Determinante/2/Rationale Funktionen/1/Aufgabe/Lösung

Es ist

${}{\begin{aligned}\det {\begin{pmatrix}{\frac {x^{2}-5}{x+3}}&{\frac {x^{3}-7}{2x}}\\{\frac {x^{2}+1}{x^{2}-4x}}&{\frac {3x^{2}-x}{x^{2}-3}}\end{pmatrix}}&={\frac {x^{2}-5}{x+3}}\cdot {\frac {3x^{2}-x}{x^{2}-3}}-{\frac {x^{3}-7}{2x}}\cdot {\frac {x^{2}+1}{x^{2}-4x}}\\&={\frac {x{\left(3x^{3}-x^{2}-15x+5\right)}}{x^{3}+3x^{2}-3x-9}}-{\frac {x^{5}+x^{3}-7x^{2}-7}{2x^{2}(x-4)}}\\&={\frac {x^{3}(2x-8){\left(3x^{3}-x^{2}-15x+5\right)}-{\left(x^{5}+x^{3}-7x^{2}-7\right)}{\left(x^{3}+3x^{2}-3x-9\right)}}{2x^{2}(x-4){\left(x^{3}+3x^{2}-3x-9\right)}}}\\&={\frac {2x^{3}{\left(3x^{4}-13x^{3}-11x^{2}+65x-20\right)}-{\left(x^{8}+3x^{7}-2x^{6}-13x^{5}-24x^{4}+5x^{3}+42x^{2}+21x+63\right)}}{2x^{2}{\left(x^{4}-x^{3}-15x^{2}+3x+36\right)}}}\\&={\frac {-x^{8}+3x^{7}-24x^{6}-9x^{5}+154x^{4}-45x^{3}-42x^{2}-21x-63}{2x^{6}-2x^{5}-30x^{4}+6x^{3}+72x^{2}}}.\,\end{aligned}}$

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