Answer
Quotient $=x^2$
Remainder $=x-2$
Work Step by Step
The given expression is
$(3x^5-x^2+x-2)\div(3x^3-1)$
Rewrite the expression as
$(3x^5+0x^4+0x^3-x^2+x-2)\div(3x^3+0x^2+0x-1)$
Perform long division:
$\begin{matrix}
& x^2 & & & && & \leftarrow &\text{Quotient}\\
&-- &-- &--&--& \\
3x^3+0x^2+0x-1) &3x^5&+0x^4&+0x^3&-x^2&+x&-2 & \\
& 3x^5 &+0x^4 &+0x^3 & -x^2& && \leftarrow &x^2(3x^3+0x^2+0x-1) \\
& -- & -- & --& --&&& \leftarrow &\text{subtract} \\
& 0 & 0& 0 &0 &+x&-2 & \leftarrow & \text{Remainder}
\end{matrix}$
Checking:
$\text{(Quotient)(divisor)+ Remainder}$
$=(x^2)(3x^3-1)+x-2$
$=3x^5-x^2+x-2$
$=\text{ Dividend}$
Hence, the quotient is $x^2$ and the remainder is $x-2$.
