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

This preview shows page 2 - 4 out of 10 pages.

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
Image of page 2
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
Image of page 3
Image of page 4

You've reached the end of your free preview.

Want to read all 10 pages?

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

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes