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Math Help - Is simplification possible (how to know)?

  1. #1
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    Is simplification possible (how to know)?

    Is it possible to simplify this expression? I'm hoping that the expression can be written into some quadratic form but I have no idea if that is possible.

    -\left(8y^3+16y^2+8y - 8y^2x + y^2x^2 - 8yx-2yx^2+2x^3y+x^2+2x^3-2x^4-2\sqrt{(16y^2+4yx^2+x^4)(y+2y-yx+1-x+x^2)^2}\right)
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  2. #2
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    Re: Is simplification possible (how to know)?

    Rewrite your function as g(x,y)+2\sqrt{h(x,y)}
    Let f_i be a function of x and y which is a multinomial (does not have square roots in it). (Note that f_i might equal zero or some other constant)
    If that function does factorise then it can be written as

    (f_1+f_2\sqrt{f_3})(f_4+f_5\sqrt{f_3})

    (f_1+f_2\sqrt{f_3})(f_4+f_5\sqrt{f_3})=f_1f_4+f_2f  _3f_5+(f_2+f_5)\sqrt{f_3}

    f_1f_4+f_2f_3f_5+(f_2+f_5)\sqrt{f_3}=g(x,y)+2\sqrt  {h(x,y)}

    From this you can determine that
    f_1f_4+f_2f_3f_5=g(x,y)

    and

    (f_2+f_5)\sqrt{f_3}=2\sqrt{h(x,y)}

    From: (f_2+f_5)\sqrt{f_3}=2\sqrt{h(x,y)}
    You can determine that
    f_2+f_5=2 and f_3=h(x,y)

    h(x,y) is simple enough that you can tell it cannot be factored any more (Well actually (16y^2+4yx^2+x^4) can be factorised but it involves imaginary numbers so ignore that)

    Since f_1f_4+f_2f_3f_5=g(x,y)
    You can rewrite that as \frac{f_1f_4}{f_3}+f_2f_5=\frac{g(x,y)}{f_3}

    You know g(x,y) and f_3 so you can do the long division. When you do the long division you will get a remainder. The remainder is equal to f_1f_4

    That's as much progress as I can make towards getting the factors. Maybe you can work out the next step. This is very involved for pre-university maths.
    Thanks from niaren
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  3. #3
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    Re: Is simplification possible (how to know)?

    First of all, thanks for your very helpful feedback. It gives me quite some things work with.

    I missed a square in the expression, it should read

    -\left(8y^3+16y^2+8y - 8y^2x + y^2x^2 - 8yx-2yx^2+2x^3y+x^2+2x^3-2x^4-2\sqrt{(16y^2+4yx^2+x^4)(y^2+2y-yx+1-x+x^2)^2}\right)

    Shouldn't your equation
    (f_1+f_2\sqrt{f_3})(f_4+f_5\sqrt{f_3})=f_1f_4+f_2f  _3f_5+(f_2+f_5)\sqrt{f_3}
    read
    (f_1+f_2\sqrt{f_3})(f_4+f_5\sqrt{f_3})=f_1f_4+f_2f  _3f_5+(f_2f_4+f_1f_5)\sqrt{f_3}
    instead?

    I was hoping that I could do the following factorization
    (f_1-\sqrt{f_2})^2 = f_1^2+f_2-2\sqrt{f_2}f_1
    but that seems like not possible.
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