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Math Help - polynomials

  1. #1
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    polynomials

    f(x) = 3x^2 - x^3 + h. For which values of h will f have three distinct zeros?

    How do I do this too.


    I guess they need a height for the graph where you get three roots. If I differentiate the function, I see that y would have to be above the x axis on x = 2, and below the x axis on x = 0.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by LeoBloom. View Post
    f(x) = 3x^2 - x^3 + h. For which values of h will f have three distinct zeros?

    How do I do this too.
    If h doesnt have to be a constant then

    -x^3+3x^2+5x\Rightarrow{-x(x^2+3x+5)}

    Setting it equal to zero we get

    x=0,x=3\pm\frac{\sqrt{29}}{2}
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  3. #3
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    Sorry, forgot to mention h has to be a constant.

    The answers in the book are

    (a) all h > 4
    (b) 0 < h < 4
    (c) all h < 0
    (d) -4 < h < 0
    (e) all h < -4

    I think that I could rule out a,c,e because then at some point in the graph, there will only be one root instead of three.
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  4. #4
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    And if h is a constant, try negative values such as -1, -2, -3....etc.
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  5. #5
    Moo
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    Hello,

    Quote Originally Posted by Mathstud28 View Post
    If h doesnt have to be a constant then

    -x^3+3x^2+5x\Rightarrow{-x(x^2+3x+5)}

    Setting it equal to zero we get

    x=0,x=3\pm\frac{\sqrt{29}}{2}
    I don't think they're asking for ONE solution, they more likely want the domain of h, such that the condition is OK.

    Plus, in géneral with this notation, h will be a constant
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  6. #6
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    Quote Originally Posted by LeoBloom. View Post
    f(x) = 3x^2 - x^3 + h. For which values of h will f have three distinct zeros?

    How do I do this too.


    I guess they need a height for the graph where you get three roots. If I differentiate the function, I see that y would have to be above the x axis on x = 2, and below the x axis on x = 0.
    youve got it spot on from that you get 2 inequalities
    f(0)<0 ie  3(0^2)+0^3+h<0

    and f(2)>0 ie 3(2^2)+2^3+h>0

    solving these two gets you the answer
    -4<h<0
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