Answer
The quotient is $x^3+2x^2+5x+10$
and the remainder is $22$.
Work Step by Step
The given expression is:-
$(x^4+x^2+2)\div (x-2)$
Rewrite as descending powers of $x$.
$(x^4+0x^3+x^2+0x+2)\div (x-2)$
The divisor is $x-2$, so the value of $c=2$.
and on the right side the coefficients of dividend in descending powers of $x$.
Perform synthetic division to obtain:
$\begin{matrix}
&-- &-- &--&--& \\
2) &1&0&1&0&2 \\
& &2 &4 &10 &20& \\
& -- & -- & --& -- &--& \\
& 1 & 2 & 5 &10 &22 &\\
\end{matrix}$
The divisor is $x-2$
The dividend is $x^4+x^2+2$
The Quotient is $x^3+2x^2+5x+10$
The remainder is $22$.
Check:-
$=\text{(Divisor)(Quotient)+Remainder}$
$=(x-2)(x^3+2x^2+5x+10)+22$
$=x^4+2x^3+5x^2+10x-2x^3-4x^2-10x-20+22$
$=x^4+x^2+2$
Hence, the quotient is $x^3+2x^2+5x+10$ and the remainder is $22$.
