Question

Divide $p\left(x\right)=x^3-3x^2+5x-3$ by $g\left(x\right)=x^2-2$, find the quotient and remainder.

Answer 1

Step 1: Use long division method:

Let $p\left(x\right)=x^3-3x^2+5x-3$ and $g\left(x\right)=x^2-2$.

Let the quotient and remainder be $q\left(x\right)\mathrm{and}r\left(x\right)$.

Divide $p\left(x\right)\mathrm{by}g\left(x\right),$ we get,

$\begin{array}{ccc}{x}^2& -& 2\end{array}\begin{array}{ccc}x& -& 3\end{array}\begin{array}{cccccc}{x}^3& -& 3x^2& +& 5x& -\begin{array}{c}3\end{array}\\ x^3& +& 0x^2& -& 2x& \\ -& -& & +& & \end{array}\begin{array}{cccccc}-& 3x^2& +& 7x& -& 3\\ -& 3x^2& +& 0x& +& 6\\ +& & -& & -& \end{array}\begin{array}{ccc}7x& -& 9\end{array}$

Therefore, $q\left(x\right)=\left(x-3\right)$ and $r\left(x\right)=\left(7x-9\right)$.

Step 2: Verify using division algorithm:

Formula:

$\text{Dividend = Divisior ×Quotient +Remainder}$

By division algorithm, we have $p\left(x\right)=g\left(x\right)\times q\left(x\right)+r\left(x\right)$.

$$\begin{gathered} \Rightarrow \left(x^2-2\right)\times \left(x-3\right)+(7x-9)=x^3-3x^2-2x+6+7x-9 \\ =x^3-3x^2+5x-3 \\ =p\left(x\right) \end{gathered}$$

Thus, division algorithm is verified.

Hence, quotient is $q\left(x\right)=\left(x-3\right)$ and and the remainder is $r\left(x\right)=\left(7x-9\right)$.