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Thread: Intermediate value problem

  1. #1
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    Intermediate value problem

    hey all,

    i got the following problem to solve:

    i got this term: 27*g^2+4*p^3<0

    and i need to proof that there are three real roots to this function:

    x^3+px+q=0

    is it enough to solve it by min/max points of the function?
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  2. #2
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    Condition for a cubic to have real roots

    Hello dannee
    Quote Originally Posted by dannee View Post
    hey all,

    i got the following problem to solve:

    i got this term: 27*g^2+4*p^3<0

    and i need to proof that there are three real roots to this function:

    x^3+px+q=0

    is it enough to solve it by min/max points of the function?
    I'm sure you mean:
    $\displaystyle 27q^2+4p^3<0$
    Then the proof goes like this:

    Every cubic has at least one real root, so let's assume that $\displaystyle x^3+px + q = 0 $ has a real root $\displaystyle x = a$. Then:
    $\displaystyle x^3+px+q \equiv (x-a)(x^2+bx+c)$, for some constants $\displaystyle b$ and $\displaystyle c$.
    $\displaystyle \equiv x^3 +(b-a)x^2+(c-ab)x-ac$
    Comparing coefficients:
    $\displaystyle \left\{\begin {array} {l}
    0=b-a\\
    p=c-ab\\
    q=-ac\\
    \end{array}\right.$

    $\displaystyle \Rightarrow \left\{\begin {array} {l}
    p=c-b^2\\
    q=-bc\\
    \end{array}\right.$
    So:
    $\displaystyle 27q^2+4p^3 < 0$

    $\displaystyle \Rightarrow 27b^2c^2 +4(c-b^2)^3<0$
    This can be expanded and factorised to give:
    $\displaystyle (4c-b^2)(c+2b^2)^2<0$

    $\displaystyle \Rightarrow 4c-b^2<0$

    $\displaystyle \Rightarrow b^2-4c>0$
    $\displaystyle \Rightarrow x^2+bx+c=0$ has two real roots, and hence $\displaystyle x^3+px+q=0$ has three real roots.

    Grandad
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  3. #3
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    i'm really really sorry, i wrote a bad syntax.... the right term should be:

    $\displaystyle 27^{(q^2)}+4 p^3<0$

    and i need to prove that there are three real roots to this function:

    $\displaystyle x^3+px+q=0$

    thanks alot and sorry again
    Last edited by CaptainBlack; Feb 13th 2010 at 07:15 AM.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by dannee View Post
    hey all,

    i got the following problem to solve:

    i got this term: 27*g^2+4*p^3<0

    and i need to proof that there are three real roots to this function:

    x^3+px+q=0

    is it enough to solve it by min/max points of the function?
    Just post the original question, even with your correction this makes no sense.

    CB
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