 2 x 3 10 x 2 18 x 9 59. P ( x ) x 6 12 x 4 23 x 2 36 60. P ( x ) x 4 x 2 16 x 2 16 61. P ( x ) 4 x 4 20 x 3 37 x 2 24 x 5 62. P ( x ) 4 x 4 8 x 3 7 x 2 30 x 50  #### You've reached the end of your free preview.

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• Summer '17
• juan alberto
• Complex number, real zeros, complex zeros
• • •  