Exercice
$\frac{x^{20}-5x^6+9x^4+5}{x-1}$
Solution étape par étape
1
Diviser $x^{20}-5x^6+9x^4+5$ par $x-1$
$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-1;}{\phantom{;}x^{19}+x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}-4x^{5}-4x^{4}+5x^{3}+5x^{2}+5x\phantom{;}+5\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-1\overline{\smash{)}\phantom{;}x^{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}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-1;}\underline{-x^{20}+x^{19}\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{-;x^n}}\\\phantom{-x^{20}+x^{19};}\phantom{;}x^{19}\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}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n;}\underline{-x^{19}+x^{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{;-x^{19}+x^{18}-;x^n;}\phantom{;}x^{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}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n;}\underline{-x^{18}+x^{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{;;-x^{18}+x^{17}-;x^n-;x^n;}\phantom{;}x^{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}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n-;x^n;}\underline{-x^{17}+x^{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{;;;-x^{17}+x^{16}-;x^n-;x^n-;x^n;}\phantom{;}x^{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}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n;}\underline{-x^{16}+x^{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{;;;;-x^{16}+x^{15}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}x^{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}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-x^{15}+x^{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{;;;;;-x^{15}+x^{14}-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}x^{14}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-x^{14}+x^{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{;;;;;;-x^{14}+x^{13}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}x^{13}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-x^{13}+x^{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{;;;;;;;-x^{13}+x^{12}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}x^{12}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-x^{12}+x^{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{;;;;;;;;-x^{12}+x^{11}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}x^{11}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-x^{11}+x^{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{;;;;;;;;;-x^{11}+x^{10}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}x^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-x^{10}+x^{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{;;;;;;;;;;-x^{10}+x^{9}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}x^{9}\phantom{-;x^n}\phantom{-;x^n}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\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{-x^{9}+x^{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{;;;;;;;;;;;-x^{9}+x^{8}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}x^{8}\phantom{-;x^n}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\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{-x^{8}+x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;-x^{8}+x^{7}-;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{;}x^{7}-5x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\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{-x^{7}+x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;;-x^{7}+x^{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-;x^n;}-4x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\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{\phantom{;}4x^{6}-4x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;;;\phantom{;}4x^{6}-4x^{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-;x^n;}-4x^{5}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\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{\phantom{;}4x^{5}-4x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;;;;\phantom{;}4x^{5}-4x^{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-;x^n;}\phantom{;}5x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\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{-5x^{4}+5x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;;;;;-5x^{4}+5x^{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-;x^n;}\phantom{;}5x^{3}\phantom{-;x^n}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\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{-5x^{3}+5x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;;;;;;-5x^{3}+5x^{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-;x^n;}\phantom{;}5x^{2}\phantom{-;x^n}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\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{-5x^{2}+5x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;;;;;;;;;;;;;;;;-5x^{2}+5x\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;}\phantom{;}5x\phantom{;}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\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-;x^n;}\underline{-5x\phantom{;}+5\phantom{;}\phantom{;}}\\\phantom{;;;;;;;;;;;;;;;;;;;-5x\phantom{;}+5\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-;x^n;}\phantom{;}10\phantom{;}\phantom{;}\\\end{array}$
$x^{19}+x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}-4x^{5}-4x^{4}+5x^{3}+5x^{2}+5x+5+\frac{10}{x-1}$
Réponse finale au problème
$x^{19}+x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}-4x^{5}-4x^{4}+5x^{3}+5x^{2}+5x+5+\frac{10}{x-1}$