Division mit Rest (Polynomring)/C/1/Aufgabe/Lösung

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Es ist

{\displaystyle {}{\begin{aligned}P-X^{2}T&=X^{4}+{\mathrm {i} }X^{3}+X^{2}+(1-3{\mathrm {i} })X+3-X^{2}{\left(X^{2}+(1-{\mathrm {i} })X+2\right)}\\&=(-1+2{\mathrm {i} })X^{3}-X^{2}+(1-3{\mathrm {i} })X+3\\&={\tilde {P}}.\end{aligned}}}

Ferner ist

{\displaystyle {}{\begin{aligned}{\tilde {P}}-(-1+2{\mathrm {i} })XT&=(-1+2{\mathrm {i} })X^{3}-X^{2}+(1-3{\mathrm {i} })X+3-(-1+2{\mathrm {i} })X{\left(X^{2}+(1-{\mathrm {i} })X+2\right)}\\&=(-2-3{\mathrm {i} })X^{2}+(3-7{\mathrm {i} })X+3\\&={\hat {P}}\end{aligned}}}

und

{\displaystyle {}{\begin{aligned}{\hat {P}}-(-2-3{\mathrm {i} })T&={\hat {P}}+(2+3{\mathrm {i} })T\\&=(-2-3{\mathrm {i} })X^{2}+(3-7{\mathrm {i} })X+3+(2+3{\mathrm {i} }){\left(X^{2}+(1-{\mathrm {i} })X+2\right)}\\&=(8-6{\mathrm {i} })X+7+6{\mathrm {i} }.\end{aligned}}}

Somit ist

${\displaystyle {}P=(X^{2}+(-1+2{\mathrm {i} })X+(2+3{\mathrm {i} }))T+(8-6{\mathrm {i} })X+7+6{\mathrm {i} }\,.}$