[latex]0 = (g+1)(g^2-3g+1)(1-b - b^4) - 2g^3(b^3+b-1)[/latex]

[latex]0 = (g^2(g+1)-3g(g+1)+1(g+1))(1-b - b^4) - 2g^3(b^3+b-1)[/latex]

[latex]0 = (g^3+g^2-3g^2-3g+g+1)(1-b - b^4) - 2g^3(b^3+b-1)[/latex]

[latex]0 = (g^3-2g^2-2g+1)(1-b - b^4) - 2g^3(b^3+b-1)[/latex]

Nach g auflösen

[latex]0 = (1-b - b^4)g^3 - 2(1-b - b^4)g^2 - 2(1-b - b^4)g + (1-b - b^4) - 2(b^3+b-1)g^3[/latex]

[latex]0 = (1-b - b^4)g^3 - (2-2b - 2b^4)g^2 - (2-2b - 2b^4)g + (1-b - b^4) - (2b^3+2b-2)g^3[/latex]

[latex]0 = (1-b - b^4 - (2b^3+2b-2))g^3 - (2-2b - 2b^4)g^2 - (2-2b - 2b^4)g + (1-b - b^4)[/latex]

[latex]0 = (1-b - b^4 - 2b^3-2b+2)g^3 - (2-2b - 2b^4)g^2 - (2-2b - 2b^4)g + (1-b - b^4)[/latex]

[latex]0 = (-b^4-2b^3-3b+3)g^3 - (-2b^4-2b+2)g^2 - (-2b^4-2b+2)g + (-b^4-b+1)[/latex]

[latex]0 = (-b^4-2b^3-3b+3)g^3 + (2b^4+2b-2)g^2 + (2b^4+2b-2)g + (-b^4-b+1)[/latex]

[latex]A = -b^4-2b^3-3b+3[/latex]

[latex]B = 2b^4+2b-2[/latex]

[latex]C = 2b^4+2b-2[/latex]

[latex]D = -b^4-b+1[/latex]

[latex]a = \frac{B}{A} = \frac{2b^4+2b-2}{-b^4-2b^3-3b+3}[/latex]

[latex]b = \frac{C}{A} = \frac{2b^4+2b-2}{-b^4-2b^3-3b+3}[/latex]

[latex]c = \frac{D}{A} = \frac{-b^4-b+1}{-b^4-2b^3-3b+3}[/latex]

[latex]p = b - \frac{a^2}3[/latex]

[latex]q = \frac {2a^3}{27} - \frac{ab}3 + c[/latex]

[latex]p = \frac{2b^4+2b-2}{-b^4-2b^3-3b+3} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3[/latex]

[latex]q = \frac {2(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^3}{27} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3 + \frac{-b^4-b+1}{-b^4-2b^3-3b+3}[/latex]

[latex]D:=\left(\frac{\frac {2(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^3}{27} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3 + \frac{-b^4-b+1}{-b^4-2b^3-3b+3}}2\right)^2 + \left(\frac{\frac{2b^4+2b-2}{-b^4-2b^3-3b+3} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3}3\right)^3[/latex]

[latex]u = \sqrt[3]{-\frac{q}2 + \sqrt{D}}[/latex]

[latex]v = \sqrt[3]{-\frac{q}2 - \sqrt{D}}[/latex]

[latex]u = \sqrt[3]{-\frac{\frac {2(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^3}{27} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3 + \frac{-b^4-b+1}{-b^4-2b^3-3b+3}}2 + \sqrt{\left(\frac{\frac {2(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^3}{27} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3 + \frac{-b^4-b+1}{-b^4-2b^3-3b+3}}2\right)^2 + \left(\frac{\frac{2b^4+2b-2}{-b^4-2b^3-3b+3} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3}3\right)^3}}[/latex]

[latex]v = \sqrt[3]{-\frac{\frac {2(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^3}{27} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3 + \frac{-b^4-b+1}{-b^4-2b^3-3b+3}}2 - \sqrt{\left(\frac{\frac {2(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^3}{27} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3 + \frac{-b^4-b+1}{-b^4-2b^3-3b+3}}2\right)^2 + \left(\frac{\frac{2b^4+2b-2}{-b^4-2b^3-3b+3} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3}3\right)^3}}[/latex]

D > 0 ?

[latex]z = u+v = \sqrt[3]{-\frac{\frac {2(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^3}{27} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3 + \frac{-b^4-b+1}{-b^4-2b^3-3b+3}}2 + \sqrt{\left(\frac{\frac {2(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^3}{27} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3 + \frac{-b^4-b+1}{-b^4-2b^3-3b+3}}2\right)^2 + \left(\frac{\frac{2b^4+2b-2}{-b^4-2b^3-3b+3} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3}3\right)^3}} + \sqrt[3]{-\frac{\frac {2(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^3}{27} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3 + \frac{-b^4-b+1}{-b^4-2b^3-3b+3}}2 - \sqrt{\left(\frac{\frac {2(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^3}{27} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3 + \frac{-b^4-b+1}{-b^4-2b^3-3b+3}}2\right)^2 + \left(\frac{\frac{2b^4+2b-2}{-b^4-2b^3-3b+3} - \frac{(\frac{2b^4+2b-2}{-b^4-2b^3-3b+3})^2}3}3\right)^3}}[/latex]

Rücksubstitution => g

Nach b auflösen

[latex]0 = -(g^3-2g^2-2g+1)b^4 - (g^3-2g^2-2g+1)b + (g^3-2g^2-2g+1) + (- 2g^3b^3+ - 2g^3b + 2g^3)[/latex]

[latex]0 = (-g^3+2g^2+2g-1)b^4 + (-2g^3)b^3 + (-3g^3+2g^2+2g-1)b + (3g^3-2g^2-2g+1)[/latex]


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