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Math Help - Polynomial sub problem

  1. #1
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    Polynomial sub problem

    for example, there is a function f(x)= x3+ax2+bx+c=0 with roots Q,W,E
    then teachers told me that by substituting y=x2 , then we can find out f(y)=0 s.t. it has roots Q2,W2,E2

    but why such substition can help us to find out equations with root Q2,W2,E2? What are the reasons behind?
    Thanks for your kind helpings
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  2. #2
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    Re: Polynomial sub problem

    No, I don't think that works. We know that

    f(y) = f(x^2) = (x^2)^3 + a(x^2)^2 + bx^2 + c

    Letting x = \sqrt{Q} and y = Q yields a root that we already know...
    Last edited by richard1234; June 14th 2012 at 12:29 PM.
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  3. #3
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    Re: Polynomial sub problem

    oh, i mean by sub. y=x2, then x3+ax2+bx+c = 0 becomes

    x(x2+b) = -(a2+c)
    x(y+b)=-(ay+c)
    x2(y+b)2=(ay+c)2
    y(y2+2by+b2) = (a2y2+2acy+c2)
    y3+2by2+b2y = a2y2 + 2acy + c2
    y3+y2(2b-a2)+y(b2-2ac)-c2 =0
    then eqaution y3+y2(2b-a2)+y(b2-2ac)-c2 =0, has required root, sorry for the misleading of using same function notication
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  4. #4
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    Re: Polynomial sub problem

    You're way overthinking this. Also, your first step:

    x(x^2 + b) = -(a^2 + c)

    It should be

    x(x^2 + b) = -(ax^2 + c)
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  5. #5
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    Re: Polynomial sub problem

    oh ya, sorry for my careless mistake... but why it works actually ><?
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