Question

Find the quotient and remainder when $p\left( x \right) = {x^3} - 3{x^2} + 5x - 3$ is divided by $g\left( x \right) = {x^2} - 2$.

Answer 1

Hint: In the given question, we are required to divide the polynomial $p\left( x \right) = {x^3} - 3{x^2} + 5x - 3$ by another polynomial $g\left( x \right) = {x^2} - 2$ to find the quotient and remainder in the division process. So, we will use the long division algorithm for division of $p\left( x \right) = {x^3} - 3{x^2} + 5x - 3$ by $g\left( x \right) = {x^2} - 2$ where $p\left( x \right)$ is dividend and $g\left( x \right)$ is divisor and we will obtain $q\left( x \right)$ as quotient and $r\left( x \right)$ as remainder.

Complete step-by-step solution:
So, in the problem, dividend $ = p\left( x \right) = {x^3} - 3{x^2} + 5x - 3$
Divisor $ = g\left( x \right) = {x^2} - 2$
Now, we will follow the long division method to divide one polynomial by another and to find the quotient and remainder.
\[{x^2} - 2\overset{{x}-3}{\overline{\left){\begin{align}
  & {{x}^{3}}-3{{x}^{2}}+5x-3 \\
 & \underline{{{x}^{3}}-2{x}} \\
 & -3{{x}^{2}}+7x-3 \\
 & \underline{-3{{x}^{2}}+6} \\
 & 7x-9 \\
\end{align}}\right.}}\]
Hence, we have divided the polynomial $p\left( x \right) = {x^3} - 3{x^2} + 5x - 3$ by $g\left( x \right) = {x^2} - 2$ using the long division method and obtained the quotient polynomial as $q\left( x \right) = x - 3$ and remainder polynomial as $r\left( x \right) = 7x - 9$.
Now, we can also verify this division method by using the property $Dividend = Divisor \times Quotient + Remainder$.
$ \Rightarrow {x^3} - 3{x^2} + 5x - 3 = \left( {{x^2} - 2} \right) \times \left( {x - 3} \right) + \left( {7x - 9} \right)$
Finding the product of polynomials, we get,
$ \Rightarrow {x^3} - 3{x^2} + 5x - 3 = {x^3} - 2x - 3{x^2} + 6 + 7x - 9$
Adding up the like terms on right side of equation, we get,
$ \Rightarrow {x^3} - 3{x^2} + 5x - 3 = {x^3} - 3{x^2} + 5x - 3$
Since both the sides of the equation are equal. So, the division process is done correctly.
Hence, the quotient is equal to $q\left( x \right) = x - 3$ and the remaining polynomial is equal to $r\left( x \right) = 7x - 9$.

Note: The property of long division $Dividend = Divisor \times Quotient + Remainder$ holds true for division of numbers as well as polynomials. The degree of dividend given to us is $3$ and the degree of the divisor is $2$. So, we must remember that the degree of the remainder must be less than that of the divisor. Hence, in this case, the degree of remainder obtained is $1$.