By acute division, find the quotient and the remainder when the first polynomial is divided by the second polynomial: $x^{4}+1 ; x+1$
By acute division, find the quotient and the remainder when the first polynomial is divided by the second polynomial: $x^{4}+1 ; x+1$
By acute division, we have
$\begin{array}{l}x-1 |\overline {x^{4}+1} (x^{3}+x^{2}+x+1)\\ \;\;\; \;\;\;\;\;\;\; x^{4}-x^{3}\;\;\;\;\;\;\; \\ \hline \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x^{3}+1 \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\,x^{3}-x^{2} \\ \hline \;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\; x^{2}+1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x^{2}-x \;\;\; \\ \hline \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x+1 \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x-1 \\ \hline \;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\;\;\;\; 2 \end{array}$
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