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anonymous

  • 5 years ago

How do you use the given root to assist in finding the remaining roots of the equation? 2x^4 - 15x^3 + 34x^2 - 19x - 20 = 0; 1/2

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  1. anonymous
    • 5 years ago
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    divide. if \[\frac{1}{2}\] is a root then you know your polynomial is \[(x-\frac{1}{2})q(x)\] or more easily written as \[p(x)=(2x-1)q(x)\]

  2. anonymous
    • 5 years ago
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    you find \[q(x)\] by dividing by \[(x-\frac{1}{2})\] synthetic division is easiest

  3. anonymous
    • 5 years ago
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    do you know how to use synthetic division?

  4. mathteacher1729
    • 5 years ago
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    These are the synthetic division steps. You know to use synthetic division because you are given a factor of polynomial. It's like saying "458 is divisible by 2... what is the remainder when divisible by 2?) Did you copy your polynomial correctly? You are left with a third degree polynomial which has complex roots... Usually the problems give "Friendlier" answers. :)

  5. anonymous
    • 5 years ago
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    if this is correct there must be a typo somewhere because you (wolfram) got a remainder of -91/4

  6. anonymous
    • 5 years ago
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    Great

  7. anonymous
    • 5 years ago
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    remainder should be 0 of course

  8. mathteacher1729
    • 5 years ago
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    Jennifer -- I think you might have mis-copied the problem. The answer is ... too ugly for something which might be assigned for a practice/hw or even test problem.

  9. anonymous
    • 5 years ago
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    Let me check quick

  10. anonymous
    • 5 years ago
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    in any case 1/2 is not a zero according to wolfram

  11. anonymous
    • 5 years ago
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    Oh wow, after the equation, the 1/2 is a negative..I am so sorry..

  12. anonymous
    • 5 years ago
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    2x^4 - 15x^3 + 34x^2 - 19x - 20 = 0; -1/2

  13. anonymous
    • 5 years ago
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    ohhhhhhhhhhhhhhhhhhhh -1/2

  14. anonymous
    • 5 years ago
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    You sure are one smart cookie to notice that. My goodness; i sure wish I could get math as easy as you .

  15. mathteacher1729
    • 5 years ago
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    That makes more sense, but you're still going to be left with a 3rd degree polynomial... And that 3rd order polynomial will not be very easy to solve.

  16. anonymous
    • 5 years ago
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    then it factors as \[(x+\frac{1}{2})q(x)\]

  17. mathteacher1729
    • 5 years ago
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  18. anonymous
    • 5 years ago
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    btw you can find the quotient relatively easily. very easy if you can synthetic division

  19. anonymous
    • 5 years ago
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    do you know how? long division is a pain.

  20. anonymous
    • 5 years ago
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    a little bit..i'm bad..really

  21. anonymous
    • 5 years ago
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    list the coefficients they are 2 -15 34 -19 20

  22. anonymous
    • 5 years ago
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    since you root is -1/2 put that on the side. i will write -.5 2 -15 34 -19 20 -.5 ___________________________________

  23. anonymous
    • 5 years ago
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    drop the 2 2 -15 34 -19 20 -.5 ___________________________________ 2

  24. anonymous
    • 5 years ago
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    -.5*-1 2 -15 34 -19 20 -.5 -1 ___________________________________ 2

  25. anonymous
    • 5 years ago
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    -15-1=-16 2 -15 34 -19 20 -.5 -1 ___________________________________ 2 -16

  26. anonymous
    • 5 years ago
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    -16*-.5 = 8 2 -15 34 -19 20 -.5 -1 8 ___________________________________ 2 -16

  27. anonymous
    • 5 years ago
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    34 + 8 = 42 2 -15 34 -19 20 -.5 -1 8 ___________________________________ 2 -16 42

  28. anonymous
    • 5 years ago
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    -.5*42=-21 2 -15 34 -19 20 -.5 -1 8 -21 ___________________________________ 2 -16 42

  29. anonymous
    • 5 years ago
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    -19-21=-40 2 -15 34 -19 20 -.5 -1 8 -21 ___________________________________ 2 -16 42 -40

  30. anonymous
    • 5 years ago
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    -.5*-40= 20 and now i see i have a typo. the last number in the top line should be -20, not 20 easy to fix 2 -15 34 -19 - 20 -.5 -1 8 -21 20 ___________________________________ 2 -16 42 -40

  31. anonymous
    • 5 years ago
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    -20+20=0 so the remainder is 0 \ 2 -15 34 -19 - 20 -.5 -1 8 -21 20 ___________________________________ 2 -16 42 -40 0

  32. anonymous
    • 5 years ago
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    and your answer is \[p(x)=(x+\frac{1}{2})(2x^3-16x^2+42x-40)\]

  33. anonymous
    • 5 years ago
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    or \[p(x)=(2x-1)(x^3-8x^2+21x-10)\]

  34. anonymous
    • 5 years ago
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    done!

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