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Math Help - A cubic equation...

  1. #1
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    A cubic equation...

    Problem 243. A cubic equation x^3+px^2+qx+r=0 has three roots x_1, x_2, x_3 Find: (x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 as an expression containing p, q, and r. This polynomial in p, q, and r is called the discriminant of the cubic equation.

    I've mulled over it for some time and seem to get half way there but never quite manage to neatly solve this.
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  2. #2
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    Quote Originally Posted by jerry View Post
    Problem 243. A cubic equation x^3+px^2+qx+r=0 has three roots x_1, x_2, x_3 Find: (x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 as an expression containing p, q, and r. This polynomial in p, q, and r is called the discriminant of the cubic equation.

    I've mulled over it for some time and seem to get half way there but never quite manage to neatly solve this.
    I don't know if this will help you a little bit further:

    (x-x_1)(x-x_2)(x-x_3)=x^3-(x_1+x_2+x_3)x^2+(x_1x_2+x_1x_3+x_2 x_3)x-x_1x_2x_3

    Thus:

    \begin{array}{l}-p= x_1+x_2+x_3 \\ q= x_1x_2+x_1x_3+x_2 x_3 \\ -r= x_1x_2x_3\end{array}
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  3. #3
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    Thanks but that's in the book also.
    I've been doing problems involving Vieta's theorem.

    The approach I'm taking to this problem is to expand the brackets, detect the roots, and make replacements accordingly. E.g. -6a^2b^2c^2 = -6r^2.

    For the other two roots I can only find confusing patterns that make me think I'm close but I can't quite work out what to do with them.

    I'm quite sick of this problem and think I've just hit a brick wall.
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