Übung
$\frac{m^{20}-20m+19}{m^2-2m+1}$
Schritt-für-Schritt-Lösung
1
Teilen Sie $m^{20}-20m+19$ durch $m^2-2m+1$
$\begin{array}{l}\phantom{\phantom{;}m^{2}-2m\phantom{;}+1;}{\phantom{;}m^{18}+2m^{17}+3m^{16}+4m^{15}+5m^{14}+6m^{13}+7m^{12}+8m^{11}+9m^{10}+10m^{9}+11m^{8}+12m^{7}+13m^{6}+14m^{5}+15m^{4}+16m^{3}+17m^{2}+18m\phantom{;}+19\phantom{;}\phantom{;}}\\\phantom{;}m^{2}-2m\phantom{;}+1\overline{\smash{)}\phantom{;}m^{20}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1;}\underline{-m^{20}+2m^{19}-m^{18}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-m^{20}+2m^{19}-m^{18};}\phantom{;}2m^{19}-m^{18}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n;}\underline{-2m^{19}+4m^{18}-2m^{17}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-2m^{19}+4m^{18}-2m^{17}-;x^n;}\phantom{;}3m^{18}-2m^{17}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n;}\underline{-3m^{18}+6m^{17}-3m^{16}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-3m^{18}+6m^{17}-3m^{16}-;x^n-;x^n;}\phantom{;}4m^{17}-3m^{16}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n;}\underline{-4m^{17}+8m^{16}-4m^{15}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;-4m^{17}+8m^{16}-4m^{15}-;x^n-;x^n-;x^n;}\phantom{;}5m^{16}-4m^{15}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n;}\underline{-5m^{16}+10m^{15}-5m^{14}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;-5m^{16}+10m^{15}-5m^{14}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}6m^{15}-5m^{14}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-6m^{15}+12m^{14}-6m^{13}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;-6m^{15}+12m^{14}-6m^{13}-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}7m^{14}-6m^{13}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-7m^{14}+14m^{13}-7m^{12}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;-7m^{14}+14m^{13}-7m^{12}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}8m^{13}-7m^{12}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-8m^{13}+16m^{12}-8m^{11}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;-8m^{13}+16m^{12}-8m^{11}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}9m^{12}-8m^{11}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-9m^{12}+18m^{11}-9m^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;-9m^{12}+18m^{11}-9m^{10}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}10m^{11}-9m^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-10m^{11}+20m^{10}-10m^{9}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;-10m^{11}+20m^{10}-10m^{9}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}11m^{10}-10m^{9}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-11m^{10}+22m^{9}-11m^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;-11m^{10}+22m^{9}-11m^{8}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}12m^{9}-11m^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-12m^{9}+24m^{8}-12m^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;-12m^{9}+24m^{8}-12m^{7}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}13m^{8}-12m^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-13m^{8}+26m^{7}-13m^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;-13m^{8}+26m^{7}-13m^{6}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}14m^{7}-13m^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-14m^{7}+28m^{6}-14m^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;;-14m^{7}+28m^{6}-14m^{5}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}15m^{6}-14m^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-15m^{6}+30m^{5}-15m^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;;;-15m^{6}+30m^{5}-15m^{4}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}16m^{5}-15m^{4}\phantom{-;x^n}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-16m^{5}+32m^{4}-16m^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;;;;-16m^{5}+32m^{4}-16m^{3}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}17m^{4}-16m^{3}\phantom{-;x^n}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-17m^{4}+34m^{3}-17m^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;;;;;-17m^{4}+34m^{3}-17m^{2}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}18m^{3}-17m^{2}-20m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-18m^{3}+36m^{2}-18m\phantom{;}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;;;;;;-18m^{3}+36m^{2}-18m\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}19m^{2}-38m\phantom{;}+19\phantom{;}\phantom{;}\\\phantom{\phantom{;}m^{2}-2m\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-19m^{2}+38m\phantom{;}-19\phantom{;}\phantom{;}}\\\phantom{;;;;;;;;;;;;;;;;;;-19m^{2}+38m\phantom{;}-19\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\\\end{array}$
$m^{18}+2m^{17}+3m^{16}+4m^{15}+5m^{14}+6m^{13}+7m^{12}+8m^{11}+9m^{10}+10m^{9}+11m^{8}+12m^{7}+13m^{6}+14m^{5}+15m^{4}+16m^{3}+17m^{2}+18m+19$
Endgültige Antwort auf das Problem
$m^{18}+2m^{17}+3m^{16}+4m^{15}+5m^{14}+6m^{13}+7m^{12}+8m^{11}+9m^{10}+10m^{9}+11m^{8}+12m^{7}+13m^{6}+14m^{5}+15m^{4}+16m^{3}+17m^{2}+18m+19$