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P.nut1996

  • 2 years ago

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.

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  1. jim_thompson5910
    • 2 years ago
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    in f(x), how many sign changes are there?

  2. P.nut1996
    • 2 years ago
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    I don't know?

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

  4. jim_thompson5910
    • 2 years ago
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    do you see this?

  5. P.nut1996
    • 2 years ago
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    Yes

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

  7. jim_thompson5910
    • 2 years ago
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    what's another one?

  8. P.nut1996
    • 2 years ago
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    +x to -19?

  9. jim_thompson5910
    • 2 years ago
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    there's one more

  10. jim_thompson5910
    • 2 years ago
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    from -8x^2 to +x

  11. jim_thompson5910
    • 2 years ago
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    so there are 4 sign changes total in f(x)

  12. jim_thompson5910
    • 2 years ago
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    this means that there are at most 4 positive real roots

  13. P.nut1996
    • 2 years ago
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    -19 to +x?

  14. jim_thompson5910
    • 2 years ago
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    no that's the same sign change (just in reverse)

  15. P.nut1996
    • 2 years ago
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    4 sign changes=4 positive real roots

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

  17. jim_thompson5910
    • 2 years ago
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    4 is the maximum

  18. jim_thompson5910
    • 2 years ago
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    now we must find f(-x)

  19. P.nut1996
    • 2 years ago
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    I'm still really confuzzled...

  20. jim_thompson5910
    • 2 years ago
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    where at?

  21. P.nut1996
    • 2 years ago
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    All of it

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

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

  24. P.nut1996
    • 2 years ago
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    Okay

  25. jim_thompson5910
    • 2 years ago
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    making more sense?

  26. P.nut1996
    • 2 years ago
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    ...

  27. jim_thompson5910
    • 2 years ago
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    yes? no?

  28. P.nut1996
    • 2 years ago
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    I don't know how to formulate an answer from you telling me this...

  29. jim_thompson5910
    • 2 years ago
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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

  30. jim_thompson5910
    • 2 years ago
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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

  31. P.nut1996
    • 2 years ago
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    But I need a definite answer, right?

  32. P.nut1996
    • 2 years ago
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    I can't just say "maybe 4"?

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

  34. P.nut1996
    • 2 years ago
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    Okay, and how would we go about that ? :)

  35. jim_thompson5910
    • 2 years ago
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    start with f(x) then replace each x with -x and simplify

  36. P.nut1996
    • 2 years ago
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    Could you give me an example?

  37. jim_thompson5910
    • 2 years ago
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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 ???

  38. jim_thompson5910
    • 2 years ago
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    does that help?

  39. P.nut1996
    • 2 years ago
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    Yes, it does :)

  40. jim_thompson5910
    • 2 years ago
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    ok what do you get

  41. jim_thompson5910
    • 2 years ago
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    when you simplify

  42. P.nut1996
    • 2 years ago
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    3x^5=f(-x)+x(x(x8x(+25)+8)+1)+19??

  43. jim_thompson5910
    • 2 years ago
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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

  44. jim_thompson5910
    • 2 years ago
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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

  45. P.nut1996
    • 2 years ago
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    No sign changes?

  46. jim_thompson5910
    • 2 years ago
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    3x^5 is the same as +3x^5

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

  48. jim_thompson5910
    • 2 years ago
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    so this means that there is at most 1 negative real root

  49. P.nut1996
    • 2 years ago
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    Oh, I was just looking at it wrong.. Sorry :(

  50. jim_thompson5910
    • 2 years ago
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    no worries

  51. jim_thompson5910
    • 2 years ago
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    So far we found that there are at most 4 positive real roots and at most 1 negative real root

  52. P.nut1996
    • 2 years ago
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    I see

  53. jim_thompson5910
    • 2 years ago
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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

  54. jim_thompson5910
    • 2 years ago
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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

  55. jim_thompson5910
    • 2 years ago
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    so that scenario of 3 positive real roots and 1 negative real root just isn't possible

  56. P.nut1996
    • 2 years ago
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    Okay So don't worry about the last scenario?

  57. jim_thompson5910
    • 2 years ago
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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)

  58. jim_thompson5910
    • 2 years ago
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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

  59. P.nut1996
    • 2 years ago
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    I got it! :D

  60. jim_thompson5910
    • 2 years ago
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    ok great

  61. P.nut1996
    • 2 years ago
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    Thanks!

  62. jim_thompson5910
    • 2 years ago
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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

  63. jim_thompson5910
    • 2 years ago
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    sure thing

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