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Finding number of zeroes in a polynomial?

  1. Jul 17, 2012 #1
    Let's say I have the equation f(x) = 2x + 3 * (3x^2 + 3) - x^2 + 5. If my algebra is right, this is a 3rd-degree polynomial. How many zeroes does this equation have? How did you figure that out?
     
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  3. Jul 17, 2012 #2

    jedishrfu

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    if you remember: an nth degree polynomial has n factors of the form f(x) = (x-a)*(x-b)*...
    hence it has at most n zeros. ( I assumed the factors for x were 1)

    So in your case for a 3rd degree:

    f(x) = (x -a) * (x - b ) * ( x - c )
    ___ = (x^2 - (a+b)x + ab) * (x - c)
    ___ = x^3 - (a+b+c)x^2 + (ab+ac+bc)x + abc
     
    Last edited: Jul 17, 2012
  4. Jul 17, 2012 #3

    micromass

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    Am I missing something?? The equation in the OP does not have third degree...
     
  5. Jul 17, 2012 #4

    jedishrfu

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    you're right, i was addressing find the number of roots only.
     
  6. Jul 17, 2012 #5
    2x2 + 3 + (3X2 + 3) = 6x3 + 6x + 6x2 + 9. That's why I thought the degree was 3.
     
  7. Jul 17, 2012 #6

    micromass

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    I have no clue why you think this equality is true.
     
  8. Jul 17, 2012 #7

    jedishrfu

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    I think you meant (2x+3)*(3x^2+3) right?
     
  9. Jul 17, 2012 #8

    haruspex

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    Assuming it is a cubic, there will be 1, 2 or 3 real roots. There are always 3 roots altogether, but some may be complex pairs. That can reduce it to 1 real root. There is also the borderline case where two real roots coincide, making only 2 values.
     
  10. Jul 18, 2012 #9

    Mark44

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    Based on what you wrote, your algebra is incorrect. Expanding what you wrote, I get
    f(x) = 2x + 9x2 + 9 - x2 + 5
    = 8x2 + 2x + 14, which is NOT a cubic.

    I suspect that you are missing some parentheses, and actually meant
    f(x) = (2x + 3)*(3x2 + 3) - x2 + 5, which IS a cubic.
     
  11. Jul 18, 2012 #10
    It can't have two real roots.
     
  12. Jul 18, 2012 #11

    HallsofIvy

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    It can have three real roots, two of which are the same, having two distinct real roots. That is what haruspex was talking about. He was not counting "multiplicity".

    For example, [itex]x^3- x^2= 0[/itex] has two distinct roots- 0 and 1. 0 is a double root.
     
    Last edited by a moderator: Jul 18, 2012
  13. Jul 18, 2012 #12
    YEs, I see now. I just re-read his post.
     
    Last edited by a moderator: Jul 18, 2012
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