Algebraische Gleichung/x^4+3x^3-5x^2+2x-7/Eliminiere b3/Aufgabe/Lösung

Wir setzen ${\displaystyle {}x=y-{\frac {3}{4}}}$ an und drücken das Polynom bzw. die Gleichung in ${\displaystyle {}y}$ aus. Es ist

${\displaystyle {}{\begin{aligned}x^{4}&={\left(y-{\frac {3}{4}}\right)}^{4}\\&=y^{4}-4{\frac {3}{4}}y^{3}+6{\frac {9}{16}}y^{2}-4{\frac {27}{64}}y+{\frac {81}{256}}\\&=y^{4}-3y^{3}+{\frac {27}{8}}y^{2}-{\frac {27}{16}}y+{\frac {81}{256}},\end{aligned}}}$

${\displaystyle {}{\begin{aligned}3x^{3}&=3{\left(y-{\frac {3}{4}}\right)}^{3}\\&=3{\left(y^{3}-3{\frac {3}{4}}y^{2}+3{\frac {9}{16}}y-{\frac {27}{64}}\right)}\\&=3y^{3}-{\frac {27}{4}}y^{2}+{\frac {81}{16}}y-{\frac {81}{64}},\end{aligned}}}$

${\displaystyle {}-5x^{2}=-5{\left(y-{\frac {3}{4}}\right)}^{2}=-5{\left(y^{2}-{\frac {3}{2}}y+{\frac {9}{16}}\right)}=-5y^{2}+{\frac {15}{2}}y-{\frac {45}{16}}\,}$

und

${\displaystyle {}2x=2{\left(y-{\frac {3}{4}}\right)}=2y-{\frac {3}{2}}\,.}$

Insgesamt ergibt sich also

${\displaystyle {}{\begin{aligned}x^{4}+3x^{3}-5x^{2}+2x-7&=y^{4}+{\left({\frac {27}{8}}-{\frac {27}{4}}-5\right)}y^{2}+{\left(-{\frac {27}{16}}+{\frac {81}{16}}+{\frac {15}{2}}+2\right)}y+{\left({\frac {81}{256}}-{\frac {81}{64}}-{\frac {45}{16}}-{\frac {3}{2}}-7\right)}\\&=y^{4}-{\frac {67}{8}}y^{2}+{\frac {103}{8}}y-{\frac {3139}{256}}.\,\end{aligned}}}$

Eine äquivalente Gleichung ist also

${\displaystyle {}y^{4}-{\frac {67}{8}}y^{2}+{\frac {103}{8}}y-{\frac {3139}{256}}=0\,.}$