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Math Help - Cubic equations / real roots etc.

  1. #1
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    Cubic equations / real roots etc.

    Show that x=-5 is the only real root of cubic equation:

    x^3+3x^2-2x+40=0

    Can I do it this way?
    (1) Fill for x=-5 and show that it equals zero.
    (2) Divide factor (x+5) into the given cubic.This will give me a quadratic.
    (3) Then show that b^2-4ac<0 which means no other real roots

    Would this be a correct approach?
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  2. #2
    A Plied Mathematician
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    That is precisely how I would do it.
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  3. #3
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    Yup it is.. but i have a cooler method
    In any general equation..
    here,ax^3 +bx^2 + cx + d = 0
    sum of roots taken 1 at a time is -b/a
    sum of roots taken 2 at a time is c/a
    sum of roots taken 3 at a time is -d/a and so on..
    let m,n,o be the roots...
    we're given m=-5

    so,m+n+o=-3
    =>n+o=-3+5=2 (1)
    and mno=-40
    =>no=-40/-5=8 (2)
    (1)^2=(n+o)^2
    =n^2 + o^2 + 2on=2^2=4
    putting on=8
    n^2 + o^2=4-16=-12

    sum of two squares is nececarily positive.. thus,n,o are both imaginary/complex.
    Cooler,right?
    you can use this for equations with degree higher than 3 also.. u see??
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