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sara1234 Group Title

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

  • 2 years ago
  • 2 years ago

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  1. sara1234 Group Title
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    I don't understand what it's asking for is it asking to find positive negitive and complex?

    • 2 years ago
  2. ash2326 Group Title
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    @sara1234 It's asking for no. of positive or negative zeros.!!

    • 2 years ago
  3. sara1234 Group Title
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    huh? for no of?

    • 2 years ago
  4. ash2326 Group Title
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    Yeah, Using Descartes' Rule of Signs we can determine that. I'll explain you

    • 2 years ago
  5. ash2326 Group Title
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    We have the polynomial \[f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19\] First step it to count the no. of sign changes in f(x) I'll underline wherever there is a sign change between two consecutive terms i.e from + to - or - to + \[f(x) = –3x^5 \underline{– 8x^4 +25x^3} – 8x^2 +x – 19 \longrightarrow 1st\ change\] \[f(x) = –3x^5 – 8x^4 \underline{+25x^3 – 8x^2} +x – 19 \longrightarrow 2nd\ change\] \[f(x) = –3x^5 – 8x^4 +25x^3 \underline{– 8x^2 +x} – 19 \longrightarrow 3rd\ change\] \[f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 \underline{+x – 19} \longrightarrow 4th\ change\] so there are 4 sign changes, do you get this part?

    • 2 years ago
  6. sara1234 Group Title
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    isnt there 5 sign changes?

    • 2 years ago
  7. sara1234 Group Title
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    oh never miind i got it

    • 2 years ago
  8. ash2326 Group Title
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    So there could be 4 positive zero, However, some of the roots may be generated by the Quadratic Formula, and these pairs of roots may be complex. so we have to down count by 2. Either it has 4 or 2 or 0 positive roots. Do you get this?

    • 2 years ago
  9. sara1234 Group Title
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    yes

    • 2 years ago
  10. ash2326 Group Title
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    Now can you write f(-x), it's used to find no. of negative zeros

    • 2 years ago
  11. sara1234 Group Title
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    -3(-x)^5-8(-x)^4+25^3-8(-x)^2+(-x)-19

    • 2 years ago
  12. ash2326 Group Title
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    Good , we need to simplify it. I'll write that \[f(-x)=-3(-x)^5-8(-x)^4+25(-x)^3-8(-x)^2+(-x)-19\] \[f(-x)=3x^5-8x^4-25x^3-8x^2-x-19\] do you get this?

    • 2 years ago
  13. sara1234 Group Title
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    Yes i do

    • 2 years ago
  14. ash2326 Group Title
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    Now count the no. of sign changes here

    • 2 years ago
  15. sara1234 Group Title
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    only one

    • 2 years ago
  16. ash2326 Group Title
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    Yeah so it's having 1 negative zero, we can't down count by 2, it'll become -1. So it has 1 negative zero and 4 or 2 or 0 positive zeros. Do you get this?

    • 2 years ago
  17. sara1234 Group Title
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    Yeah thankssss!! Now we find complex?

    • 2 years ago
  18. ash2326 Group Title
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    We found that there are either 4 or 2 or 0 positive roots. But these could also be complex, so if we have have 4 positive roots there will be 0 complex if we have 2 positive there will be 2 complex or if we have 0 positive there will be 4 complex

    • 2 years ago
  19. sara1234 Group Title
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    why do we have to add the ) positive?

    • 2 years ago
  20. sara1234 Group Title
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    0

    • 2 years ago
  21. ash2326 Group Title
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    Because 0 could be due to 4 complex roots

    • 2 years ago
  22. sara1234 Group Title
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    theres 4 or 2 positive real zeros but why do we have to add the 0

    • 2 years ago
  23. ash2326 Group Title
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    There could be a double quadratic, with 4 roots complex. therefore we won't have any positive zero

    • 2 years ago
  24. sara1234 Group Title
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    thanks for all your help @ash2326 you really helped me :)

    • 2 years ago
  25. ash2326 Group Title
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    you're welcome. did you understand all the things?

    • 2 years ago
  26. sara1234 Group Title
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    Yessss :D

    • 2 years ago
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