Question

Use synthetic division to find the quotient Q and remainder R when dividing
$f\left(x\right)=x^5+x^4+x^3+2x-5$ by $x+i$. ($i=\sqrt{(}-1)$ imaginary unit)

Answer 1

Let divide $f\left(x\right)=x^5+x^{34}+x^3+2x-5$ by $x+i$ using synthetic division and we have

$$\begin{array}{cc}\begin{array}{c}\\ -i\\ \end{array}& \begin{array}{cccccc}1& 1& 1& 0& 2& -5\\ & -i& -i-1& -1& i& 1-2i\\ 1& 1-i& -i& -1& 2+i& -4-2i\end{array}\end{array}$$

$\Rightarrow Quotient=x^4+(1-i)x^3-i{x}^2-x+2+i$

and $Remainder=-4-2i$