Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder in each of the following:
(i) $p(x)=x^{3}-3 x^{2}+5 x-3, \quad g(x)=x^{2}-2$
(ii) $p(x)=x^{4}-3 x^{2}+4 x+5, \quad g(x)=x^{2}+1-x$
(iii) $p(x)=x^{4}-5 x+6, \quad g(x)=2-x^{2}$

Question

Accepted Answer

(i) $\quad p(x)=x^{3}-3 x^{2}+5 x-3$
$$
q(x)=x^{2}-2
$$
$$
\begin{array}{r}
x ^ { 2 } - 2 \longdiv { \begin{array} { l r }
{ x ^ { 3 } - 3 x ^ { 2 } + 5 x - 3 } \
{ x ^ { 3 } + 2 x } \
{ - \quad + }
\end{array} } \
\hline \begin{array}{r}
-3 x^{2}+7 x-3 \
-3 x^{2}+6 \
+\quad-
\end{array} \
\hline \begin{array}{r}
7 x-9 \
+
\end{array}
\end{array}
$$

Quotient $=x-3$

Remainder $=7 x-9$
(ii) $p(x)=x^{4}-3 x^{2}+4 x+5=x^{4}+0 x^{3}-3 x^{2}+4 x+5$
$$
q(x)=x^{2}+1-x=x^{2}-x+1
$$
$$
\begin{aligned}
& x ^ { 2 } - x + 1 \longdiv { x ^ { 4 } + 0 . x ^ { 3 } - 3 x ^ { 2 } + 4 x + 5 } x^{4}-x^{3}+x^{2} \
&-\quad+\quad- x^{3}-4 x^{2}+4 x+5 \
& x^{3}-x^{2}+x-\quad+\quad- \
& \hline \begin{array}{r}
-3 x^{2}+3 x+5 \
-3 x^{2}+3 x-3 \
+\quad+
\end{array} \
& \hline
\end{aligned}
$$

Quotient $=x^{2}+x-3$

Remainder $=8$
(iii) $\quad p(x)=x^{4}-5 x+6=x^{4}+0 x^{2}-5 x+6$
$$
q(x)=2-x^{2}=-x^{2}+2
$$
$$
\begin{aligned}
& - x ^ { 2 } + 2 \longdiv { - x ^ { 2 } - 2 } \
& \begin{array}{l}
x^{4}+0 x^{2}-5 x+6 \
x^{4}-2 x^{2} \
\frac{-+}{2 x^{2}-5 x+6} \
2 x^{2}-4 \
\frac{-\quad+}{-5 x+10}
\end{array}
\end{aligned}
$$

Quotient $=-x^{2}-2$
Remainder $=-5 x+10$