Answer
The quotient is $x^4-x^3+x^2-x+1$
and the remainder is $0$.
Work Step by Step
The given expression is:-
$(x^5+1)\div (x+1)$
Rewrite as descending powers of $x$.
$(x^5+0x^4+0x^3+0x^2+0x+1)\div (x+1)$
The divisor is $x+1$, so the value of $c=-1$.
and on the right side the coefficients of dividend in descending powers of $x$.
Perform synthetic division to obtain:
$\begin{matrix}
&-- &-- &--&--&--&-- \\
-1) &1&0&0&0&0&1& \\
& &-1 &1 &- 1 &1 &-1\\
& -- & -- & --&-- &--&--& \\
& 1 & -1 & 1 & -1 &1 &0&\\
\end{matrix}$
The divisor is $x+1$
The dividend is $x^5+1$
The Quotient is $x^4-x^3+x^2-x+1$
The remainder is $0$.
Check:-
$\text{(Divisor)(Quotient)+Remainder}$
$=(x+1)(x^4-x^3+x^2-x+1)+0$
$=x^5-x^4+x^3-x^2+x+x^4-x^3+x^2-x+1+0$
$=x^5+1$
Hence, the quotient is $x^4-x^3+x^2-x+1$ and the remainder is $0$.
