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P.nut1996

  • one year 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
    • one year ago
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    in f(x), how many sign changes are there?

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

  3. jim_thompson5910
    • one year 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
    • one year ago
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    do you see this?

  5. P.nut1996
    • one year ago
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    Yes

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

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

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

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

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

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

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

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

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

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

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

  17. jim_thompson5910
    • one year ago
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    4 is the maximum

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

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

  20. jim_thompson5910
    • one year ago
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    where at?

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

  22. jim_thompson5910
    • one year 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
    • one year 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
    • one year ago
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    Okay

  25. jim_thompson5910
    • one year ago
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    making more sense?

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

  27. jim_thompson5910
    • one year ago
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    yes? no?

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

  29. jim_thompson5910
    • one year 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
    • one year 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
    • one year ago
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    But I need a definite answer, right?

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

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

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

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

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

  37. jim_thompson5910
    • one year 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
    • one year ago
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    does that help?

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

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

  41. jim_thompson5910
    • one year ago
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    when you simplify

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

  43. jim_thompson5910
    • one year 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
    • one year 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
    • one year ago
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    No sign changes?

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

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

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

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

  50. jim_thompson5910
    • one year ago
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    no worries

  51. jim_thompson5910
    • one year 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
    • one year ago
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    I see

  53. jim_thompson5910
    • one year 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
    • one year 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
    • one year 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
    • one year ago
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    Okay So don't worry about the last scenario?

  57. jim_thompson5910
    • one year 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
    • one year 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
    • one year ago
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    I got it! :D

  60. jim_thompson5910
    • one year ago
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    ok great

  61. P.nut1996
    • one year ago
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    Thanks!

  62. jim_thompson5910
    • one year 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
    • one year ago
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    sure thing

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