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P.nut1996 Group Title

Help? Using complete sentences, describe how you would analyze the zeros of the polynomial function f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19 using Descartes’ Rule of Signs.

  • one year ago
  • one year ago

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  1. jim_thompson5910 Group Title
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    in f(x), how many sign changes are there?

    • one year ago
  2. P.nut1996 Group Title
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    I don't know?

    • one year ago
  3. jim_thompson5910 Group Title
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    notice that going from -8x^4 to +25x^3, there's a sign change from negative to positive

    • one year ago
  4. jim_thompson5910 Group Title
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    do you see this?

    • one year ago
  5. P.nut1996 Group Title
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    Yes

    • one year ago
  6. jim_thompson5910 Group Title
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    ok there's another from +25x^3 to -8x^2

    • one year ago
  7. jim_thompson5910 Group Title
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    what's another one?

    • one year ago
  8. P.nut1996 Group Title
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    +x to -19?

    • one year ago
  9. jim_thompson5910 Group Title
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    there's one more

    • one year ago
  10. jim_thompson5910 Group Title
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    from -8x^2 to +x

    • one year ago
  11. jim_thompson5910 Group Title
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    so there are 4 sign changes total in f(x)

    • one year ago
  12. jim_thompson5910 Group Title
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    this means that there are at most 4 positive real roots

    • one year ago
  13. P.nut1996 Group Title
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    -19 to +x?

    • one year ago
  14. jim_thompson5910 Group Title
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    no that's the same sign change (just in reverse)

    • one year ago
  15. P.nut1996 Group Title
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    4 sign changes=4 positive real roots

    • one year ago
  16. jim_thompson5910 Group Title
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    at most 4 (there could be 0, 1, 2, 3, or 4 positive real roots)

    • one year ago
  17. jim_thompson5910 Group Title
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    4 is the maximum

    • one year ago
  18. jim_thompson5910 Group Title
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    now we must find f(-x)

    • one year ago
  19. P.nut1996 Group Title
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    I'm still really confuzzled...

    • one year ago
  20. jim_thompson5910 Group Title
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    where at?

    • one year ago
  21. P.nut1996 Group Title
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    All of it

    • one year ago
  22. jim_thompson5910 Group Title
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    the rule is if f(x) has n sign changes, then there are AT MOST n positive real roots

    • one year ago
  23. jim_thompson5910 Group Title
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    if n = 4, then if f(x) has 4 sign changes, then there are AT MOST 4 positive real roots

    • one year ago
  24. P.nut1996 Group Title
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    Okay

    • one year ago
  25. jim_thompson5910 Group Title
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    making more sense?

    • one year ago
  26. P.nut1996 Group Title
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    ...

    • one year ago
  27. jim_thompson5910 Group Title
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    yes? no?

    • one year ago
  28. P.nut1996 Group Title
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    I don't know how to formulate an answer from you telling me this...

    • one year ago
  29. jim_thompson5910 Group Title
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    you don't need to find the actual roots you just need to be able to count the possible number and type

    • one year ago
  30. jim_thompson5910 Group Title
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    more specifically, how many positive real roots you could have (in this case, at most 4) you don't need to find the 4 or find out how many actual positive roots there are

    • one year ago
  31. P.nut1996 Group Title
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    But I need a definite answer, right?

    • one year ago
  32. P.nut1996 Group Title
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    I can't just say "maybe 4"?

    • one year ago
  33. jim_thompson5910 Group Title
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    well that's part of the answer, we still have to find f(-x)

    • one year ago
  34. P.nut1996 Group Title
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    Okay, and how would we go about that ? :)

    • one year ago
  35. jim_thompson5910 Group Title
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    start with f(x) then replace each x with -x and simplify

    • one year ago
  36. P.nut1996 Group Title
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    Could you give me an example?

    • one year ago
  37. jim_thompson5910 Group Title
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    start with this f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19 then replace each 'x' with '-x' to get f(-x) = –3(-x)^5 – 8(-x)^4 +25(-x)^3 – 8(-x)^2 +(-x) – 19 then simplify to get ???

    • one year ago
  38. jim_thompson5910 Group Title
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    does that help?

    • one year ago
  39. P.nut1996 Group Title
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    Yes, it does :)

    • one year ago
  40. jim_thompson5910 Group Title
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    ok what do you get

    • one year ago
  41. jim_thompson5910 Group Title
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    when you simplify

    • one year ago
  42. P.nut1996 Group Title
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    3x^5=f(-x)+x(x(x8x(+25)+8)+1)+19??

    • one year ago
  43. jim_thompson5910 Group Title
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    f(-x) = –3(-x)^5 – 8(-x)^4 +25(-x)^3 – 8(-x)^2 +(-x) – 19 would simplify to f(-x) = 3x^5 – 8x^4 - 25x^3 – 8x^2 - x – 19

    • one year ago
  44. jim_thompson5910 Group Title
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    now count the sign changes in f(-x) = 3x^5 – 8x^4 - 25x^3 – 8x^2 - x – 19 and tell me how many you got

    • one year ago
  45. P.nut1996 Group Title
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    No sign changes?

    • one year ago
  46. jim_thompson5910 Group Title
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    3x^5 is the same as +3x^5

    • one year ago
  47. jim_thompson5910 Group Title
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    so there's only one sign change from +3x^5 to -8x^4

    • one year ago
  48. jim_thompson5910 Group Title
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    so this means that there is at most 1 negative real root

    • one year ago
  49. P.nut1996 Group Title
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    Oh, I was just looking at it wrong.. Sorry :(

    • one year ago
  50. jim_thompson5910 Group Title
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    no worries

    • one year ago
  51. jim_thompson5910 Group Title
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    So far we found that there are at most 4 positive real roots and at most 1 negative real root

    • one year ago
  52. P.nut1996 Group Title
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    I see

    • one year ago
  53. jim_thompson5910 Group Title
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    so one possibility is that there are 4+1 =5 real roots (4 positive, 1 negative) BUT this is not the only possibility

    • one year ago
  54. jim_thompson5910 Group Title
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    we could have 3 positive real roots and 1 negative real root to have 3+1 = 4 real roots total but the problem with this scenario is that you would only have 5-1 = 1 complex root...when complex roots should come in pairs

    • one year ago
  55. jim_thompson5910 Group Title
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    so that scenario of 3 positive real roots and 1 negative real root just isn't possible

    • one year ago
  56. P.nut1996 Group Title
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    Okay So don't worry about the last scenario?

    • one year ago
  57. jim_thompson5910 Group Title
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    anyways, there are a ton of possibilities the good news is that you can sum them all up by saying there are at most 4 positive real roots (you could have 0, 1, 2, 3, or 4 positive real roots) there is at most 1 negative real root (you could have 0 or 1 negative real roots) there are at most 5 complex roots (if there are no real roots at all)

    • one year ago
  58. jim_thompson5910 Group Title
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    in short there are at most 4 positive real roots there is at most 1 negative real root there are at most 5 complex roots

    • one year ago
  59. P.nut1996 Group Title
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    I got it! :D

    • one year ago
  60. jim_thompson5910 Group Title
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    ok great

    • one year ago
  61. P.nut1996 Group Title
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    Thanks!

    • one year ago
  62. jim_thompson5910 Group Title
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    we can stop there because we don't need to find the actual roots or the counts of the types or roots...just the max of all of the possible types of roots

    • one year ago
  63. jim_thompson5910 Group Title
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    sure thing

    • one year ago
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